Research ArticleA Bivariate Chebyshev Spectral Collocation QuasilinearizationMethod for Nonlinear Evolution Parabolic Equations

S S Motsa V M Magagula and P Sibanda

School of Mathematics Statistics and Computer Science University of KwaZulu-Natal Private Bag X01Scottsville Pietermaritzburg 3209 South Africa

Correspondence should be addressed to S S Motsa sandilemotsagmailcom

Received 15 July 2014 Accepted 12 August 2014 Published 27 August 2014

Academic Editor Hassan Saberi Nik

Copyright copy 2014 S S Motsa et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a newmethod for solving higher order nonlinear evolution partial differential equations (NPDEs)Themethodcombines quasilinearisation the Chebyshev spectral collocationmethod and bivariate Lagrange interpolation In this paper we usethe method to solve several nonlinear evolution equations such as the modified KdV-Burgers equation highly nonlinear modifiedKdV equation Fisherrsquos equation Burgers-Fisher equation Burgers-Huxley equation and the Fitzhugh-Nagumo equation Theresults are compared with known exact analytical solutions from literature to confirm accuracy convergence and effectivenessof the method There is congruence between the numerical results and the exact solutions to a high order of accuracy Tables weregenerated to present the order of accuracy of themethod convergence graphs to verify convergence of themethod and error graphsare presented to show the excellent agreement between the results from this study and the known results from literature

1 Introduction

Nonlinearity exists everywhere and in general nature is non-linear Nonlinear evolution partial differential equations ariseinmany fields of science particularly in physics engineeringchemistry finance and biological systems They are widelyused to describe complex phenomena in various fields of sci-ences such as wave propagation phenomena fluid mechan-ics plasma physics quantum mechanics nonlinear opticssolid state physics chemical kinematics physical chemistrypopulation dynamics financial industry and numerous areasof mathematical modeling The development of both numer-ical and analytical methods for solving complicated highlynonlinear evolution partial differential equations continuesto be an area of interest to scientists whose research aimis to enrich deep understanding of such alluring nonlinearproblems

Innumerable number of methods for obtaining analyticaland approximate solutions to nonlinear evolution equationshave been proposed Someof the analyticalmethods that havebeen used to solve evolution nonlinear partial differentialequations include Adomianrsquos decomposition method [1ndash3]

homotopy analysis method [4ndash7] tanh-function method [8ndash10] Haar wavelet method [11ndash13] and Exp-function method[14ndash16] Several numerical methods have been used tosolve nonlinear evolution partial differential equationsTheseinclude the explicit-implicit method [17] Chebyshev finitedifference methods [18] finite difference methods [19] finiteelement methods [20] and pseudospectral methods [21 22]

Some drawbacks of approximate analytical methodsinclude slow convergence particularly for large time (119905 gt 1)Theymay also be cumbersome to use as some involvemanualintegration of approximate series solutions and hence it isdifficult to find closed solutions sometimes On the otherhand some numerical methods may not work in some casesfor example when the required solution has to be foundnear a singularity Certain numerical methods for examplefinite differences require many grid points to achieve goodaccuracy and hence require a lot of computer memory andcomputational time Conventional first-order finite differ-ence methods may result in monotonic and stable solutionsbut they are strongly dissipative causing the solution of thestrongly convective partial differential equations to becomesmeared out and often grossly inaccurate On the other hand

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 581987 13 pageshttpdxdoiorg1011552014581987

2 The Scientific World Journal

higher order difference methods are less dissipative but areprone to numerical instabilities

Spectral methods have been used successfully in manydifferent fields in sciences and engineering because of theirability to give accurate solutions of differential equationsKhater et al [23] applied the Chebyshev spectral collocationmethod to solve Burgers type of equations in space and finitedifferences to approximate the time derivative The Cheby-shev spectral collocationmethod has been used together withthe fourth-order Runge-Kutta method to solve the nonlinearPDEs in this studyTheChebyshev spectral collocation is firstapplied to the NPDE and this yields a system of ordinarydifferential equations which are solved using the fourth-order Runge-Kutta method Olmos and Shizgal [24] Javidi[25 26] Dehghan and Fakhar-Izadi [27] Driscoll [28] andDriscoll [28] solved the Fisher Burgers-Fisher Burgers-Huxley Fitzhugh-Nagumo and KdV equations respectivelyusing a combination of the Chebyshev spectral collocationmethod and fourth-order Runge-Kutta method Darvishi etal [29 30] solved the KdV and the Burgers-Huxley equationsusing a combination of the Chebyshev spectral collocationmethod and Darvishirsquos preconditioning Jacobs and Harley[31] and Tohidi and Kilicman [32] used spectral collocationdirectly for solving linear partial differential equations Accu-racy will be compromised if they implement their approachin solving nonlinear partial differential equations since theyuse Kronecker multiplication

Chebyshev spectral methods are defined everywhere inthe computational domain Therefore it is easy to get anaccurate value of the function under consideration at anypoint of the domain beside the collocation points Thisproperty is often exploited in particular to get a significantgraphic representation of the solution making the possibleoscillations due to a wrong approximation of the derivativeapparent Spectral collocationmethods are easy to implementand are adaptable to various problems including variablecoefficient and nonlinear differential equations The errorassociated with the Chebyshev approximation is O(1119873

119903

)

where 119873 refers to the truncation and 119903 is connected tothe number of continuous derivatives of the function Theinterest in using Chebyshev spectral methods in solvingnonlinear PDEs stems from the fact that these methodsrequire less grid points to achieve accurate results Theyare computational and efficient compared to traditionalmethods like finite difference and finite element methodsChebyshev spectral collocation method has been used inconjunction with additional methods which may have theirown drawbacks Here we provide an alternative method thatis not dependent on another method to approximate thesolution

The main objective of this work is to introduce a newmethod that uses Chebyshev spectral collocation bivariateLagrange interpolation polynomials together with quasilin-earisation techniques The nonlinear evolution equationsare first linearized using the quasilinearisation method TheChebyshev spectral collocation method with Lagrange inter-polation polynomials are applied independently in space andtime variables of the linearized evolution partial differentialequation This new method is termed bivariate interpolated

spectral quasilinearisation method (BI-SQLM) We presentthe BI-SQLM algorithm in a general setting where it can beused to solve any 119903th order nonlinear evolution equationsThe applicability accuracy and reliability of the proposedBI-SQLM are confirmed by solving the modified KdV-Burger equation highly nonlinear modified KdV equationthe Cahn-Hillard equationthe fourth-order KdV equationFisherrsquos Burgers-Fisher Burger-Huxley and the Fitzhugh-Nagumo equationsThe results of the BI-SQLMare comparedagainst known exact solutions that have been reported in thescientific literature It is observed that the method achieveshigh accuracy with relatively fewer spatial grid points It alsoconverges fast to the exact solution and approximates thesolution of the problem in a computationally efficientmannerwith simulations completed in fractions of a second in allcases Tables are generated to show the order of accuracyof the method and time taken to compute the solutions Itis observed that as the number of grid points is increasedthe error decreases Error graphs and graphs showing theexcellent agreement of the exact and analytical solutions forall the nonlinear evolution equations are also presented

The paper is organized as follows In Section 2 weintroduce the BI-SQLM algorithm for a general nonlinearevolution PDE In Section 3 we describe the applicationof the BI-SQLM to selected test problems The numericalsimulations and results are presented in Section 4 Finally weconclude in Section 5

2 Bivariate Interpolated SpectralQuasilinearization Method (BI-SQLM)

In this section we introduce the Bivariate InterpolatedSpectral Quasilinearization Method (BI-SQLM) for findingsolutions to nonlinear evolution PDEs Without loss ofgenerality we consider nonlinear PDEs of the form

120597119906

120597120591= 119867(119906

120597119906

1205971205781205972

119906

1205971205782

120597119899

119906

120597120578119899)

with the physical region 120591 isin [0 119879] 120578 isin [119886 119887]

(1)

where 119899 is the order of differentiation 119906(120578 120591) is the requiredsolution and 119867 is a nonlinear operator which contains allthe spatial derivatives of 119906 The given physical region 120591 isin

[0 119879] is converted to the region 119905 isin [minus1 1] using the lineartransformation 120591 = 119879(119905 + 1)2 and 120578 isin [119886 119887] is converted tothe region 119909 isin [minus1 1] using the linear transformation

120578 =1

2(119887 minus 119886) 119909 +

1

2(119887 + 119886) (2)

Equation (1) can be expressed as

120597119906

120597119905= 119867(119906

120597119906

1205971199091205972

119906

1205971199092

120597119899

119906

120597119909119899) 119905 isin [minus1 1] 119909 isin [minus1 1]

(3)

The Scientific World Journal 3

The solution procedure assumes that the solution can beapproximated by a bivariate Lagrange interpolation polyno-mial of the form

119906 (119909 119905) asymp

119873119909

sum

119894=0

119873119905

sum

119895=0

119906 (119909119894 119905119895) 119871119894(119909) 119871119895(119905) (4)

which interpolates 119906(119909 119905) at selected points in both the 119909 and119905 directions defined by

119909119894 = cos( 120587119894

119873119909

)

119873119909

119894=0

119905119895 = cos(

120587119895

119873119905

)

119873119905

119895=0

(5)

The choice of the Chebyshev-Gauss-Lobatto grid points (5)ensures that there is a simple conversion of the continuousderivatives in both space and time to discrete derivativesat the grid points The functions 119871

119894(119909) are the characteristic

Lagrange cardinal polynomials

119871119894(119909) =

119873119909

prod

119894=0

119894 =119896

119909 minus 119909119896

119909119894minus 119909119896

(6)

where

119871119894(119909119896) = 120575119894119896

= 0 if 119894 = 119896

1 if 119894 = 119896(7)

The function 119871119895(119905) is defined in a similar manner Before

linearizing (3) it is convenient to split 119867 into its linear andnonlinear components and rewrite the governing equation inthe form

119865 [119906 1199061015840

119906(119899)

] + 119866 [119906 1199061015840

119906(119899)

] minus = 0 (8)

where the dot and primes denote the time and space deriva-tives respectively 119865 is a linear operator and 119866 is a nonlinearoperator Assuming that the difference 119906

119903+1minus 119906119903and all itrsquos

space derivative is small we first approximate the nonlinearoperator 119866 using the linear terms of the Taylor series andhence

119866 [119906 1199061015840

119906(119899)

] asymp 119866 [119906119903 1199061015840

119903 119906

(119899)

119903]

+

119899

sum

119896=0

120597119866

120597119906(119896)

(119906(119896)

119903+1minus 119906(119896)

119903)

(9)

where 119903 and 119903 + 1 denote previous and current iterationsrespectively We remark that this quasilinearization method(QLM) approach is a generalisation of the Newton-Raphsonmethod and was first proposed by Bellman and Kalaba [33]for solving nonlinear boundary value problems

Equation (9) can be expressed as

119866 [119906 1199061015840

119906(119899)

] asymp 119866 [119906119903 1199061015840

119903 119906

(119899)

119903]

+

119899

sum

119896=0

120601119896119903

[119906119903 1199061015840

119903 119906

(119899)

119903] 119906(119896)

119903+1

minus

119899

sum

119896=0

120601119896119903

[119906119903 1199061015840

119903 119906

(119899)

119903] 119906(119896)

119903

(10)

where

120601119896119903

[119906119903 1199061015840

119903 119906

(119899)

119903] =

120597119866

120597119906(119896)

[119906119903 1199061015840

119903 119906

(119899)

119903] (11)

Substituting (10) into (8) we get

119865 [119906119903+1

1199061015840

119903+1 119906

(119899)

119903+1] +

119899

sum

119896=0

120601119896119903119906(119896)

119903+1minus 119903+1

= 119877119903[119906119903 1199061015840

119903 119906

(119899)

119903]

(12)

where

119877119903[119906119903 1199061015840

119903 119906

(119899)

119903] =

119899

sum

119896=0

120601119896119903119906(119896)

119903minus 119866 [119906

119903 1199061015840

119903 119906

(119899)

119903] (13)

A crucial step in the implementation of the solution proce-dure is the evaluation of the time derivative at the grid points119905119895(119895 = 0 1 119873

119905) and the spatial derivatives at the grid

points 119909119894(119894 = 0 1 119873

119909) The values of the time derivatives

at the Chebyshev-Gauss-Lobatto points (119909119894 119905119895) are computed

as (for 119895 = 0 1 2 119873119905)

120597119906

120597119905

10038161003816100381610038161003816100381610038161003816119909=119909119894119905=119905119895

=

119873119909

sum

119901=0

119873119905

sum

119896=0

119906 (119909119901 119905119896) 119871119901(119909119894)

119889119871119896(119905119895)

119889119905

=

119873119905

sum

119896=0

119906 (119909119894 119905119896) 119889119895119896

=

119873119905

sum

119896=0

119889119895119896119906 (119909119894 119905119896)

(14)

where 119889119895119896

= 119889119871119896(119905119895)119889119905 is the standard first derivative Che-

byshev differentiation matrix of size (119873119905+ 1) times (119873

119905+ 1) as

defined in [34] The values of the space derivatives at theChebyshev-Gauss-Lobatto points (119909

119894 119905119895) (119894 = 0 1 2 119873

119909)

are computed as

120597119906

120597119909

10038161003816100381610038161003816100381610038161003816119909=119909119894119905=119905119895

=

119873119909

sum

119901=0

119873119905

sum

119896=0

119906 (119909119901 119905119896)

119889119871119901(119909119894)

119889119909119871119896(119905119895)

=

119873119909

sum

119901=0

119906 (119909119901 119905119895)119863119894119901

=

119873119909

sum

119901=0

119863119894119901119906 (119909119901 119905119895)

(15)

where 119863119894119901

= 119889119871119901(119909119894)119889119909 is the standard first derivative

Chebyshev differentiation matrix of size (119873119909+ 1) times (119873

119909+ 1)

Similarly for an 119899th order derivative we have

120597119899

119906

120597119909119899

10038161003816100381610038161003816100381610038161003816119909=119909119894119905=119905119895

=

119873119909

sum

119901=0

119863119899

119894119901119906 (119909119901 119905119895) = D119899U

119895

119894 = 0 1 2 119873119909

(16)

where the vector U119895is defined as

U119895= [119906119895(1199090) 119906119895(1199091) 119906

119895(119909119873119909

)]119879 (17)

4 The Scientific World Journal

and the superscript 119879 denotes matrix transpose Substituting(16) into (12) we get

119865 [U119903+1119895

U1015840119903+1119895

U(119899)119903+1119895

] +

119899

sum

119896=0

Φ119896119903U(119896)119903+1119895

minus

119873119905

sum

119896=0

119889119895119896U119903+1119896

= 119877119903[U119903119895U1015840119903119895 U(119899)

119903119895]

(18)

for 119895 = 0 1 2 119873119905 where

U(119899)119903+1119895

= D119899U119903+1119895

Φ119896119903

=

[[[[

[

120601119896119903

(1199090 119905119895)

120601119896119903

(1199091 119905119895)

d120601119896119903

(119909119873119909

119905119895)

]]]]

]

(19)

The initial condition for (3) corresponds to 120591119873119905

= minus1 andhence we express (18) as

119865 [U119903+1119895

U1015840119903+1119895

U(119899)119903+1119895

]

+

119899

sum

119896=0

Φ119896119903U(119896)119903+1119895

minus

119873119905minus1

sum

119896=0

119889119895119896U119903+1119896

= R119895

(20)

where

R119895= 119877119903[U119903119895U1015840119903119895 U(119899)

119903119895] + 119889119895119873119905

U119873119905

119895 = 0 1 2 119873119905minus 1

(21)

Equation (20) can be expressed as the following119873119905(119873119909+1) times

119873119905(119873119909+ 1)matrix system

[[[[

[

11986000

11986001

sdot sdot sdot 1198600119873119905minus1

11986010

11986011

sdot sdot sdot 1198601119873119905minus1

d

119860119873119905minus10

119860119873119905minus11

sdot sdot sdot 119860119873119905minus1119873119905minus1

]]]]

]

[[[[

[

U0

U1

U119873119905minus1

]]]]

]

=

[[[[

[

R0

R1

R119873119905minus1

]]]]

]

(22)

where

119860119894119894

= 119865 [ID D(119899)] +119899

sum

119896=0

Φ119896119903D(119896) minus 119889

119894119894I

119860119894119895

= minus119889119894119895I when 119894 = 119895

(23)

and I is the identity matrix of size (119873119909+1)times (119873

119909+1) Solving

(19) gives 119906(119909119894 119905119895) and hence we use (4) to approximate

119906(119909 119905)

3 Numerical Experiments

We apply the proposed algorithm to well-known nonlinearPDEs of the form (3) with exact solutions In order todetermine the level of accuracy of the BI-SQLM approximatesolution at a particular time level in comparison with theexact solution we report maximum error which is defined by

119864119873

= max119903

1003816100381610038161003816119906 (119909119903 119905) minus (119909

119903 119905)

1003816100381610038161003816 0 le 119903 le 119873 (24)

where (119909119903 119905) is the approximate solution and is the 119906(119909

119903 119905)

exact solution at the time level 119905

Example 1 Weconsider the generalizedBurgers-Fisher equa-tion [35]

120597119906

120597119905+ 120572119906120575120597119906

120597119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (25)

with initial condition

119906 (119909 0) = 1

2+

1

2tanh(

minus120572120575

2(120575 + 1)119909)

1120575

(26)

and exact solution

119906 (119909 119905)

= 1

2+

1

2tanh(

minus120572120575

2 (120575 + 1)

times [119909 minus (120572

120575 + 1+

120573 (120575 + 1)

120572) 119905])

1120575

(27)

where 120572 120573 and 120575 are parameters For illustration purposesthese parameters are chosen to be120572 = 120573 = 120575 = 1 in this paperThe linear operator 119865 and nonlinear operator119866 are chosen as

119865 (119906) = 11990610158401015840

+ 119906 119866 (119906) = minus1199061199061015840

minus 1199062

(28)

We first linearize the nonlinear operator 119866 We approximate119866 using the equation

119866 asymp 119866 [119906119903 1199061015840

119903 11990610158401015840

119903] +

2

sum

119896=0

120601119896119903119906(119896)

119903+1minus

2

sum

119896=0

120601119896119903119906(119896)

119903 (29)

The coefficients are given by

1206010119903

=120597119866

120597119906[119906119903 1199061015840

119903 11990610158401015840

119903] = minus (119906

1015840

119903+ 2119906119903)

1206011119903

=120597119866

1205971199061015840[119906119903 1199061015840

119903 11990610158401015840

119903] = minus119906

119903

1206012119903

=120597119866

12059711990610158401015840[119906119903 1199061015840

119903 11990610158401015840

119903] = 0

119877119903=

2

sum

119896=0

120601119896119903119906(119896)

119903minus 119866 [119906

119903 1199061015840

119903 11990610158401015840

119903] = minus119906

2

119903minus 1199061199031199061015840

119903

(30)

The Scientific World Journal 5

Therefore the linearized equation can be expressed as

11990610158401015840

119903+1+ 12060111199031199061015840

119903+1+ 1206010119903119906119903+1

+ 119906119903+1

minus = 119877119903 (31)

Applying the spectral method both in 119909 and 119905 and initialcondition we get

D2U119903+1119894

+Φ1119903DU119903+1119894

+Φ0119903U119903+1119894

+ U119903+1119894

minus 2

119873119905minus1

sum

119895=0

119889119894119895U119903+1119895

= R119894

(32)

Equation (32) can be expressed as

[[[[

[

11986000

11986001

sdot sdot sdot 1198600119873119905minus1

11986010

11986011

sdot sdot sdot 1198601119873119905minus1

d

119860119873119905minus10

119860119873119905minus11

sdot sdot sdot 119860119873119905minus1119873119905minus1

]]]]

]

[[[[

[

U0

U1

U119873119905minus1

]]]]

]

=

[[[[

[

R0

R1

R119873119905minus1

]]]]

]

(33)

where

119860119894119894

= D2 +Φ(119894)1119903D +Φ

(119894)

0119903+ (1 minus 2119889

119894119894) I

119860119894119895

= minus 2119889119894119895I when 119894 = 119895

R119894= 119877119903+ 2119889119894119873119905

U119903119873119905

(34)

The boundary conditions are implemented in the first andlast row of the matrices 119860

119894119895and the column vectors R

119894for

119894 = 0 1 119873119905minus 1 and 119895 = 0 1 119873

119905minus 1 The procedure

for finding the variable coefficients 120601119894and matrices for the

remaining examples is similar

Example 2 We consider Fisherrsquos equation

120597119906

120597119905=

1205972

119906

1205971199092+ 120572119906 (1 minus 119906) (35)

subject to the initial condition

119906 (119909 0) =1

(1 + 119890radic1205726119909

)2 (36)

and exact solution [36]

119906 (119909 119905) =1

(1 + 119890radic1205726119909minus51205721199056

)2 (37)

where 120572 is a constant The Fisher equation represents areactive-diffusive system and is encountered in chemicalkinetics and population dynamics applications For thisexample the appropriate linear operator 119865 and nonlinearoperator 119866 are chosen as

119865 (119906) = 11990610158401015840

+ 120572119906 119866 (119906) = minus1205721199062

(38)

Table 1 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1001 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

02 3934119890 minus 008 3121119890 minus 011 1552119890 minus 012 1561119890 minus 012

03 5577119890 minus 008 4864119890 minus 011 1004119890 minus 012 1005119890 minus 012

04 6997119890 minus 008 6802119890 minus 011 7895119890 minus 013 8124119890 minus 013

05 8107119890 minus 008 7971119890 minus 011 1088119890 minus 012 1027119890 minus 012

06 8891119890 minus 008 8560119890 minus 011 8805119890 minus 013 7847119890 minus 013

07 9344119890 minus 008 8953119890 minus 011 6418119890 minus 013 6463119890 minus 013

08 9431119890 minus 008 8759119890 minus 011 6199119890 minus 013 6164119890 minus 013

09 9178119890 minus 008 8325119890 minus 011 3978119890 minus 013 3695119890 minus 013

10 8787119890 minus 008 7421119890 minus 011 7988119890 minus 014 5596119890 minus 014

CPUtime(sec)

0019942 0025988 0027756 0029436

Table 2Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 120574 = 120575 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 1142119890 minus 007 1369119890 minus 010 5891119890 minus 012 6143119890 minus 012

02 1178119890 minus 007 1373119890 minus 010 9570119890 minus 012 1013119890 minus 011

03 1186119890 minus 007 1479119890 minus 010 1489119890 minus 011 1512119890 minus 011

04 1069119890 minus 007 9450119890 minus 011 1703119890 minus 011 1702119890 minus 011

05 9030119890 minus 008 7944119890 minus 011 5283119890 minus 012 5736119890 minus 012

06 6963119890 minus 008 6618119890 minus 011 1639119890 minus 011 1626119890 minus 011

07 4638119890 minus 008 1579119890 minus 011 1362119890 minus 011 1364119890 minus 011

08 2457119890 minus 008 4030119890 minus 011 3934119890 minus 012 3852119890 minus 012

09 2028119890 minus 008 6006119890 minus 011 4466119890 minus 012 4727119890 minus 012

10 3147119890 minus 008 7708119890 minus 011 7757119890 minus 013 7261119890 minus 013

CPUTime(sec)

0010152 0015387 0019163 0021564

Example 3 Consider the Fitzhugh-Nagumo equation

120597119906

120597119905=

1205972

119906

1205971199092+ 119906 (119906 minus 120572) (1 minus 119906) (39)

with initial condition

119906 (119909 0) =1

2[1 minus coth(minus

119909

2radic2

)] (40)

This equation has the exact solution [37]

119906 (119909 119905) =1

2[1 minus coth(minus

119909

2radic2

+2120572 minus 1

4119905)] (41)

where 120572 is a parameter In this example the linear operator 119865and nonlinear operator 119866 are chosen as

119865 (119906) = 11990610158401015840

minus 120572119906 119866 (119906) = (1 + 120572) 1199062

minus 1199063

(42)

6 The Scientific World Journal

Table 3 Maximum errors 119864119873for the Fitzhug-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 5719119890 minus 007 1196119890 minus 009 2367119890 minus 012 9881119890 minus 014

02 6193119890 minus 007 1299119890 minus 009 2761119890 minus 012 3952119890 minus 014

03 6662119890 minus 007 1463119890 minus 009 3259119890 minus 012 8216119890 minus 014

04 6779119890 minus 007 1448119890 minus 009 3341119890 minus 012 8094119890 minus 014

05 6920119890 minus 007 1526119890 minus 009 3587119890 minus 012 5063119890 minus 014

06 7019119890 minus 007 1573119890 minus 009 3729119890 minus 012 3775119890 minus 014

07 6933119890 minus 007 1516119890 minus 009 3660119890 minus 012 8915119890 minus 014

08 6828119890 minus 007 81535119890 minus 009 3635119890 minus 012 7594119890 minus 014

09 6765119890 minus 007 1528119890 minus 009 3519119890 minus 012 3242119890 minus 013

10 6687119890 minus 007 1490119890 minus 009 3405119890 minus 012 1688119890 minus 013

CPUtime(sec)

0024281 0024901 0026810 0032389

Table 4Maximum errors119864119873for the Burger-Huxley equation when

120574 = 075 120573 = 1 and119873119905= 10

119905 119873119909

4 6 8 1001 2217119890 minus 006 8482119890 minus 009 2166119890 minus 011 7822119890 minus 014

02 2596119890 minus 006 9369119890 minus 009 2536119890 minus 011 1184119890 minus 013

03 2859119890 minus 006 1073119890 minus 008 3201119890 minus 011 1049119890 minus 013

04 3001119890 minus 006 1112119890 minus 008 3652119890 minus 011 9426119890 minus 014

05 3137119890 minus 006 1213119890 minus 008 4262119890 minus 011 1510119890 minus 013

06 3270119890 minus 006 1311119890 minus 008 4842119890 minus 011 2127119890 minus 013

07 3367119890 minus 006 1359119890 minus 008 5289119890 minus 011 1230119890 minus 013

08 3467119890 minus 006 1438119890 minus 008 5803119890 minus 011 1549119890 minus 013

09 3562119890 minus 006 1504119890 minus 008 6260119890 minus 011 3063119890 minus 013

10 3640119890 minus 006 1559119890 minus 008 6674119890 minus 011 2951119890 minus 013

CPUtime(sec)

0023822 0024901 002685 0032806

Example 4 Consider the Burgers-Huxley equation

120597119906

120597119905+ 120572119906120575

119906119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (119906120575

minus 120574) (43)

where 120572 120573 ge 0 are constant parameters 120575 is a positive integer(set to be 120575 = 1 in this study) and 120574 isin (0 1) The exactsolution subject to the initial condition

119906 (119909 0) =1

2minus

1

2tanh [

120573

119903 minus 120572119909] (44)

is reported in [38 39] as

119906 (119909 119905) =1

2minus

1

2tanh [

120573

119903 minus 120572(119909 minus 119888119905)] (45)

where

119903 = radic1205722+ 8120573 119888 =

(120572 minus 119903) (2120574 minus 1) + 2120572

4

(46)

Table 5 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1001 1803119890 minus 007 3419119890 minus 010 4449119890 minus 013 1572119890 minus 013

02 2614119890 minus 007 4347119890 minus 010 5049119890 minus 013 5992119890 minus 014

03 2717119890 minus 007 4677119890 minus 010 5532119890 minus 013 8128119890 minus 013

04 2009119890 minus 007 3663119890 minus 010 4771119890 minus 013 6158119890 minus 013

05 2580119890 minus 007 4410119890 minus 010 7518119890 minus 013 2555119890 minus 013

06 2653119890 minus 007 4606119890 minus 010 8738119890 minus 013 5756119890 minus 013

07 2248119890 minus 007 4039119890 minus 010 6210119890 minus 013 2393119890 minus 013

08 2572119890 minus 007 4476119890 minus 010 5432119890 minus 013 6812119890 minus 013

09 2436119890 minus 007 4351119890 minus 010 6111119890 minus 013 6287119890 minus 013

10 8275119890 minus 008 3721119890 minus 010 7569119890 minus 013 1087119890 minus 007

CPUtime(sec)

0015646 0021226 0030159 0035675

Table 6 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1001 7788119890 minus 005 3553119890 minus 007 7601119890 minus 010 2080119890 minus 010

02 1153119890 minus 004 4000119890 minus 007 5684119890 minus 010 1189119890 minus 010

03 1011119890 minus 004 3739119890 minus 007 4471119890 minus 010 4503119890 minus 010

04 3926119890 minus 005 1785119890 minus 007 6544119890 minus 010 4987119890 minus 010

05 6727119890 minus 005 2342119890 minus 007 2638119890 minus 010 1528119890 minus 010

06 6065119890 minus 005 2207119890 minus 007 4565119890 minus 010 4568119890 minus 010

07 2511119890 minus 005 1105119890 minus 007 4749119890 minus 010 3748119890 minus 010

08 4074119890 minus 005 1427119890 minus 007 1062119890 minus 010 1604119890 minus 010

09 2386119890 minus 005 1018119890 minus 007 2343119890 minus 010 8114119890 minus 011

10 1440119890 minus 006 7256119890 minus 008 1436119890 minus 009 1513119890 minus 011

CPUtime(sec)

0020609 0021241 0030617 0032816

The general solution (45) was reported in [40 41] In thisexample the linear operator 119865 and nonlinear operator 119866 arechosen as

119865 (119906) = 11990610158401015840

minus 120573120574119906

119866 (119906) = minus1205721199061199061015840

+ 120573 (1 + 120574) 1199062

minus 1205731199063

(47)

Example 5 We consider the modified KdV-Burgers equation

120597119906

120597119905=

1205973

119906

1205971199093minus

1205972

119906

1205971199092minus 61199062120597119906

120597119909

(48)

subject to the initial condition

119906 (119909 0) =1

6+

1

6tanh(

119909

6) (49)

and exact solution [42]

119906 (119909 119905) =1

6+

1

6tanh(

119909

6minus

119905

27) (50)

The Scientific World Journal 7

Table 7 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1002 1119119890 minus 011 7398119890 minus 013 8266119890 minus 013 3808119890 minus 014

04 3121119890 minus 011 1552119890 minus 012 7378119890 minus 013 3780119890 minus 014

06 4864119890 minus 011 1004119890 minus 012 3402119890 minus 012 7283119890 minus 014

08 6802119890 minus 011 7895119890 minus 013 1118119890 minus 012 3714119890 minus 014

10 7971119890 minus 011 1088119890 minus 012 1473119890 minus 012 1691119890 minus 013

12 8560119890 minus 011 8805119890 minus 013 2611119890 minus 012 3119119890 minus 013

14 8953119890 minus 011 6418119890 minus 013 6671119890 minus 012 1796119890 minus 013

16 8759119890 minus 011 6199119890 minus 013 1118119890 minus 011 1097119890 minus 013

18 8325119890 minus 011 3978119890 minus 013 7515119890 minus 013 6273119890 minus 014

20 7421119890 minus 011 7988119890 minus 014 3682119890 minus 012 2311119890 minus 013

CPUtime(sec)

0013542 0022967 0023792 0024758

Table 8Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 1223119890 minus 007 1400119890 minus 008 1402119890 minus 008 1094119890 minus 012

04 1145119890 minus 007 1919119890 minus 008 1918119890 minus 008 3919119890 minus 012

06 9192119890 minus 008 2082119890 minus 008 2085119890 minus 008 1953119890 minus 012

08 2293119890 minus 008 1793119890 minus 008 1793119890 minus 008 6340119890 minus 013

10 2395119890 minus 008 1337119890 minus 008 1339119890 minus 008 2381119890 minus 012

12 5778119890 minus 008 1954119890 minus 008 1930119890 minus 008 1005119890 minus 011

14 6045119890 minus 008 1620119890 minus 008 1620119890 minus 008 3535119890 minus 012

16 5244119890 minus 008 7218119890 minus 009 7345119890 minus 009 5765119890 minus 012

18 4395119890 minus 008 6828119890 minus 009 6784119890 minus 009 3983119890 minus 012

20 2944119890 minus 008 9406119890 minus 010 8820119890 minus 010 3812119890 minus 012

CPUtime(sec)

0019942 0025988 0027756 0029436

The modified KdV-Burgers equation describes various kindsof phenomena such as a mathematical model of turbulence[43] and the approximate theory of flow through a shockwavetraveling in viscous fluid [44] For this example the linearoperator 119865 and nonlinear operator 119866 are chosen as

119865 (119906) = 119906101584010158401015840

minus 11990610158401015840

119866 (119906) = minus61199061015840

1199062

(51)

Example 6 We consider the high nonlinear modified KdVequation

120597119906

120597119905=

1205973

119906

1205971199093+ (

120597119906

120597119909)

2

minus 1199062 (52)

subject to the initial condition

119906 (119909 0) =1

2+

119890minus119909

4

(53)

Table 9 Maximum errors 119864119873for the Fitzhugh-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 6326119890 minus 007 1311119890 minus 009 2886119890 minus 012 1131119890 minus 012

04 6721119890 minus 007 1467119890 minus 009 3310119890 minus 012 1564119890 minus 012

06 7140119890 minus 007 1602119890 minus 009 3617119890 minus 012 1936119890 minus 012

08 6730119890 minus 007 1496119890 minus 009 4707119890 minus 012 1196119890 minus 012

10 6660119890 minus 007 1487119890 minus 009 3675119890 minus 012 1264119890 minus 012

12 6449119890 minus 007 1366119890 minus 009 1897119890 minus 012 1727119890 minus 012

14 5690119890 minus 007 1083119890 minus 009 2972119890 minus 012 1200119890 minus 012

16 4931119890 minus 007 8010119890 minus 010 1519119890 minus 012 8590119890 minus 013

18 3986119890 minus 007 4658119890 minus 010 1068119890 minus 012 6790119890 minus 013

20 2904119890 minus 007 2968119890 minus 010 1592119890 minus 012 1770119890 minus 013

CPUtime(sec)

0041048 0049629 0055008 0053863

Table 10 Maximum errors 119864119873

for the Burgers-Huxley equationwhen 120574 = 05 120573 = 1 and119873

119905= 10

119905 119873119909

4 6 8 1002 2866119890 minus 006 1119119890 minus 008 3670119890 minus 011 1150119890 minus 012

04 3401119890 minus 006 1420119890 minus 008 5744119890 minus 011 1638119890 minus 012

06 3814119890 minus 006 1687119890 minus 008 7426119890 minus 011 1958119890 minus 012

08 3915119890 minus 006 1729119890 minus 008 8171119890 minus 011 7002119890 minus 013

10 3938119890 minus 006 1738119890 minus 008 8157119890 minus 011 1267119890 minus 012

12 3808119890 minus 006 1624119890 minus 008 7687119890 minus 011 1710119890 minus 012

14 3456119890 minus 006 1527119890 minus 008 6965119890 minus 011 5109119890 minus 013

16 3230119890 minus 006 1349119890 minus 008 5535119890 minus 011 8203119890 minus 013

18 2925119890 minus 006 1078119890 minus 008 3598119890 minus 011 8294119890 minus 013

20 2497119890 minus 006 7505119890 minus 009 2265119890 minus 011 9726119890 minus 014

CPUtime(sec)

0023822 0024901 002685 0032806

and exact solution

119906 (119909 119905) =1

119905 + 2+

119890minus(119909+119905)

(119905 + 2)2 (54)

For this example the linear operator 119865 and nonlinear opera-tor 119866 are chosen as

119865 (119906) = 119906101584010158401015840

119866 (119906) = (1199061015840

)2

minus 1199062

(55)

4 Results and Discussion

In this section we present the numerical solutions obtainedusing the BI-SQLM algorithm The number of collocationpoints in the space 119909 variable used to generate the results is119873119909

= 10 in all cases Similarly the number of collocationpoints in the time 119905 variable used is 119873

119905= 10 in all cases It

was found that sufficient accuracy was achieved using thesevalues in all numerical simulations

8 The Scientific World Journal

Table 11 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1002 2137119890 minus 007 3820119890 minus 010 4846119890 minus 013 9998119890 minus 013

04 2480119890 minus 007 4267119890 minus 010 5596119890 minus 013 8775119890 minus 013

06 2691119890 minus 007 4676119890 minus 010 6565119890 minus 013 2054119890 minus 012

08 2214119890 minus 007 3979119890 minus 010 8776119890 minus 013 1168119890 minus 012

10 2538119890 minus 007 4463119890 minus 010 9650119890 minus 013 8410119890 minus 013

12 2650119890 minus 007 4680119890 minus 010 7450119890 minus 013 5113119890 minus 013

14 2383119890 minus 007 4296119890 minus 010 7500119890 minus 013 1110119890 minus 012

16 2568119890 minus 007 4572119890 minus 010 9704119890 minus 013 2837119890 minus 013

18 2520119890 minus 007 4529119890 minus 010 7443119890 minus 013 5353119890 minus 013

20 2370119890 minus 007 4438119890 minus 010 2719119890 minus 013 8849119890 minus 013

CPUtime(sec)

0062066 0081646 0080718 010775

Table 12 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1002 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

04 8010119890 minus 005 3577119890 minus 007 3902119890 minus 008 1979119890 minus 010

06 7235119890 minus 005 2549119890 minus 007 2016119890 minus 008 4899119890 minus 010

08 6284119890 minus 005 1663119890 minus 007 1155119890 minus 007 2679119890 minus 010

10 1642119890 minus 005 1620119890 minus 007 1243119890 minus 007 2474119890 minus 010

12 2753119890 minus 005 1073119890 minus 007 1073119890 minus 007 1679119890 minus 010

14 3738119890 minus 006 8971119890 minus 008 8598119890 minus 008 4788119890 minus 011

16 1223119890 minus 005 2153119890 minus 008 2503119890 minus 008 2941119890 minus 011

18 5836119890 minus 006 2986119890 minus 008 9127119890 minus 009 5177119890 minus 011

20 9310119890 minus 006 6548119890 minus 008 7277119890 minus 008 1453119890 minus 009

CPUtime(sec)

0020609 0021241 0030617 0032816

In Tables 1 2 3 4 5 and 6 we give the maximumerrors between the exact and BI-SQLM results for the Fisherequation Burgers-Fisher equation Fitzhugh-Nagumo equa-tion Burgers-Huxley equation the modified KdV-Burgersequation and the modified KdV equation respectively at119905 isin [01 1] The results were computed in the space domain119909 isin [0 1] To give a sense of the computational efficiency ofthe method the computational time to generate the resultsis also given Tables 1ndash6 clearly show the accuracy of themethod The accuracy is seen to improve with an increasein the number of collocation points 119873

119909 It is remarkable to

note that accurate results with errors of order up to 10minus14

are obtained using very few collocation points in both the 119909

and 119905 variables 119873119905le 10 119873

119909le 10 This is a clear indication

that the BI-SQLM is powerful method that is appropriatein solving nonlinear evolution PDEs We remark also thatthe BI-SQLM is computationally fast as accurate results aregenerated in a fraction of a second in all the examplesconsidered in this work

0 05 1 15 206

062

064

066

068

07

072

Time (s)

The analytical solution versus the approximate solution ofFishers equation

ApproximateAnalytic

u(xt)

Figure 1 Fishers equation analytical solution graph

088

0885

089

0895

09

0905

091

0915

092

0925

093

The analytical solution versus the approximate solution of theBurgers-Fisher equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 2 Burger-Fishers equation analytical solution graph

In Tables 7 8 9 10 11 and 12 we give the maxi-mum errors of the BI-SQLM results for the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and themodified KdV equation respectively at selectedvalues of 119905 = 2 for different collocation points 119873

119905 in the

119905-variable The results in Tables 7ndash12 were computed on thespace domain 119909 isin [0 1] We note that the accuracy does notdetoriate when 119905 gt 1 for this method as is often the case withnumerical schemes such as finite differences

The Scientific World Journal 9

0 05 1 15 2Time (s)

ApproximateAnalytic

026

028

03

032

034

036

038

04

042

044

The analytical solution versus the approximate solution of theFitzhurg-Nagumo equation

u(xt)-a

xis

Figure 3 Fitzhugh-Nagumo equation analytical solution graph

025

035

045

055

06

05

04

03

The analytical solution and the approximate solution of theBurger-Huxley Equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 4 Burgers-Huxley equation analytical solution graph

Figures 1 2 3 4 5 and 6 show a comparison of theanalytical and approximate solutions of the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and the modified KdV equation respectively when 119905 =

2 The approximate solutions are in excellent agreement withthe analytical solutions and this demonstrates the accuracyof the algorithm presented in this study

In Figures 7 8 9 10 11 and 12 we present error analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation the

0 05 1 15 2Time (s)

ApproximateAnalytic

015

0155

016

0165

017

0175

018

0185

The analytical solution and the approximate solution of themodified KdV-Burger

u(xt)

Figure 5 Modified KdV-Burger equation analytical solution graph

0 05 1 15 2Time (s)

ApproximateAnalytic

0253

0254

0255

0256

0257

0258

0259

026

The analytical solution versus the approximate solution ofmodified KdV

u(xt)

Figure 6 Modified KdV equation analytical solution graph

modified KdV-Burgers equation and the modified KdVequation respectively when 119905 = 2

In Figures 13 14 15 16 17 and 18 convergence analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation themodified KdV-Burgers equation and the modified KdVequation respectively The figures present a variation of theerror norm at a fixed value of time (119905 = 1) with iterationsof the BI-SQLM scheme It can be seen that in almost allthe examples considered the iteration scheme takes about3 or 4 iterations to converge fully Beyond the point wherefull convergence is reached error norm levels off and does

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

higher order difference methods are less dissipative but areprone to numerical instabilities

Spectral methods have been used successfully in manydifferent fields in sciences and engineering because of theirability to give accurate solutions of differential equationsKhater et al [23] applied the Chebyshev spectral collocationmethod to solve Burgers type of equations in space and finitedifferences to approximate the time derivative The Cheby-shev spectral collocationmethod has been used together withthe fourth-order Runge-Kutta method to solve the nonlinearPDEs in this studyTheChebyshev spectral collocation is firstapplied to the NPDE and this yields a system of ordinarydifferential equations which are solved using the fourth-order Runge-Kutta method Olmos and Shizgal [24] Javidi[25 26] Dehghan and Fakhar-Izadi [27] Driscoll [28] andDriscoll [28] solved the Fisher Burgers-Fisher Burgers-Huxley Fitzhugh-Nagumo and KdV equations respectivelyusing a combination of the Chebyshev spectral collocationmethod and fourth-order Runge-Kutta method Darvishi etal [29 30] solved the KdV and the Burgers-Huxley equationsusing a combination of the Chebyshev spectral collocationmethod and Darvishirsquos preconditioning Jacobs and Harley[31] and Tohidi and Kilicman [32] used spectral collocationdirectly for solving linear partial differential equations Accu-racy will be compromised if they implement their approachin solving nonlinear partial differential equations since theyuse Kronecker multiplication

Chebyshev spectral methods are defined everywhere inthe computational domain Therefore it is easy to get anaccurate value of the function under consideration at anypoint of the domain beside the collocation points Thisproperty is often exploited in particular to get a significantgraphic representation of the solution making the possibleoscillations due to a wrong approximation of the derivativeapparent Spectral collocationmethods are easy to implementand are adaptable to various problems including variablecoefficient and nonlinear differential equations The errorassociated with the Chebyshev approximation is O(1119873

119903

)

where 119873 refers to the truncation and 119903 is connected tothe number of continuous derivatives of the function Theinterest in using Chebyshev spectral methods in solvingnonlinear PDEs stems from the fact that these methodsrequire less grid points to achieve accurate results Theyare computational and efficient compared to traditionalmethods like finite difference and finite element methodsChebyshev spectral collocation method has been used inconjunction with additional methods which may have theirown drawbacks Here we provide an alternative method thatis not dependent on another method to approximate thesolution

The main objective of this work is to introduce a newmethod that uses Chebyshev spectral collocation bivariateLagrange interpolation polynomials together with quasilin-earisation techniques The nonlinear evolution equationsare first linearized using the quasilinearisation method TheChebyshev spectral collocation method with Lagrange inter-polation polynomials are applied independently in space andtime variables of the linearized evolution partial differentialequation This new method is termed bivariate interpolated

spectral quasilinearisation method (BI-SQLM) We presentthe BI-SQLM algorithm in a general setting where it can beused to solve any 119903th order nonlinear evolution equationsThe applicability accuracy and reliability of the proposedBI-SQLM are confirmed by solving the modified KdV-Burger equation highly nonlinear modified KdV equationthe Cahn-Hillard equationthe fourth-order KdV equationFisherrsquos Burgers-Fisher Burger-Huxley and the Fitzhugh-Nagumo equationsThe results of the BI-SQLMare comparedagainst known exact solutions that have been reported in thescientific literature It is observed that the method achieveshigh accuracy with relatively fewer spatial grid points It alsoconverges fast to the exact solution and approximates thesolution of the problem in a computationally efficientmannerwith simulations completed in fractions of a second in allcases Tables are generated to show the order of accuracyof the method and time taken to compute the solutions Itis observed that as the number of grid points is increasedthe error decreases Error graphs and graphs showing theexcellent agreement of the exact and analytical solutions forall the nonlinear evolution equations are also presented

The paper is organized as follows In Section 2 weintroduce the BI-SQLM algorithm for a general nonlinearevolution PDE In Section 3 we describe the applicationof the BI-SQLM to selected test problems The numericalsimulations and results are presented in Section 4 Finally weconclude in Section 5

2 Bivariate Interpolated SpectralQuasilinearization Method (BI-SQLM)

In this section we introduce the Bivariate InterpolatedSpectral Quasilinearization Method (BI-SQLM) for findingsolutions to nonlinear evolution PDEs Without loss ofgenerality we consider nonlinear PDEs of the form

120597119906

120597120591= 119867(119906

120597119906

1205971205781205972

119906

1205971205782

120597119899

119906

120597120578119899)

with the physical region 120591 isin [0 119879] 120578 isin [119886 119887]

(1)

where 119899 is the order of differentiation 119906(120578 120591) is the requiredsolution and 119867 is a nonlinear operator which contains allthe spatial derivatives of 119906 The given physical region 120591 isin

[0 119879] is converted to the region 119905 isin [minus1 1] using the lineartransformation 120591 = 119879(119905 + 1)2 and 120578 isin [119886 119887] is converted tothe region 119909 isin [minus1 1] using the linear transformation

120578 =1

2(119887 minus 119886) 119909 +

1

2(119887 + 119886) (2)

Equation (1) can be expressed as

120597119906

120597119905= 119867(119906

120597119906

1205971199091205972

119906

1205971199092

120597119899

119906

120597119909119899) 119905 isin [minus1 1] 119909 isin [minus1 1]

(3)

The Scientific World Journal 3

The solution procedure assumes that the solution can beapproximated by a bivariate Lagrange interpolation polyno-mial of the form

119906 (119909 119905) asymp

119873119909

sum

119894=0

119873119905

sum

119895=0

119906 (119909119894 119905119895) 119871119894(119909) 119871119895(119905) (4)

which interpolates 119906(119909 119905) at selected points in both the 119909 and119905 directions defined by

119909119894 = cos( 120587119894

119873119909

)

119873119909

119894=0

119905119895 = cos(

120587119895

119873119905

)

119873119905

119895=0

(5)

The choice of the Chebyshev-Gauss-Lobatto grid points (5)ensures that there is a simple conversion of the continuousderivatives in both space and time to discrete derivativesat the grid points The functions 119871

119894(119909) are the characteristic

Lagrange cardinal polynomials

119871119894(119909) =

119873119909

prod

119894=0

119894 =119896

119909 minus 119909119896

119909119894minus 119909119896

(6)

where

119871119894(119909119896) = 120575119894119896

= 0 if 119894 = 119896

1 if 119894 = 119896(7)

The function 119871119895(119905) is defined in a similar manner Before

linearizing (3) it is convenient to split 119867 into its linear andnonlinear components and rewrite the governing equation inthe form

119865 [119906 1199061015840

119906(119899)

] + 119866 [119906 1199061015840

119906(119899)

] minus = 0 (8)

where the dot and primes denote the time and space deriva-tives respectively 119865 is a linear operator and 119866 is a nonlinearoperator Assuming that the difference 119906

119903+1minus 119906119903and all itrsquos

space derivative is small we first approximate the nonlinearoperator 119866 using the linear terms of the Taylor series andhence

119866 [119906 1199061015840

119906(119899)

] asymp 119866 [119906119903 1199061015840

119903 119906

(119899)

119903]

+

119899

sum

119896=0

120597119866

120597119906(119896)

(119906(119896)

119903+1minus 119906(119896)

119903)

(9)

where 119903 and 119903 + 1 denote previous and current iterationsrespectively We remark that this quasilinearization method(QLM) approach is a generalisation of the Newton-Raphsonmethod and was first proposed by Bellman and Kalaba [33]for solving nonlinear boundary value problems

Equation (9) can be expressed as

119866 [119906 1199061015840

119906(119899)

] asymp 119866 [119906119903 1199061015840

119903 119906

(119899)

119903]

+

119899

sum

119896=0

120601119896119903

[119906119903 1199061015840

119903 119906

(119899)

119903] 119906(119896)

119903+1

minus

119899

sum

119896=0

120601119896119903

[119906119903 1199061015840

119903 119906

(119899)

119903] 119906(119896)

119903

(10)

where

120601119896119903

[119906119903 1199061015840

119903 119906

(119899)

119903] =

120597119866

120597119906(119896)

[119906119903 1199061015840

119903 119906

(119899)

119903] (11)

Substituting (10) into (8) we get

119865 [119906119903+1

1199061015840

119903+1 119906

(119899)

119903+1] +

119899

sum

119896=0

120601119896119903119906(119896)

119903+1minus 119903+1

= 119877119903[119906119903 1199061015840

119903 119906

(119899)

119903]

(12)

where

119877119903[119906119903 1199061015840

119903 119906

(119899)

119903] =

119899

sum

119896=0

120601119896119903119906(119896)

119903minus 119866 [119906

119903 1199061015840

119903 119906

(119899)

119903] (13)

A crucial step in the implementation of the solution proce-dure is the evaluation of the time derivative at the grid points119905119895(119895 = 0 1 119873

119905) and the spatial derivatives at the grid

points 119909119894(119894 = 0 1 119873

119909) The values of the time derivatives

at the Chebyshev-Gauss-Lobatto points (119909119894 119905119895) are computed

as (for 119895 = 0 1 2 119873119905)

120597119906

120597119905

10038161003816100381610038161003816100381610038161003816119909=119909119894119905=119905119895

=

119873119909

sum

119901=0

119873119905

sum

119896=0

119906 (119909119901 119905119896) 119871119901(119909119894)

119889119871119896(119905119895)

119889119905

=

119873119905

sum

119896=0

119906 (119909119894 119905119896) 119889119895119896

=

119873119905

sum

119896=0

119889119895119896119906 (119909119894 119905119896)

(14)

where 119889119895119896

= 119889119871119896(119905119895)119889119905 is the standard first derivative Che-

byshev differentiation matrix of size (119873119905+ 1) times (119873

119905+ 1) as

defined in [34] The values of the space derivatives at theChebyshev-Gauss-Lobatto points (119909

119894 119905119895) (119894 = 0 1 2 119873

119909)

are computed as

120597119906

120597119909

10038161003816100381610038161003816100381610038161003816119909=119909119894119905=119905119895

=

119873119909

sum

119901=0

119873119905

sum

119896=0

119906 (119909119901 119905119896)

119889119871119901(119909119894)

119889119909119871119896(119905119895)

=

119873119909

sum

119901=0

119906 (119909119901 119905119895)119863119894119901

=

119873119909

sum

119901=0

119863119894119901119906 (119909119901 119905119895)

(15)

where 119863119894119901

= 119889119871119901(119909119894)119889119909 is the standard first derivative

Chebyshev differentiation matrix of size (119873119909+ 1) times (119873

119909+ 1)

Similarly for an 119899th order derivative we have

120597119899

119906

120597119909119899

10038161003816100381610038161003816100381610038161003816119909=119909119894119905=119905119895

=

119873119909

sum

119901=0

119863119899

119894119901119906 (119909119901 119905119895) = D119899U

119895

119894 = 0 1 2 119873119909

(16)

where the vector U119895is defined as

U119895= [119906119895(1199090) 119906119895(1199091) 119906

119895(119909119873119909

)]119879 (17)

4 The Scientific World Journal

and the superscript 119879 denotes matrix transpose Substituting(16) into (12) we get

119865 [U119903+1119895

U1015840119903+1119895

U(119899)119903+1119895

] +

119899

sum

119896=0

Φ119896119903U(119896)119903+1119895

minus

119873119905

sum

119896=0

119889119895119896U119903+1119896

= 119877119903[U119903119895U1015840119903119895 U(119899)

119903119895]

(18)

for 119895 = 0 1 2 119873119905 where

U(119899)119903+1119895

= D119899U119903+1119895

Φ119896119903

=

[[[[

[

120601119896119903

(1199090 119905119895)

120601119896119903

(1199091 119905119895)

d120601119896119903

(119909119873119909

119905119895)

]]]]

]

(19)

The initial condition for (3) corresponds to 120591119873119905

= minus1 andhence we express (18) as

119865 [U119903+1119895

U1015840119903+1119895

U(119899)119903+1119895

]

+

119899

sum

119896=0

Φ119896119903U(119896)119903+1119895

minus

119873119905minus1

sum

119896=0

119889119895119896U119903+1119896

= R119895

(20)

where

R119895= 119877119903[U119903119895U1015840119903119895 U(119899)

119903119895] + 119889119895119873119905

U119873119905

119895 = 0 1 2 119873119905minus 1

(21)

Equation (20) can be expressed as the following119873119905(119873119909+1) times

119873119905(119873119909+ 1)matrix system

[[[[

[

11986000

11986001

sdot sdot sdot 1198600119873119905minus1

11986010

11986011

sdot sdot sdot 1198601119873119905minus1

d

119860119873119905minus10

119860119873119905minus11

sdot sdot sdot 119860119873119905minus1119873119905minus1

]]]]

]

[[[[

[

U0

U1

U119873119905minus1

]]]]

]

=

[[[[

[

R0

R1

R119873119905minus1

]]]]

]

(22)

where

119860119894119894

= 119865 [ID D(119899)] +119899

sum

119896=0

Φ119896119903D(119896) minus 119889

119894119894I

119860119894119895

= minus119889119894119895I when 119894 = 119895

(23)

and I is the identity matrix of size (119873119909+1)times (119873

119909+1) Solving

(19) gives 119906(119909119894 119905119895) and hence we use (4) to approximate

119906(119909 119905)

3 Numerical Experiments

We apply the proposed algorithm to well-known nonlinearPDEs of the form (3) with exact solutions In order todetermine the level of accuracy of the BI-SQLM approximatesolution at a particular time level in comparison with theexact solution we report maximum error which is defined by

119864119873

= max119903

1003816100381610038161003816119906 (119909119903 119905) minus (119909

119903 119905)

1003816100381610038161003816 0 le 119903 le 119873 (24)

where (119909119903 119905) is the approximate solution and is the 119906(119909

119903 119905)

exact solution at the time level 119905

Example 1 Weconsider the generalizedBurgers-Fisher equa-tion [35]

120597119906

120597119905+ 120572119906120575120597119906

120597119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (25)

with initial condition

119906 (119909 0) = 1

2+

1

2tanh(

minus120572120575

2(120575 + 1)119909)

1120575

(26)

and exact solution

119906 (119909 119905)

= 1

2+

1

2tanh(

minus120572120575

2 (120575 + 1)

times [119909 minus (120572

120575 + 1+

120573 (120575 + 1)

120572) 119905])

1120575

(27)

where 120572 120573 and 120575 are parameters For illustration purposesthese parameters are chosen to be120572 = 120573 = 120575 = 1 in this paperThe linear operator 119865 and nonlinear operator119866 are chosen as

119865 (119906) = 11990610158401015840

+ 119906 119866 (119906) = minus1199061199061015840

minus 1199062

(28)

We first linearize the nonlinear operator 119866 We approximate119866 using the equation

119866 asymp 119866 [119906119903 1199061015840

119903 11990610158401015840

119903] +

2

sum

119896=0

120601119896119903119906(119896)

119903+1minus

2

sum

119896=0

120601119896119903119906(119896)

119903 (29)

The coefficients are given by

1206010119903

=120597119866

120597119906[119906119903 1199061015840

119903 11990610158401015840

119903] = minus (119906

1015840

119903+ 2119906119903)

1206011119903

=120597119866

1205971199061015840[119906119903 1199061015840

119903 11990610158401015840

119903] = minus119906

119903

1206012119903

=120597119866

12059711990610158401015840[119906119903 1199061015840

119903 11990610158401015840

119903] = 0

119877119903=

2

sum

119896=0

120601119896119903119906(119896)

119903minus 119866 [119906

119903 1199061015840

119903 11990610158401015840

119903] = minus119906

2

119903minus 1199061199031199061015840

119903

(30)

The Scientific World Journal 5

Therefore the linearized equation can be expressed as

11990610158401015840

119903+1+ 12060111199031199061015840

119903+1+ 1206010119903119906119903+1

+ 119906119903+1

minus = 119877119903 (31)

Applying the spectral method both in 119909 and 119905 and initialcondition we get

D2U119903+1119894

+Φ1119903DU119903+1119894

+Φ0119903U119903+1119894

+ U119903+1119894

minus 2

119873119905minus1

sum

119895=0

119889119894119895U119903+1119895

= R119894

(32)

Equation (32) can be expressed as

[[[[

[

11986000

11986001

sdot sdot sdot 1198600119873119905minus1

11986010

11986011

sdot sdot sdot 1198601119873119905minus1

d

119860119873119905minus10

119860119873119905minus11

sdot sdot sdot 119860119873119905minus1119873119905minus1

]]]]

]

[[[[

[

U0

U1

U119873119905minus1

]]]]

]

=

[[[[

[

R0

R1

R119873119905minus1

]]]]

]

(33)

where

119860119894119894

= D2 +Φ(119894)1119903D +Φ

(119894)

0119903+ (1 minus 2119889

119894119894) I

119860119894119895

= minus 2119889119894119895I when 119894 = 119895

R119894= 119877119903+ 2119889119894119873119905

U119903119873119905

(34)

The boundary conditions are implemented in the first andlast row of the matrices 119860

119894119895and the column vectors R

119894for

119894 = 0 1 119873119905minus 1 and 119895 = 0 1 119873

119905minus 1 The procedure

for finding the variable coefficients 120601119894and matrices for the

remaining examples is similar

Example 2 We consider Fisherrsquos equation

120597119906

120597119905=

1205972

119906

1205971199092+ 120572119906 (1 minus 119906) (35)

subject to the initial condition

119906 (119909 0) =1

(1 + 119890radic1205726119909

)2 (36)

and exact solution [36]

119906 (119909 119905) =1

(1 + 119890radic1205726119909minus51205721199056

)2 (37)

where 120572 is a constant The Fisher equation represents areactive-diffusive system and is encountered in chemicalkinetics and population dynamics applications For thisexample the appropriate linear operator 119865 and nonlinearoperator 119866 are chosen as

119865 (119906) = 11990610158401015840

+ 120572119906 119866 (119906) = minus1205721199062

(38)

Table 1 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1001 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

02 3934119890 minus 008 3121119890 minus 011 1552119890 minus 012 1561119890 minus 012

03 5577119890 minus 008 4864119890 minus 011 1004119890 minus 012 1005119890 minus 012

04 6997119890 minus 008 6802119890 minus 011 7895119890 minus 013 8124119890 minus 013

05 8107119890 minus 008 7971119890 minus 011 1088119890 minus 012 1027119890 minus 012

06 8891119890 minus 008 8560119890 minus 011 8805119890 minus 013 7847119890 minus 013

07 9344119890 minus 008 8953119890 minus 011 6418119890 minus 013 6463119890 minus 013

08 9431119890 minus 008 8759119890 minus 011 6199119890 minus 013 6164119890 minus 013

09 9178119890 minus 008 8325119890 minus 011 3978119890 minus 013 3695119890 minus 013

10 8787119890 minus 008 7421119890 minus 011 7988119890 minus 014 5596119890 minus 014

CPUtime(sec)

0019942 0025988 0027756 0029436

Table 2Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 120574 = 120575 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 1142119890 minus 007 1369119890 minus 010 5891119890 minus 012 6143119890 minus 012

02 1178119890 minus 007 1373119890 minus 010 9570119890 minus 012 1013119890 minus 011

03 1186119890 minus 007 1479119890 minus 010 1489119890 minus 011 1512119890 minus 011

04 1069119890 minus 007 9450119890 minus 011 1703119890 minus 011 1702119890 minus 011

05 9030119890 minus 008 7944119890 minus 011 5283119890 minus 012 5736119890 minus 012

06 6963119890 minus 008 6618119890 minus 011 1639119890 minus 011 1626119890 minus 011

07 4638119890 minus 008 1579119890 minus 011 1362119890 minus 011 1364119890 minus 011

08 2457119890 minus 008 4030119890 minus 011 3934119890 minus 012 3852119890 minus 012

09 2028119890 minus 008 6006119890 minus 011 4466119890 minus 012 4727119890 minus 012

10 3147119890 minus 008 7708119890 minus 011 7757119890 minus 013 7261119890 minus 013

CPUTime(sec)

0010152 0015387 0019163 0021564

Example 3 Consider the Fitzhugh-Nagumo equation

120597119906

120597119905=

1205972

119906

1205971199092+ 119906 (119906 minus 120572) (1 minus 119906) (39)

with initial condition

119906 (119909 0) =1

2[1 minus coth(minus

119909

2radic2

)] (40)

This equation has the exact solution [37]

119906 (119909 119905) =1

2[1 minus coth(minus

119909

2radic2

+2120572 minus 1

4119905)] (41)

where 120572 is a parameter In this example the linear operator 119865and nonlinear operator 119866 are chosen as

119865 (119906) = 11990610158401015840

minus 120572119906 119866 (119906) = (1 + 120572) 1199062

minus 1199063

(42)

6 The Scientific World Journal

Table 3 Maximum errors 119864119873for the Fitzhug-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 5719119890 minus 007 1196119890 minus 009 2367119890 minus 012 9881119890 minus 014

02 6193119890 minus 007 1299119890 minus 009 2761119890 minus 012 3952119890 minus 014

03 6662119890 minus 007 1463119890 minus 009 3259119890 minus 012 8216119890 minus 014

04 6779119890 minus 007 1448119890 minus 009 3341119890 minus 012 8094119890 minus 014

05 6920119890 minus 007 1526119890 minus 009 3587119890 minus 012 5063119890 minus 014

06 7019119890 minus 007 1573119890 minus 009 3729119890 minus 012 3775119890 minus 014

07 6933119890 minus 007 1516119890 minus 009 3660119890 minus 012 8915119890 minus 014

08 6828119890 minus 007 81535119890 minus 009 3635119890 minus 012 7594119890 minus 014

09 6765119890 minus 007 1528119890 minus 009 3519119890 minus 012 3242119890 minus 013

10 6687119890 minus 007 1490119890 minus 009 3405119890 minus 012 1688119890 minus 013

CPUtime(sec)

0024281 0024901 0026810 0032389

Table 4Maximum errors119864119873for the Burger-Huxley equation when

120574 = 075 120573 = 1 and119873119905= 10

119905 119873119909

4 6 8 1001 2217119890 minus 006 8482119890 minus 009 2166119890 minus 011 7822119890 minus 014

02 2596119890 minus 006 9369119890 minus 009 2536119890 minus 011 1184119890 minus 013

03 2859119890 minus 006 1073119890 minus 008 3201119890 minus 011 1049119890 minus 013

04 3001119890 minus 006 1112119890 minus 008 3652119890 minus 011 9426119890 minus 014

05 3137119890 minus 006 1213119890 minus 008 4262119890 minus 011 1510119890 minus 013

06 3270119890 minus 006 1311119890 minus 008 4842119890 minus 011 2127119890 minus 013

07 3367119890 minus 006 1359119890 minus 008 5289119890 minus 011 1230119890 minus 013

08 3467119890 minus 006 1438119890 minus 008 5803119890 minus 011 1549119890 minus 013

09 3562119890 minus 006 1504119890 minus 008 6260119890 minus 011 3063119890 minus 013

10 3640119890 minus 006 1559119890 minus 008 6674119890 minus 011 2951119890 minus 013

CPUtime(sec)

0023822 0024901 002685 0032806

Example 4 Consider the Burgers-Huxley equation

120597119906

120597119905+ 120572119906120575

119906119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (119906120575

minus 120574) (43)

where 120572 120573 ge 0 are constant parameters 120575 is a positive integer(set to be 120575 = 1 in this study) and 120574 isin (0 1) The exactsolution subject to the initial condition

119906 (119909 0) =1

2minus

1

2tanh [

120573

119903 minus 120572119909] (44)

is reported in [38 39] as

119906 (119909 119905) =1

2minus

1

2tanh [

120573

119903 minus 120572(119909 minus 119888119905)] (45)

where

119903 = radic1205722+ 8120573 119888 =

(120572 minus 119903) (2120574 minus 1) + 2120572

4

(46)

Table 5 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1001 1803119890 minus 007 3419119890 minus 010 4449119890 minus 013 1572119890 minus 013

02 2614119890 minus 007 4347119890 minus 010 5049119890 minus 013 5992119890 minus 014

03 2717119890 minus 007 4677119890 minus 010 5532119890 minus 013 8128119890 minus 013

04 2009119890 minus 007 3663119890 minus 010 4771119890 minus 013 6158119890 minus 013

05 2580119890 minus 007 4410119890 minus 010 7518119890 minus 013 2555119890 minus 013

06 2653119890 minus 007 4606119890 minus 010 8738119890 minus 013 5756119890 minus 013

07 2248119890 minus 007 4039119890 minus 010 6210119890 minus 013 2393119890 minus 013

08 2572119890 minus 007 4476119890 minus 010 5432119890 minus 013 6812119890 minus 013

09 2436119890 minus 007 4351119890 minus 010 6111119890 minus 013 6287119890 minus 013

10 8275119890 minus 008 3721119890 minus 010 7569119890 minus 013 1087119890 minus 007

CPUtime(sec)

0015646 0021226 0030159 0035675

Table 6 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1001 7788119890 minus 005 3553119890 minus 007 7601119890 minus 010 2080119890 minus 010

02 1153119890 minus 004 4000119890 minus 007 5684119890 minus 010 1189119890 minus 010

03 1011119890 minus 004 3739119890 minus 007 4471119890 minus 010 4503119890 minus 010

04 3926119890 minus 005 1785119890 minus 007 6544119890 minus 010 4987119890 minus 010

05 6727119890 minus 005 2342119890 minus 007 2638119890 minus 010 1528119890 minus 010

06 6065119890 minus 005 2207119890 minus 007 4565119890 minus 010 4568119890 minus 010

07 2511119890 minus 005 1105119890 minus 007 4749119890 minus 010 3748119890 minus 010

08 4074119890 minus 005 1427119890 minus 007 1062119890 minus 010 1604119890 minus 010

09 2386119890 minus 005 1018119890 minus 007 2343119890 minus 010 8114119890 minus 011

10 1440119890 minus 006 7256119890 minus 008 1436119890 minus 009 1513119890 minus 011

CPUtime(sec)

0020609 0021241 0030617 0032816

The general solution (45) was reported in [40 41] In thisexample the linear operator 119865 and nonlinear operator 119866 arechosen as

119865 (119906) = 11990610158401015840

minus 120573120574119906

119866 (119906) = minus1205721199061199061015840

+ 120573 (1 + 120574) 1199062

minus 1205731199063

(47)

Example 5 We consider the modified KdV-Burgers equation

120597119906

120597119905=

1205973

119906

1205971199093minus

1205972

119906

1205971199092minus 61199062120597119906

120597119909

(48)

subject to the initial condition

119906 (119909 0) =1

6+

1

6tanh(

119909

6) (49)

and exact solution [42]

119906 (119909 119905) =1

6+

1

6tanh(

119909

6minus

119905

27) (50)

The Scientific World Journal 7

Table 7 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1002 1119119890 minus 011 7398119890 minus 013 8266119890 minus 013 3808119890 minus 014

04 3121119890 minus 011 1552119890 minus 012 7378119890 minus 013 3780119890 minus 014

06 4864119890 minus 011 1004119890 minus 012 3402119890 minus 012 7283119890 minus 014

08 6802119890 minus 011 7895119890 minus 013 1118119890 minus 012 3714119890 minus 014

10 7971119890 minus 011 1088119890 minus 012 1473119890 minus 012 1691119890 minus 013

12 8560119890 minus 011 8805119890 minus 013 2611119890 minus 012 3119119890 minus 013

14 8953119890 minus 011 6418119890 minus 013 6671119890 minus 012 1796119890 minus 013

16 8759119890 minus 011 6199119890 minus 013 1118119890 minus 011 1097119890 minus 013

18 8325119890 minus 011 3978119890 minus 013 7515119890 minus 013 6273119890 minus 014

20 7421119890 minus 011 7988119890 minus 014 3682119890 minus 012 2311119890 minus 013

CPUtime(sec)

0013542 0022967 0023792 0024758

Table 8Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 1223119890 minus 007 1400119890 minus 008 1402119890 minus 008 1094119890 minus 012

04 1145119890 minus 007 1919119890 minus 008 1918119890 minus 008 3919119890 minus 012

06 9192119890 minus 008 2082119890 minus 008 2085119890 minus 008 1953119890 minus 012

08 2293119890 minus 008 1793119890 minus 008 1793119890 minus 008 6340119890 minus 013

10 2395119890 minus 008 1337119890 minus 008 1339119890 minus 008 2381119890 minus 012

12 5778119890 minus 008 1954119890 minus 008 1930119890 minus 008 1005119890 minus 011

14 6045119890 minus 008 1620119890 minus 008 1620119890 minus 008 3535119890 minus 012

16 5244119890 minus 008 7218119890 minus 009 7345119890 minus 009 5765119890 minus 012

18 4395119890 minus 008 6828119890 minus 009 6784119890 minus 009 3983119890 minus 012

20 2944119890 minus 008 9406119890 minus 010 8820119890 minus 010 3812119890 minus 012

CPUtime(sec)

0019942 0025988 0027756 0029436

The modified KdV-Burgers equation describes various kindsof phenomena such as a mathematical model of turbulence[43] and the approximate theory of flow through a shockwavetraveling in viscous fluid [44] For this example the linearoperator 119865 and nonlinear operator 119866 are chosen as

119865 (119906) = 119906101584010158401015840

minus 11990610158401015840

119866 (119906) = minus61199061015840

1199062

(51)

Example 6 We consider the high nonlinear modified KdVequation

120597119906

120597119905=

1205973

119906

1205971199093+ (

120597119906

120597119909)

2

minus 1199062 (52)

subject to the initial condition

119906 (119909 0) =1

2+

119890minus119909

4

(53)

Table 9 Maximum errors 119864119873for the Fitzhugh-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 6326119890 minus 007 1311119890 minus 009 2886119890 minus 012 1131119890 minus 012

04 6721119890 minus 007 1467119890 minus 009 3310119890 minus 012 1564119890 minus 012

06 7140119890 minus 007 1602119890 minus 009 3617119890 minus 012 1936119890 minus 012

08 6730119890 minus 007 1496119890 minus 009 4707119890 minus 012 1196119890 minus 012

10 6660119890 minus 007 1487119890 minus 009 3675119890 minus 012 1264119890 minus 012

12 6449119890 minus 007 1366119890 minus 009 1897119890 minus 012 1727119890 minus 012

14 5690119890 minus 007 1083119890 minus 009 2972119890 minus 012 1200119890 minus 012

16 4931119890 minus 007 8010119890 minus 010 1519119890 minus 012 8590119890 minus 013

18 3986119890 minus 007 4658119890 minus 010 1068119890 minus 012 6790119890 minus 013

20 2904119890 minus 007 2968119890 minus 010 1592119890 minus 012 1770119890 minus 013

CPUtime(sec)

0041048 0049629 0055008 0053863

Table 10 Maximum errors 119864119873

for the Burgers-Huxley equationwhen 120574 = 05 120573 = 1 and119873

119905= 10

119905 119873119909

4 6 8 1002 2866119890 minus 006 1119119890 minus 008 3670119890 minus 011 1150119890 minus 012

04 3401119890 minus 006 1420119890 minus 008 5744119890 minus 011 1638119890 minus 012

06 3814119890 minus 006 1687119890 minus 008 7426119890 minus 011 1958119890 minus 012

08 3915119890 minus 006 1729119890 minus 008 8171119890 minus 011 7002119890 minus 013

10 3938119890 minus 006 1738119890 minus 008 8157119890 minus 011 1267119890 minus 012

12 3808119890 minus 006 1624119890 minus 008 7687119890 minus 011 1710119890 minus 012

14 3456119890 minus 006 1527119890 minus 008 6965119890 minus 011 5109119890 minus 013

16 3230119890 minus 006 1349119890 minus 008 5535119890 minus 011 8203119890 minus 013

18 2925119890 minus 006 1078119890 minus 008 3598119890 minus 011 8294119890 minus 013

20 2497119890 minus 006 7505119890 minus 009 2265119890 minus 011 9726119890 minus 014

CPUtime(sec)

0023822 0024901 002685 0032806

and exact solution

119906 (119909 119905) =1

119905 + 2+

119890minus(119909+119905)

(119905 + 2)2 (54)

For this example the linear operator 119865 and nonlinear opera-tor 119866 are chosen as

119865 (119906) = 119906101584010158401015840

119866 (119906) = (1199061015840

)2

minus 1199062

(55)

4 Results and Discussion

In this section we present the numerical solutions obtainedusing the BI-SQLM algorithm The number of collocationpoints in the space 119909 variable used to generate the results is119873119909

= 10 in all cases Similarly the number of collocationpoints in the time 119905 variable used is 119873

119905= 10 in all cases It

was found that sufficient accuracy was achieved using thesevalues in all numerical simulations

8 The Scientific World Journal

Table 11 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1002 2137119890 minus 007 3820119890 minus 010 4846119890 minus 013 9998119890 minus 013

04 2480119890 minus 007 4267119890 minus 010 5596119890 minus 013 8775119890 minus 013

06 2691119890 minus 007 4676119890 minus 010 6565119890 minus 013 2054119890 minus 012

08 2214119890 minus 007 3979119890 minus 010 8776119890 minus 013 1168119890 minus 012

10 2538119890 minus 007 4463119890 minus 010 9650119890 minus 013 8410119890 minus 013

12 2650119890 minus 007 4680119890 minus 010 7450119890 minus 013 5113119890 minus 013

14 2383119890 minus 007 4296119890 minus 010 7500119890 minus 013 1110119890 minus 012

16 2568119890 minus 007 4572119890 minus 010 9704119890 minus 013 2837119890 minus 013

18 2520119890 minus 007 4529119890 minus 010 7443119890 minus 013 5353119890 minus 013

20 2370119890 minus 007 4438119890 minus 010 2719119890 minus 013 8849119890 minus 013

CPUtime(sec)

0062066 0081646 0080718 010775

Table 12 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1002 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

04 8010119890 minus 005 3577119890 minus 007 3902119890 minus 008 1979119890 minus 010

06 7235119890 minus 005 2549119890 minus 007 2016119890 minus 008 4899119890 minus 010

08 6284119890 minus 005 1663119890 minus 007 1155119890 minus 007 2679119890 minus 010

10 1642119890 minus 005 1620119890 minus 007 1243119890 minus 007 2474119890 minus 010

12 2753119890 minus 005 1073119890 minus 007 1073119890 minus 007 1679119890 minus 010

14 3738119890 minus 006 8971119890 minus 008 8598119890 minus 008 4788119890 minus 011

16 1223119890 minus 005 2153119890 minus 008 2503119890 minus 008 2941119890 minus 011

18 5836119890 minus 006 2986119890 minus 008 9127119890 minus 009 5177119890 minus 011

20 9310119890 minus 006 6548119890 minus 008 7277119890 minus 008 1453119890 minus 009

CPUtime(sec)

0020609 0021241 0030617 0032816

In Tables 1 2 3 4 5 and 6 we give the maximumerrors between the exact and BI-SQLM results for the Fisherequation Burgers-Fisher equation Fitzhugh-Nagumo equa-tion Burgers-Huxley equation the modified KdV-Burgersequation and the modified KdV equation respectively at119905 isin [01 1] The results were computed in the space domain119909 isin [0 1] To give a sense of the computational efficiency ofthe method the computational time to generate the resultsis also given Tables 1ndash6 clearly show the accuracy of themethod The accuracy is seen to improve with an increasein the number of collocation points 119873

119909 It is remarkable to

note that accurate results with errors of order up to 10minus14

are obtained using very few collocation points in both the 119909

and 119905 variables 119873119905le 10 119873

119909le 10 This is a clear indication

that the BI-SQLM is powerful method that is appropriatein solving nonlinear evolution PDEs We remark also thatthe BI-SQLM is computationally fast as accurate results aregenerated in a fraction of a second in all the examplesconsidered in this work

0 05 1 15 206

062

064

066

068

07

072

Time (s)

The analytical solution versus the approximate solution ofFishers equation

ApproximateAnalytic

u(xt)

Figure 1 Fishers equation analytical solution graph

088

0885

089

0895

09

0905

091

0915

092

0925

093

The analytical solution versus the approximate solution of theBurgers-Fisher equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 2 Burger-Fishers equation analytical solution graph

In Tables 7 8 9 10 11 and 12 we give the maxi-mum errors of the BI-SQLM results for the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and themodified KdV equation respectively at selectedvalues of 119905 = 2 for different collocation points 119873

119905 in the

119905-variable The results in Tables 7ndash12 were computed on thespace domain 119909 isin [0 1] We note that the accuracy does notdetoriate when 119905 gt 1 for this method as is often the case withnumerical schemes such as finite differences

The Scientific World Journal 9

0 05 1 15 2Time (s)

ApproximateAnalytic

026

028

03

032

034

036

038

04

042

044

The analytical solution versus the approximate solution of theFitzhurg-Nagumo equation

u(xt)-a

xis

Figure 3 Fitzhugh-Nagumo equation analytical solution graph

025

035

045

055

06

05

04

03

The analytical solution and the approximate solution of theBurger-Huxley Equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 4 Burgers-Huxley equation analytical solution graph

Figures 1 2 3 4 5 and 6 show a comparison of theanalytical and approximate solutions of the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and the modified KdV equation respectively when 119905 =

2 The approximate solutions are in excellent agreement withthe analytical solutions and this demonstrates the accuracyof the algorithm presented in this study

In Figures 7 8 9 10 11 and 12 we present error analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation the

0 05 1 15 2Time (s)

ApproximateAnalytic

015

0155

016

0165

017

0175

018

0185

The analytical solution and the approximate solution of themodified KdV-Burger

u(xt)

Figure 5 Modified KdV-Burger equation analytical solution graph

0 05 1 15 2Time (s)

ApproximateAnalytic

0253

0254

0255

0256

0257

0258

0259

026

The analytical solution versus the approximate solution ofmodified KdV

u(xt)

Figure 6 Modified KdV equation analytical solution graph

modified KdV-Burgers equation and the modified KdVequation respectively when 119905 = 2

In Figures 13 14 15 16 17 and 18 convergence analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation themodified KdV-Burgers equation and the modified KdVequation respectively The figures present a variation of theerror norm at a fixed value of time (119905 = 1) with iterationsof the BI-SQLM scheme It can be seen that in almost allthe examples considered the iteration scheme takes about3 or 4 iterations to converge fully Beyond the point wherefull convergence is reached error norm levels off and does

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

The solution procedure assumes that the solution can beapproximated by a bivariate Lagrange interpolation polyno-mial of the form

119906 (119909 119905) asymp

119873119909

sum

119894=0

119873119905

sum

119895=0

119906 (119909119894 119905119895) 119871119894(119909) 119871119895(119905) (4)

which interpolates 119906(119909 119905) at selected points in both the 119909 and119905 directions defined by

119909119894 = cos( 120587119894

119873119909

)

119873119909

119894=0

119905119895 = cos(

120587119895

119873119905

)

119873119905

119895=0

(5)

The choice of the Chebyshev-Gauss-Lobatto grid points (5)ensures that there is a simple conversion of the continuousderivatives in both space and time to discrete derivativesat the grid points The functions 119871

119894(119909) are the characteristic

Lagrange cardinal polynomials

119871119894(119909) =

119873119909

prod

119894=0

119894 =119896

119909 minus 119909119896

119909119894minus 119909119896

(6)

where

119871119894(119909119896) = 120575119894119896

= 0 if 119894 = 119896

1 if 119894 = 119896(7)

The function 119871119895(119905) is defined in a similar manner Before

linearizing (3) it is convenient to split 119867 into its linear andnonlinear components and rewrite the governing equation inthe form

119865 [119906 1199061015840

119906(119899)

] + 119866 [119906 1199061015840

119906(119899)

] minus = 0 (8)

where the dot and primes denote the time and space deriva-tives respectively 119865 is a linear operator and 119866 is a nonlinearoperator Assuming that the difference 119906

119903+1minus 119906119903and all itrsquos

space derivative is small we first approximate the nonlinearoperator 119866 using the linear terms of the Taylor series andhence

119866 [119906 1199061015840

119906(119899)

] asymp 119866 [119906119903 1199061015840

119903 119906

(119899)

119903]

+

119899

sum

119896=0

120597119866

120597119906(119896)

(119906(119896)

119903+1minus 119906(119896)

119903)

(9)

where 119903 and 119903 + 1 denote previous and current iterationsrespectively We remark that this quasilinearization method(QLM) approach is a generalisation of the Newton-Raphsonmethod and was first proposed by Bellman and Kalaba [33]for solving nonlinear boundary value problems

Equation (9) can be expressed as

119866 [119906 1199061015840

119906(119899)

] asymp 119866 [119906119903 1199061015840

119903 119906

(119899)

119903]

+

119899

sum

119896=0

120601119896119903

[119906119903 1199061015840

119903 119906

(119899)

119903] 119906(119896)

119903+1

minus

119899

sum

119896=0

120601119896119903

[119906119903 1199061015840

119903 119906

(119899)

119903] 119906(119896)

119903

(10)

where

120601119896119903

[119906119903 1199061015840

119903 119906

(119899)

119903] =

120597119866

120597119906(119896)

[119906119903 1199061015840

119903 119906

(119899)

119903] (11)

Substituting (10) into (8) we get

119865 [119906119903+1

1199061015840

119903+1 119906

(119899)

119903+1] +

119899

sum

119896=0

120601119896119903119906(119896)

119903+1minus 119903+1

= 119877119903[119906119903 1199061015840

119903 119906

(119899)

119903]

(12)

where

119877119903[119906119903 1199061015840

119903 119906

(119899)

119903] =

119899

sum

119896=0

120601119896119903119906(119896)

119903minus 119866 [119906

119903 1199061015840

119903 119906

(119899)

119903] (13)

A crucial step in the implementation of the solution proce-dure is the evaluation of the time derivative at the grid points119905119895(119895 = 0 1 119873

119905) and the spatial derivatives at the grid

points 119909119894(119894 = 0 1 119873

119909) The values of the time derivatives

at the Chebyshev-Gauss-Lobatto points (119909119894 119905119895) are computed

as (for 119895 = 0 1 2 119873119905)

120597119906

120597119905

10038161003816100381610038161003816100381610038161003816119909=119909119894119905=119905119895

=

119873119909

sum

119901=0

119873119905

sum

119896=0

119906 (119909119901 119905119896) 119871119901(119909119894)

119889119871119896(119905119895)

119889119905

=

119873119905

sum

119896=0

119906 (119909119894 119905119896) 119889119895119896

=

119873119905

sum

119896=0

119889119895119896119906 (119909119894 119905119896)

(14)

where 119889119895119896

= 119889119871119896(119905119895)119889119905 is the standard first derivative Che-

byshev differentiation matrix of size (119873119905+ 1) times (119873

119905+ 1) as

defined in [34] The values of the space derivatives at theChebyshev-Gauss-Lobatto points (119909

119894 119905119895) (119894 = 0 1 2 119873

119909)

are computed as

120597119906

120597119909

10038161003816100381610038161003816100381610038161003816119909=119909119894119905=119905119895

=

119873119909

sum

119901=0

119873119905

sum

119896=0

119906 (119909119901 119905119896)

119889119871119901(119909119894)

119889119909119871119896(119905119895)

=

119873119909

sum

119901=0

119906 (119909119901 119905119895)119863119894119901

=

119873119909

sum

119901=0

119863119894119901119906 (119909119901 119905119895)

(15)

where 119863119894119901

= 119889119871119901(119909119894)119889119909 is the standard first derivative

Chebyshev differentiation matrix of size (119873119909+ 1) times (119873

119909+ 1)

Similarly for an 119899th order derivative we have

120597119899

119906

120597119909119899

10038161003816100381610038161003816100381610038161003816119909=119909119894119905=119905119895

=

119873119909

sum

119901=0

119863119899

119894119901119906 (119909119901 119905119895) = D119899U

119895

119894 = 0 1 2 119873119909

(16)

where the vector U119895is defined as

U119895= [119906119895(1199090) 119906119895(1199091) 119906

119895(119909119873119909

)]119879 (17)

4 The Scientific World Journal

and the superscript 119879 denotes matrix transpose Substituting(16) into (12) we get

119865 [U119903+1119895

U1015840119903+1119895

U(119899)119903+1119895

] +

119899

sum

119896=0

Φ119896119903U(119896)119903+1119895

minus

119873119905

sum

119896=0

119889119895119896U119903+1119896

= 119877119903[U119903119895U1015840119903119895 U(119899)

119903119895]

(18)

for 119895 = 0 1 2 119873119905 where

U(119899)119903+1119895

= D119899U119903+1119895

Φ119896119903

=

[[[[

[

120601119896119903

(1199090 119905119895)

120601119896119903

(1199091 119905119895)

d120601119896119903

(119909119873119909

119905119895)

]]]]

]

(19)

The initial condition for (3) corresponds to 120591119873119905

= minus1 andhence we express (18) as

119865 [U119903+1119895

U1015840119903+1119895

U(119899)119903+1119895

]

+

119899

sum

119896=0

Φ119896119903U(119896)119903+1119895

minus

119873119905minus1

sum

119896=0

119889119895119896U119903+1119896

= R119895

(20)

where

R119895= 119877119903[U119903119895U1015840119903119895 U(119899)

119903119895] + 119889119895119873119905

U119873119905

119895 = 0 1 2 119873119905minus 1

(21)

Equation (20) can be expressed as the following119873119905(119873119909+1) times

119873119905(119873119909+ 1)matrix system

[[[[

[

11986000

11986001

sdot sdot sdot 1198600119873119905minus1

11986010

11986011

sdot sdot sdot 1198601119873119905minus1

d

119860119873119905minus10

119860119873119905minus11

sdot sdot sdot 119860119873119905minus1119873119905minus1

]]]]

]

[[[[

[

U0

U1

U119873119905minus1

]]]]

]

=

[[[[

[

R0

R1

R119873119905minus1

]]]]

]

(22)

where

119860119894119894

= 119865 [ID D(119899)] +119899

sum

119896=0

Φ119896119903D(119896) minus 119889

119894119894I

119860119894119895

= minus119889119894119895I when 119894 = 119895

(23)

and I is the identity matrix of size (119873119909+1)times (119873

119909+1) Solving

(19) gives 119906(119909119894 119905119895) and hence we use (4) to approximate

119906(119909 119905)

3 Numerical Experiments

We apply the proposed algorithm to well-known nonlinearPDEs of the form (3) with exact solutions In order todetermine the level of accuracy of the BI-SQLM approximatesolution at a particular time level in comparison with theexact solution we report maximum error which is defined by

119864119873

= max119903

1003816100381610038161003816119906 (119909119903 119905) minus (119909

119903 119905)

1003816100381610038161003816 0 le 119903 le 119873 (24)

where (119909119903 119905) is the approximate solution and is the 119906(119909

119903 119905)

exact solution at the time level 119905

Example 1 Weconsider the generalizedBurgers-Fisher equa-tion [35]

120597119906

120597119905+ 120572119906120575120597119906

120597119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (25)

with initial condition

119906 (119909 0) = 1

2+

1

2tanh(

minus120572120575

2(120575 + 1)119909)

1120575

(26)

and exact solution

119906 (119909 119905)

= 1

2+

1

2tanh(

minus120572120575

2 (120575 + 1)

times [119909 minus (120572

120575 + 1+

120573 (120575 + 1)

120572) 119905])

1120575

(27)

where 120572 120573 and 120575 are parameters For illustration purposesthese parameters are chosen to be120572 = 120573 = 120575 = 1 in this paperThe linear operator 119865 and nonlinear operator119866 are chosen as

119865 (119906) = 11990610158401015840

+ 119906 119866 (119906) = minus1199061199061015840

minus 1199062

(28)

We first linearize the nonlinear operator 119866 We approximate119866 using the equation

119866 asymp 119866 [119906119903 1199061015840

119903 11990610158401015840

119903] +

2

sum

119896=0

120601119896119903119906(119896)

119903+1minus

2

sum

119896=0

120601119896119903119906(119896)

119903 (29)

The coefficients are given by

1206010119903

=120597119866

120597119906[119906119903 1199061015840

119903 11990610158401015840

119903] = minus (119906

1015840

119903+ 2119906119903)

1206011119903

=120597119866

1205971199061015840[119906119903 1199061015840

119903 11990610158401015840

119903] = minus119906

119903

1206012119903

=120597119866

12059711990610158401015840[119906119903 1199061015840

119903 11990610158401015840

119903] = 0

119877119903=

2

sum

119896=0

120601119896119903119906(119896)

119903minus 119866 [119906

119903 1199061015840

119903 11990610158401015840

119903] = minus119906

2

119903minus 1199061199031199061015840

119903

(30)

The Scientific World Journal 5

Therefore the linearized equation can be expressed as

11990610158401015840

119903+1+ 12060111199031199061015840

119903+1+ 1206010119903119906119903+1

+ 119906119903+1

minus = 119877119903 (31)

Applying the spectral method both in 119909 and 119905 and initialcondition we get

D2U119903+1119894

+Φ1119903DU119903+1119894

+Φ0119903U119903+1119894

+ U119903+1119894

minus 2

119873119905minus1

sum

119895=0

119889119894119895U119903+1119895

= R119894

(32)

Equation (32) can be expressed as

[[[[

[

11986000

11986001

sdot sdot sdot 1198600119873119905minus1

11986010

11986011

sdot sdot sdot 1198601119873119905minus1

d

119860119873119905minus10

119860119873119905minus11

sdot sdot sdot 119860119873119905minus1119873119905minus1

]]]]

]

[[[[

[

U0

U1

U119873119905minus1

]]]]

]

=

[[[[

[

R0

R1

R119873119905minus1

]]]]

]

(33)

where

119860119894119894

= D2 +Φ(119894)1119903D +Φ

(119894)

0119903+ (1 minus 2119889

119894119894) I

119860119894119895

= minus 2119889119894119895I when 119894 = 119895

R119894= 119877119903+ 2119889119894119873119905

U119903119873119905

(34)

The boundary conditions are implemented in the first andlast row of the matrices 119860

119894119895and the column vectors R

119894for

119894 = 0 1 119873119905minus 1 and 119895 = 0 1 119873

119905minus 1 The procedure

for finding the variable coefficients 120601119894and matrices for the

remaining examples is similar

Example 2 We consider Fisherrsquos equation

120597119906

120597119905=

1205972

119906

1205971199092+ 120572119906 (1 minus 119906) (35)

subject to the initial condition

119906 (119909 0) =1

(1 + 119890radic1205726119909

)2 (36)

and exact solution [36]

119906 (119909 119905) =1

(1 + 119890radic1205726119909minus51205721199056

)2 (37)

where 120572 is a constant The Fisher equation represents areactive-diffusive system and is encountered in chemicalkinetics and population dynamics applications For thisexample the appropriate linear operator 119865 and nonlinearoperator 119866 are chosen as

119865 (119906) = 11990610158401015840

+ 120572119906 119866 (119906) = minus1205721199062

(38)

Table 1 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1001 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

02 3934119890 minus 008 3121119890 minus 011 1552119890 minus 012 1561119890 minus 012

03 5577119890 minus 008 4864119890 minus 011 1004119890 minus 012 1005119890 minus 012

04 6997119890 minus 008 6802119890 minus 011 7895119890 minus 013 8124119890 minus 013

05 8107119890 minus 008 7971119890 minus 011 1088119890 minus 012 1027119890 minus 012

06 8891119890 minus 008 8560119890 minus 011 8805119890 minus 013 7847119890 minus 013

07 9344119890 minus 008 8953119890 minus 011 6418119890 minus 013 6463119890 minus 013

08 9431119890 minus 008 8759119890 minus 011 6199119890 minus 013 6164119890 minus 013

09 9178119890 minus 008 8325119890 minus 011 3978119890 minus 013 3695119890 minus 013

10 8787119890 minus 008 7421119890 minus 011 7988119890 minus 014 5596119890 minus 014

CPUtime(sec)

0019942 0025988 0027756 0029436

Table 2Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 120574 = 120575 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 1142119890 minus 007 1369119890 minus 010 5891119890 minus 012 6143119890 minus 012

02 1178119890 minus 007 1373119890 minus 010 9570119890 minus 012 1013119890 minus 011

03 1186119890 minus 007 1479119890 minus 010 1489119890 minus 011 1512119890 minus 011

04 1069119890 minus 007 9450119890 minus 011 1703119890 minus 011 1702119890 minus 011

05 9030119890 minus 008 7944119890 minus 011 5283119890 minus 012 5736119890 minus 012

06 6963119890 minus 008 6618119890 minus 011 1639119890 minus 011 1626119890 minus 011

07 4638119890 minus 008 1579119890 minus 011 1362119890 minus 011 1364119890 minus 011

08 2457119890 minus 008 4030119890 minus 011 3934119890 minus 012 3852119890 minus 012

09 2028119890 minus 008 6006119890 minus 011 4466119890 minus 012 4727119890 minus 012

10 3147119890 minus 008 7708119890 minus 011 7757119890 minus 013 7261119890 minus 013

CPUTime(sec)

0010152 0015387 0019163 0021564

Example 3 Consider the Fitzhugh-Nagumo equation

120597119906

120597119905=

1205972

119906

1205971199092+ 119906 (119906 minus 120572) (1 minus 119906) (39)

with initial condition

119906 (119909 0) =1

2[1 minus coth(minus

119909

2radic2

)] (40)

This equation has the exact solution [37]

119906 (119909 119905) =1

2[1 minus coth(minus

119909

2radic2

+2120572 minus 1

4119905)] (41)

where 120572 is a parameter In this example the linear operator 119865and nonlinear operator 119866 are chosen as

119865 (119906) = 11990610158401015840

minus 120572119906 119866 (119906) = (1 + 120572) 1199062

minus 1199063

(42)

6 The Scientific World Journal

Table 3 Maximum errors 119864119873for the Fitzhug-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 5719119890 minus 007 1196119890 minus 009 2367119890 minus 012 9881119890 minus 014

02 6193119890 minus 007 1299119890 minus 009 2761119890 minus 012 3952119890 minus 014

03 6662119890 minus 007 1463119890 minus 009 3259119890 minus 012 8216119890 minus 014

04 6779119890 minus 007 1448119890 minus 009 3341119890 minus 012 8094119890 minus 014

05 6920119890 minus 007 1526119890 minus 009 3587119890 minus 012 5063119890 minus 014

06 7019119890 minus 007 1573119890 minus 009 3729119890 minus 012 3775119890 minus 014

07 6933119890 minus 007 1516119890 minus 009 3660119890 minus 012 8915119890 minus 014

08 6828119890 minus 007 81535119890 minus 009 3635119890 minus 012 7594119890 minus 014

09 6765119890 minus 007 1528119890 minus 009 3519119890 minus 012 3242119890 minus 013

10 6687119890 minus 007 1490119890 minus 009 3405119890 minus 012 1688119890 minus 013

CPUtime(sec)

0024281 0024901 0026810 0032389

Table 4Maximum errors119864119873for the Burger-Huxley equation when

120574 = 075 120573 = 1 and119873119905= 10

119905 119873119909

4 6 8 1001 2217119890 minus 006 8482119890 minus 009 2166119890 minus 011 7822119890 minus 014

02 2596119890 minus 006 9369119890 minus 009 2536119890 minus 011 1184119890 minus 013

03 2859119890 minus 006 1073119890 minus 008 3201119890 minus 011 1049119890 minus 013

04 3001119890 minus 006 1112119890 minus 008 3652119890 minus 011 9426119890 minus 014

05 3137119890 minus 006 1213119890 minus 008 4262119890 minus 011 1510119890 minus 013

06 3270119890 minus 006 1311119890 minus 008 4842119890 minus 011 2127119890 minus 013

07 3367119890 minus 006 1359119890 minus 008 5289119890 minus 011 1230119890 minus 013

08 3467119890 minus 006 1438119890 minus 008 5803119890 minus 011 1549119890 minus 013

09 3562119890 minus 006 1504119890 minus 008 6260119890 minus 011 3063119890 minus 013

10 3640119890 minus 006 1559119890 minus 008 6674119890 minus 011 2951119890 minus 013

CPUtime(sec)

0023822 0024901 002685 0032806

Example 4 Consider the Burgers-Huxley equation

120597119906

120597119905+ 120572119906120575

119906119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (119906120575

minus 120574) (43)

where 120572 120573 ge 0 are constant parameters 120575 is a positive integer(set to be 120575 = 1 in this study) and 120574 isin (0 1) The exactsolution subject to the initial condition

119906 (119909 0) =1

2minus

1

2tanh [

120573

119903 minus 120572119909] (44)

is reported in [38 39] as

119906 (119909 119905) =1

2minus

1

2tanh [

120573

119903 minus 120572(119909 minus 119888119905)] (45)

where

119903 = radic1205722+ 8120573 119888 =

(120572 minus 119903) (2120574 minus 1) + 2120572

4

(46)

Table 5 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1001 1803119890 minus 007 3419119890 minus 010 4449119890 minus 013 1572119890 minus 013

02 2614119890 minus 007 4347119890 minus 010 5049119890 minus 013 5992119890 minus 014

03 2717119890 minus 007 4677119890 minus 010 5532119890 minus 013 8128119890 minus 013

04 2009119890 minus 007 3663119890 minus 010 4771119890 minus 013 6158119890 minus 013

05 2580119890 minus 007 4410119890 minus 010 7518119890 minus 013 2555119890 minus 013

06 2653119890 minus 007 4606119890 minus 010 8738119890 minus 013 5756119890 minus 013

07 2248119890 minus 007 4039119890 minus 010 6210119890 minus 013 2393119890 minus 013

08 2572119890 minus 007 4476119890 minus 010 5432119890 minus 013 6812119890 minus 013

09 2436119890 minus 007 4351119890 minus 010 6111119890 minus 013 6287119890 minus 013

10 8275119890 minus 008 3721119890 minus 010 7569119890 minus 013 1087119890 minus 007

CPUtime(sec)

0015646 0021226 0030159 0035675

Table 6 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1001 7788119890 minus 005 3553119890 minus 007 7601119890 minus 010 2080119890 minus 010

02 1153119890 minus 004 4000119890 minus 007 5684119890 minus 010 1189119890 minus 010

03 1011119890 minus 004 3739119890 minus 007 4471119890 minus 010 4503119890 minus 010

04 3926119890 minus 005 1785119890 minus 007 6544119890 minus 010 4987119890 minus 010

05 6727119890 minus 005 2342119890 minus 007 2638119890 minus 010 1528119890 minus 010

06 6065119890 minus 005 2207119890 minus 007 4565119890 minus 010 4568119890 minus 010

07 2511119890 minus 005 1105119890 minus 007 4749119890 minus 010 3748119890 minus 010

08 4074119890 minus 005 1427119890 minus 007 1062119890 minus 010 1604119890 minus 010

09 2386119890 minus 005 1018119890 minus 007 2343119890 minus 010 8114119890 minus 011

10 1440119890 minus 006 7256119890 minus 008 1436119890 minus 009 1513119890 minus 011

CPUtime(sec)

0020609 0021241 0030617 0032816

The general solution (45) was reported in [40 41] In thisexample the linear operator 119865 and nonlinear operator 119866 arechosen as

119865 (119906) = 11990610158401015840

minus 120573120574119906

119866 (119906) = minus1205721199061199061015840

+ 120573 (1 + 120574) 1199062

minus 1205731199063

(47)

Example 5 We consider the modified KdV-Burgers equation

120597119906

120597119905=

1205973

119906

1205971199093minus

1205972

119906

1205971199092minus 61199062120597119906

120597119909

(48)

subject to the initial condition

119906 (119909 0) =1

6+

1

6tanh(

119909

6) (49)

and exact solution [42]

119906 (119909 119905) =1

6+

1

6tanh(

119909

6minus

119905

27) (50)

The Scientific World Journal 7

Table 7 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1002 1119119890 minus 011 7398119890 minus 013 8266119890 minus 013 3808119890 minus 014

04 3121119890 minus 011 1552119890 minus 012 7378119890 minus 013 3780119890 minus 014

06 4864119890 minus 011 1004119890 minus 012 3402119890 minus 012 7283119890 minus 014

08 6802119890 minus 011 7895119890 minus 013 1118119890 minus 012 3714119890 minus 014

10 7971119890 minus 011 1088119890 minus 012 1473119890 minus 012 1691119890 minus 013

12 8560119890 minus 011 8805119890 minus 013 2611119890 minus 012 3119119890 minus 013

14 8953119890 minus 011 6418119890 minus 013 6671119890 minus 012 1796119890 minus 013

16 8759119890 minus 011 6199119890 minus 013 1118119890 minus 011 1097119890 minus 013

18 8325119890 minus 011 3978119890 minus 013 7515119890 minus 013 6273119890 minus 014

20 7421119890 minus 011 7988119890 minus 014 3682119890 minus 012 2311119890 minus 013

CPUtime(sec)

0013542 0022967 0023792 0024758

Table 8Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 1223119890 minus 007 1400119890 minus 008 1402119890 minus 008 1094119890 minus 012

04 1145119890 minus 007 1919119890 minus 008 1918119890 minus 008 3919119890 minus 012

06 9192119890 minus 008 2082119890 minus 008 2085119890 minus 008 1953119890 minus 012

08 2293119890 minus 008 1793119890 minus 008 1793119890 minus 008 6340119890 minus 013

10 2395119890 minus 008 1337119890 minus 008 1339119890 minus 008 2381119890 minus 012

12 5778119890 minus 008 1954119890 minus 008 1930119890 minus 008 1005119890 minus 011

14 6045119890 minus 008 1620119890 minus 008 1620119890 minus 008 3535119890 minus 012

16 5244119890 minus 008 7218119890 minus 009 7345119890 minus 009 5765119890 minus 012

18 4395119890 minus 008 6828119890 minus 009 6784119890 minus 009 3983119890 minus 012

20 2944119890 minus 008 9406119890 minus 010 8820119890 minus 010 3812119890 minus 012

CPUtime(sec)

0019942 0025988 0027756 0029436

The modified KdV-Burgers equation describes various kindsof phenomena such as a mathematical model of turbulence[43] and the approximate theory of flow through a shockwavetraveling in viscous fluid [44] For this example the linearoperator 119865 and nonlinear operator 119866 are chosen as

119865 (119906) = 119906101584010158401015840

minus 11990610158401015840

119866 (119906) = minus61199061015840

1199062

(51)

Example 6 We consider the high nonlinear modified KdVequation

120597119906

120597119905=

1205973

119906

1205971199093+ (

120597119906

120597119909)

2

minus 1199062 (52)

subject to the initial condition

119906 (119909 0) =1

2+

119890minus119909

4

(53)

Table 9 Maximum errors 119864119873for the Fitzhugh-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 6326119890 minus 007 1311119890 minus 009 2886119890 minus 012 1131119890 minus 012

04 6721119890 minus 007 1467119890 minus 009 3310119890 minus 012 1564119890 minus 012

06 7140119890 minus 007 1602119890 minus 009 3617119890 minus 012 1936119890 minus 012

08 6730119890 minus 007 1496119890 minus 009 4707119890 minus 012 1196119890 minus 012

10 6660119890 minus 007 1487119890 minus 009 3675119890 minus 012 1264119890 minus 012

12 6449119890 minus 007 1366119890 minus 009 1897119890 minus 012 1727119890 minus 012

14 5690119890 minus 007 1083119890 minus 009 2972119890 minus 012 1200119890 minus 012

16 4931119890 minus 007 8010119890 minus 010 1519119890 minus 012 8590119890 minus 013

18 3986119890 minus 007 4658119890 minus 010 1068119890 minus 012 6790119890 minus 013

20 2904119890 minus 007 2968119890 minus 010 1592119890 minus 012 1770119890 minus 013

CPUtime(sec)

0041048 0049629 0055008 0053863

Table 10 Maximum errors 119864119873

for the Burgers-Huxley equationwhen 120574 = 05 120573 = 1 and119873

119905= 10

119905 119873119909

4 6 8 1002 2866119890 minus 006 1119119890 minus 008 3670119890 minus 011 1150119890 minus 012

04 3401119890 minus 006 1420119890 minus 008 5744119890 minus 011 1638119890 minus 012

06 3814119890 minus 006 1687119890 minus 008 7426119890 minus 011 1958119890 minus 012

08 3915119890 minus 006 1729119890 minus 008 8171119890 minus 011 7002119890 minus 013

10 3938119890 minus 006 1738119890 minus 008 8157119890 minus 011 1267119890 minus 012

12 3808119890 minus 006 1624119890 minus 008 7687119890 minus 011 1710119890 minus 012

14 3456119890 minus 006 1527119890 minus 008 6965119890 minus 011 5109119890 minus 013

16 3230119890 minus 006 1349119890 minus 008 5535119890 minus 011 8203119890 minus 013

18 2925119890 minus 006 1078119890 minus 008 3598119890 minus 011 8294119890 minus 013

20 2497119890 minus 006 7505119890 minus 009 2265119890 minus 011 9726119890 minus 014

CPUtime(sec)

0023822 0024901 002685 0032806

and exact solution

119906 (119909 119905) =1

119905 + 2+

119890minus(119909+119905)

(119905 + 2)2 (54)

For this example the linear operator 119865 and nonlinear opera-tor 119866 are chosen as

119865 (119906) = 119906101584010158401015840

119866 (119906) = (1199061015840

)2

minus 1199062

(55)

4 Results and Discussion

In this section we present the numerical solutions obtainedusing the BI-SQLM algorithm The number of collocationpoints in the space 119909 variable used to generate the results is119873119909

= 10 in all cases Similarly the number of collocationpoints in the time 119905 variable used is 119873

119905= 10 in all cases It

was found that sufficient accuracy was achieved using thesevalues in all numerical simulations

8 The Scientific World Journal

Table 11 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1002 2137119890 minus 007 3820119890 minus 010 4846119890 minus 013 9998119890 minus 013

04 2480119890 minus 007 4267119890 minus 010 5596119890 minus 013 8775119890 minus 013

06 2691119890 minus 007 4676119890 minus 010 6565119890 minus 013 2054119890 minus 012

08 2214119890 minus 007 3979119890 minus 010 8776119890 minus 013 1168119890 minus 012

10 2538119890 minus 007 4463119890 minus 010 9650119890 minus 013 8410119890 minus 013

12 2650119890 minus 007 4680119890 minus 010 7450119890 minus 013 5113119890 minus 013

14 2383119890 minus 007 4296119890 minus 010 7500119890 minus 013 1110119890 minus 012

16 2568119890 minus 007 4572119890 minus 010 9704119890 minus 013 2837119890 minus 013

18 2520119890 minus 007 4529119890 minus 010 7443119890 minus 013 5353119890 minus 013

20 2370119890 minus 007 4438119890 minus 010 2719119890 minus 013 8849119890 minus 013

CPUtime(sec)

0062066 0081646 0080718 010775

Table 12 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1002 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

04 8010119890 minus 005 3577119890 minus 007 3902119890 minus 008 1979119890 minus 010

06 7235119890 minus 005 2549119890 minus 007 2016119890 minus 008 4899119890 minus 010

08 6284119890 minus 005 1663119890 minus 007 1155119890 minus 007 2679119890 minus 010

10 1642119890 minus 005 1620119890 minus 007 1243119890 minus 007 2474119890 minus 010

12 2753119890 minus 005 1073119890 minus 007 1073119890 minus 007 1679119890 minus 010

14 3738119890 minus 006 8971119890 minus 008 8598119890 minus 008 4788119890 minus 011

16 1223119890 minus 005 2153119890 minus 008 2503119890 minus 008 2941119890 minus 011

18 5836119890 minus 006 2986119890 minus 008 9127119890 minus 009 5177119890 minus 011

20 9310119890 minus 006 6548119890 minus 008 7277119890 minus 008 1453119890 minus 009

CPUtime(sec)

0020609 0021241 0030617 0032816

In Tables 1 2 3 4 5 and 6 we give the maximumerrors between the exact and BI-SQLM results for the Fisherequation Burgers-Fisher equation Fitzhugh-Nagumo equa-tion Burgers-Huxley equation the modified KdV-Burgersequation and the modified KdV equation respectively at119905 isin [01 1] The results were computed in the space domain119909 isin [0 1] To give a sense of the computational efficiency ofthe method the computational time to generate the resultsis also given Tables 1ndash6 clearly show the accuracy of themethod The accuracy is seen to improve with an increasein the number of collocation points 119873

119909 It is remarkable to

note that accurate results with errors of order up to 10minus14

are obtained using very few collocation points in both the 119909

and 119905 variables 119873119905le 10 119873

119909le 10 This is a clear indication

that the BI-SQLM is powerful method that is appropriatein solving nonlinear evolution PDEs We remark also thatthe BI-SQLM is computationally fast as accurate results aregenerated in a fraction of a second in all the examplesconsidered in this work

0 05 1 15 206

062

064

066

068

07

072

Time (s)

The analytical solution versus the approximate solution ofFishers equation

ApproximateAnalytic

u(xt)

Figure 1 Fishers equation analytical solution graph

088

0885

089

0895

09

0905

091

0915

092

0925

093

The analytical solution versus the approximate solution of theBurgers-Fisher equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 2 Burger-Fishers equation analytical solution graph

In Tables 7 8 9 10 11 and 12 we give the maxi-mum errors of the BI-SQLM results for the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and themodified KdV equation respectively at selectedvalues of 119905 = 2 for different collocation points 119873

119905 in the

119905-variable The results in Tables 7ndash12 were computed on thespace domain 119909 isin [0 1] We note that the accuracy does notdetoriate when 119905 gt 1 for this method as is often the case withnumerical schemes such as finite differences

The Scientific World Journal 9

0 05 1 15 2Time (s)

ApproximateAnalytic

026

028

03

032

034

036

038

04

042

044

The analytical solution versus the approximate solution of theFitzhurg-Nagumo equation

u(xt)-a

xis

Figure 3 Fitzhugh-Nagumo equation analytical solution graph

025

035

045

055

06

05

04

03

The analytical solution and the approximate solution of theBurger-Huxley Equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 4 Burgers-Huxley equation analytical solution graph

Figures 1 2 3 4 5 and 6 show a comparison of theanalytical and approximate solutions of the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and the modified KdV equation respectively when 119905 =

2 The approximate solutions are in excellent agreement withthe analytical solutions and this demonstrates the accuracyof the algorithm presented in this study

In Figures 7 8 9 10 11 and 12 we present error analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation the

0 05 1 15 2Time (s)

ApproximateAnalytic

015

0155

016

0165

017

0175

018

0185

The analytical solution and the approximate solution of themodified KdV-Burger

u(xt)

Figure 5 Modified KdV-Burger equation analytical solution graph

0 05 1 15 2Time (s)

ApproximateAnalytic

0253

0254

0255

0256

0257

0258

0259

026

The analytical solution versus the approximate solution ofmodified KdV

u(xt)

Figure 6 Modified KdV equation analytical solution graph

modified KdV-Burgers equation and the modified KdVequation respectively when 119905 = 2

In Figures 13 14 15 16 17 and 18 convergence analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation themodified KdV-Burgers equation and the modified KdVequation respectively The figures present a variation of theerror norm at a fixed value of time (119905 = 1) with iterationsof the BI-SQLM scheme It can be seen that in almost allthe examples considered the iteration scheme takes about3 or 4 iterations to converge fully Beyond the point wherefull convergence is reached error norm levels off and does

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

and the superscript 119879 denotes matrix transpose Substituting(16) into (12) we get

119865 [U119903+1119895

U1015840119903+1119895

U(119899)119903+1119895

] +

119899

sum

119896=0

Φ119896119903U(119896)119903+1119895

minus

119873119905

sum

119896=0

119889119895119896U119903+1119896

= 119877119903[U119903119895U1015840119903119895 U(119899)

119903119895]

(18)

for 119895 = 0 1 2 119873119905 where

U(119899)119903+1119895

= D119899U119903+1119895

Φ119896119903

=

[[[[

[

120601119896119903

(1199090 119905119895)

120601119896119903

(1199091 119905119895)

d120601119896119903

(119909119873119909

119905119895)

]]]]

]

(19)

The initial condition for (3) corresponds to 120591119873119905

= minus1 andhence we express (18) as

119865 [U119903+1119895

U1015840119903+1119895

U(119899)119903+1119895

]

+

119899

sum

119896=0

Φ119896119903U(119896)119903+1119895

minus

119873119905minus1

sum

119896=0

119889119895119896U119903+1119896

= R119895

(20)

where

R119895= 119877119903[U119903119895U1015840119903119895 U(119899)

119903119895] + 119889119895119873119905

U119873119905

119895 = 0 1 2 119873119905minus 1

(21)

Equation (20) can be expressed as the following119873119905(119873119909+1) times

119873119905(119873119909+ 1)matrix system

[[[[

[

11986000

11986001

sdot sdot sdot 1198600119873119905minus1

11986010

11986011

sdot sdot sdot 1198601119873119905minus1

d

119860119873119905minus10

119860119873119905minus11

sdot sdot sdot 119860119873119905minus1119873119905minus1

]]]]

]

[[[[

[

U0

U1

U119873119905minus1

]]]]

]

=

[[[[

[

R0

R1

R119873119905minus1

]]]]

]

(22)

where

119860119894119894

= 119865 [ID D(119899)] +119899

sum

119896=0

Φ119896119903D(119896) minus 119889

119894119894I

119860119894119895

= minus119889119894119895I when 119894 = 119895

(23)

and I is the identity matrix of size (119873119909+1)times (119873

119909+1) Solving

(19) gives 119906(119909119894 119905119895) and hence we use (4) to approximate

119906(119909 119905)

3 Numerical Experiments

We apply the proposed algorithm to well-known nonlinearPDEs of the form (3) with exact solutions In order todetermine the level of accuracy of the BI-SQLM approximatesolution at a particular time level in comparison with theexact solution we report maximum error which is defined by

119864119873

= max119903

1003816100381610038161003816119906 (119909119903 119905) minus (119909

119903 119905)

1003816100381610038161003816 0 le 119903 le 119873 (24)

where (119909119903 119905) is the approximate solution and is the 119906(119909

119903 119905)

exact solution at the time level 119905

Example 1 Weconsider the generalizedBurgers-Fisher equa-tion [35]

120597119906

120597119905+ 120572119906120575120597119906

120597119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (25)

with initial condition

119906 (119909 0) = 1

2+

1

2tanh(

minus120572120575

2(120575 + 1)119909)

1120575

(26)

and exact solution

119906 (119909 119905)

= 1

2+

1

2tanh(

minus120572120575

2 (120575 + 1)

times [119909 minus (120572

120575 + 1+

120573 (120575 + 1)

120572) 119905])

1120575

(27)

where 120572 120573 and 120575 are parameters For illustration purposesthese parameters are chosen to be120572 = 120573 = 120575 = 1 in this paperThe linear operator 119865 and nonlinear operator119866 are chosen as

119865 (119906) = 11990610158401015840

+ 119906 119866 (119906) = minus1199061199061015840

minus 1199062

(28)

We first linearize the nonlinear operator 119866 We approximate119866 using the equation

119866 asymp 119866 [119906119903 1199061015840

119903 11990610158401015840

119903] +

2

sum

119896=0

120601119896119903119906(119896)

119903+1minus

2

sum

119896=0

120601119896119903119906(119896)

119903 (29)

The coefficients are given by

1206010119903

=120597119866

120597119906[119906119903 1199061015840

119903 11990610158401015840

119903] = minus (119906

1015840

119903+ 2119906119903)

1206011119903

=120597119866

1205971199061015840[119906119903 1199061015840

119903 11990610158401015840

119903] = minus119906

119903

1206012119903

=120597119866

12059711990610158401015840[119906119903 1199061015840

119903 11990610158401015840

119903] = 0

119877119903=

2

sum

119896=0

120601119896119903119906(119896)

119903minus 119866 [119906

119903 1199061015840

119903 11990610158401015840

119903] = minus119906

2

119903minus 1199061199031199061015840

119903

(30)

The Scientific World Journal 5

Therefore the linearized equation can be expressed as

11990610158401015840

119903+1+ 12060111199031199061015840

119903+1+ 1206010119903119906119903+1

+ 119906119903+1

minus = 119877119903 (31)

Applying the spectral method both in 119909 and 119905 and initialcondition we get

D2U119903+1119894

+Φ1119903DU119903+1119894

+Φ0119903U119903+1119894

+ U119903+1119894

minus 2

119873119905minus1

sum

119895=0

119889119894119895U119903+1119895

= R119894

(32)

Equation (32) can be expressed as

[[[[

[

11986000

11986001

sdot sdot sdot 1198600119873119905minus1

11986010

11986011

sdot sdot sdot 1198601119873119905minus1

d

119860119873119905minus10

119860119873119905minus11

sdot sdot sdot 119860119873119905minus1119873119905minus1

]]]]

]

[[[[

[

U0

U1

U119873119905minus1

]]]]

]

=

[[[[

[

R0

R1

R119873119905minus1

]]]]

]

(33)

where

119860119894119894

= D2 +Φ(119894)1119903D +Φ

(119894)

0119903+ (1 minus 2119889

119894119894) I

119860119894119895

= minus 2119889119894119895I when 119894 = 119895

R119894= 119877119903+ 2119889119894119873119905

U119903119873119905

(34)

The boundary conditions are implemented in the first andlast row of the matrices 119860

119894119895and the column vectors R

119894for

119894 = 0 1 119873119905minus 1 and 119895 = 0 1 119873

119905minus 1 The procedure

for finding the variable coefficients 120601119894and matrices for the

remaining examples is similar

Example 2 We consider Fisherrsquos equation

120597119906

120597119905=

1205972

119906

1205971199092+ 120572119906 (1 minus 119906) (35)

subject to the initial condition

119906 (119909 0) =1

(1 + 119890radic1205726119909

)2 (36)

and exact solution [36]

119906 (119909 119905) =1

(1 + 119890radic1205726119909minus51205721199056

)2 (37)

where 120572 is a constant The Fisher equation represents areactive-diffusive system and is encountered in chemicalkinetics and population dynamics applications For thisexample the appropriate linear operator 119865 and nonlinearoperator 119866 are chosen as

119865 (119906) = 11990610158401015840

+ 120572119906 119866 (119906) = minus1205721199062

(38)

Table 1 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1001 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

02 3934119890 minus 008 3121119890 minus 011 1552119890 minus 012 1561119890 minus 012

03 5577119890 minus 008 4864119890 minus 011 1004119890 minus 012 1005119890 minus 012

04 6997119890 minus 008 6802119890 minus 011 7895119890 minus 013 8124119890 minus 013

05 8107119890 minus 008 7971119890 minus 011 1088119890 minus 012 1027119890 minus 012

06 8891119890 minus 008 8560119890 minus 011 8805119890 minus 013 7847119890 minus 013

07 9344119890 minus 008 8953119890 minus 011 6418119890 minus 013 6463119890 minus 013

08 9431119890 minus 008 8759119890 minus 011 6199119890 minus 013 6164119890 minus 013

09 9178119890 minus 008 8325119890 minus 011 3978119890 minus 013 3695119890 minus 013

10 8787119890 minus 008 7421119890 minus 011 7988119890 minus 014 5596119890 minus 014

CPUtime(sec)

0019942 0025988 0027756 0029436

Table 2Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 120574 = 120575 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 1142119890 minus 007 1369119890 minus 010 5891119890 minus 012 6143119890 minus 012

02 1178119890 minus 007 1373119890 minus 010 9570119890 minus 012 1013119890 minus 011

03 1186119890 minus 007 1479119890 minus 010 1489119890 minus 011 1512119890 minus 011

04 1069119890 minus 007 9450119890 minus 011 1703119890 minus 011 1702119890 minus 011

05 9030119890 minus 008 7944119890 minus 011 5283119890 minus 012 5736119890 minus 012

06 6963119890 minus 008 6618119890 minus 011 1639119890 minus 011 1626119890 minus 011

07 4638119890 minus 008 1579119890 minus 011 1362119890 minus 011 1364119890 minus 011

08 2457119890 minus 008 4030119890 minus 011 3934119890 minus 012 3852119890 minus 012

09 2028119890 minus 008 6006119890 minus 011 4466119890 minus 012 4727119890 minus 012

10 3147119890 minus 008 7708119890 minus 011 7757119890 minus 013 7261119890 minus 013

CPUTime(sec)

0010152 0015387 0019163 0021564

Example 3 Consider the Fitzhugh-Nagumo equation

120597119906

120597119905=

1205972

119906

1205971199092+ 119906 (119906 minus 120572) (1 minus 119906) (39)

with initial condition

119906 (119909 0) =1

2[1 minus coth(minus

119909

2radic2

)] (40)

This equation has the exact solution [37]

119906 (119909 119905) =1

2[1 minus coth(minus

119909

2radic2

+2120572 minus 1

4119905)] (41)

where 120572 is a parameter In this example the linear operator 119865and nonlinear operator 119866 are chosen as

119865 (119906) = 11990610158401015840

minus 120572119906 119866 (119906) = (1 + 120572) 1199062

minus 1199063

(42)

6 The Scientific World Journal

Table 3 Maximum errors 119864119873for the Fitzhug-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 5719119890 minus 007 1196119890 minus 009 2367119890 minus 012 9881119890 minus 014

02 6193119890 minus 007 1299119890 minus 009 2761119890 minus 012 3952119890 minus 014

03 6662119890 minus 007 1463119890 minus 009 3259119890 minus 012 8216119890 minus 014

04 6779119890 minus 007 1448119890 minus 009 3341119890 minus 012 8094119890 minus 014

05 6920119890 minus 007 1526119890 minus 009 3587119890 minus 012 5063119890 minus 014

06 7019119890 minus 007 1573119890 minus 009 3729119890 minus 012 3775119890 minus 014

07 6933119890 minus 007 1516119890 minus 009 3660119890 minus 012 8915119890 minus 014

08 6828119890 minus 007 81535119890 minus 009 3635119890 minus 012 7594119890 minus 014

09 6765119890 minus 007 1528119890 minus 009 3519119890 minus 012 3242119890 minus 013

10 6687119890 minus 007 1490119890 minus 009 3405119890 minus 012 1688119890 minus 013

CPUtime(sec)

0024281 0024901 0026810 0032389

Table 4Maximum errors119864119873for the Burger-Huxley equation when

120574 = 075 120573 = 1 and119873119905= 10

119905 119873119909

4 6 8 1001 2217119890 minus 006 8482119890 minus 009 2166119890 minus 011 7822119890 minus 014

02 2596119890 minus 006 9369119890 minus 009 2536119890 minus 011 1184119890 minus 013

03 2859119890 minus 006 1073119890 minus 008 3201119890 minus 011 1049119890 minus 013

04 3001119890 minus 006 1112119890 minus 008 3652119890 minus 011 9426119890 minus 014

05 3137119890 minus 006 1213119890 minus 008 4262119890 minus 011 1510119890 minus 013

06 3270119890 minus 006 1311119890 minus 008 4842119890 minus 011 2127119890 minus 013

07 3367119890 minus 006 1359119890 minus 008 5289119890 minus 011 1230119890 minus 013

08 3467119890 minus 006 1438119890 minus 008 5803119890 minus 011 1549119890 minus 013

09 3562119890 minus 006 1504119890 minus 008 6260119890 minus 011 3063119890 minus 013

10 3640119890 minus 006 1559119890 minus 008 6674119890 minus 011 2951119890 minus 013

CPUtime(sec)

0023822 0024901 002685 0032806

Example 4 Consider the Burgers-Huxley equation

120597119906

120597119905+ 120572119906120575

119906119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (119906120575

minus 120574) (43)

where 120572 120573 ge 0 are constant parameters 120575 is a positive integer(set to be 120575 = 1 in this study) and 120574 isin (0 1) The exactsolution subject to the initial condition

119906 (119909 0) =1

2minus

1

2tanh [

120573

119903 minus 120572119909] (44)

is reported in [38 39] as

119906 (119909 119905) =1

2minus

1

2tanh [

120573

119903 minus 120572(119909 minus 119888119905)] (45)

where

119903 = radic1205722+ 8120573 119888 =

(120572 minus 119903) (2120574 minus 1) + 2120572

4

(46)

Table 5 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1001 1803119890 minus 007 3419119890 minus 010 4449119890 minus 013 1572119890 minus 013

02 2614119890 minus 007 4347119890 minus 010 5049119890 minus 013 5992119890 minus 014

03 2717119890 minus 007 4677119890 minus 010 5532119890 minus 013 8128119890 minus 013

04 2009119890 minus 007 3663119890 minus 010 4771119890 minus 013 6158119890 minus 013

05 2580119890 minus 007 4410119890 minus 010 7518119890 minus 013 2555119890 minus 013

06 2653119890 minus 007 4606119890 minus 010 8738119890 minus 013 5756119890 minus 013

07 2248119890 minus 007 4039119890 minus 010 6210119890 minus 013 2393119890 minus 013

08 2572119890 minus 007 4476119890 minus 010 5432119890 minus 013 6812119890 minus 013

09 2436119890 minus 007 4351119890 minus 010 6111119890 minus 013 6287119890 minus 013

10 8275119890 minus 008 3721119890 minus 010 7569119890 minus 013 1087119890 minus 007

CPUtime(sec)

0015646 0021226 0030159 0035675

Table 6 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1001 7788119890 minus 005 3553119890 minus 007 7601119890 minus 010 2080119890 minus 010

02 1153119890 minus 004 4000119890 minus 007 5684119890 minus 010 1189119890 minus 010

03 1011119890 minus 004 3739119890 minus 007 4471119890 minus 010 4503119890 minus 010

04 3926119890 minus 005 1785119890 minus 007 6544119890 minus 010 4987119890 minus 010

05 6727119890 minus 005 2342119890 minus 007 2638119890 minus 010 1528119890 minus 010

06 6065119890 minus 005 2207119890 minus 007 4565119890 minus 010 4568119890 minus 010

07 2511119890 minus 005 1105119890 minus 007 4749119890 minus 010 3748119890 minus 010

08 4074119890 minus 005 1427119890 minus 007 1062119890 minus 010 1604119890 minus 010

09 2386119890 minus 005 1018119890 minus 007 2343119890 minus 010 8114119890 minus 011

10 1440119890 minus 006 7256119890 minus 008 1436119890 minus 009 1513119890 minus 011

CPUtime(sec)

0020609 0021241 0030617 0032816

The general solution (45) was reported in [40 41] In thisexample the linear operator 119865 and nonlinear operator 119866 arechosen as

119865 (119906) = 11990610158401015840

minus 120573120574119906

119866 (119906) = minus1205721199061199061015840

+ 120573 (1 + 120574) 1199062

minus 1205731199063

(47)

Example 5 We consider the modified KdV-Burgers equation

120597119906

120597119905=

1205973

119906

1205971199093minus

1205972

119906

1205971199092minus 61199062120597119906

120597119909

(48)

subject to the initial condition

119906 (119909 0) =1

6+

1

6tanh(

119909

6) (49)

and exact solution [42]

119906 (119909 119905) =1

6+

1

6tanh(

119909

6minus

119905

27) (50)

The Scientific World Journal 7

Table 7 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1002 1119119890 minus 011 7398119890 minus 013 8266119890 minus 013 3808119890 minus 014

04 3121119890 minus 011 1552119890 minus 012 7378119890 minus 013 3780119890 minus 014

06 4864119890 minus 011 1004119890 minus 012 3402119890 minus 012 7283119890 minus 014

08 6802119890 minus 011 7895119890 minus 013 1118119890 minus 012 3714119890 minus 014

10 7971119890 minus 011 1088119890 minus 012 1473119890 minus 012 1691119890 minus 013

12 8560119890 minus 011 8805119890 minus 013 2611119890 minus 012 3119119890 minus 013

14 8953119890 minus 011 6418119890 minus 013 6671119890 minus 012 1796119890 minus 013

16 8759119890 minus 011 6199119890 minus 013 1118119890 minus 011 1097119890 minus 013

18 8325119890 minus 011 3978119890 minus 013 7515119890 minus 013 6273119890 minus 014

20 7421119890 minus 011 7988119890 minus 014 3682119890 minus 012 2311119890 minus 013

CPUtime(sec)

0013542 0022967 0023792 0024758

Table 8Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 1223119890 minus 007 1400119890 minus 008 1402119890 minus 008 1094119890 minus 012

04 1145119890 minus 007 1919119890 minus 008 1918119890 minus 008 3919119890 minus 012

06 9192119890 minus 008 2082119890 minus 008 2085119890 minus 008 1953119890 minus 012

08 2293119890 minus 008 1793119890 minus 008 1793119890 minus 008 6340119890 minus 013

10 2395119890 minus 008 1337119890 minus 008 1339119890 minus 008 2381119890 minus 012

12 5778119890 minus 008 1954119890 minus 008 1930119890 minus 008 1005119890 minus 011

14 6045119890 minus 008 1620119890 minus 008 1620119890 minus 008 3535119890 minus 012

16 5244119890 minus 008 7218119890 minus 009 7345119890 minus 009 5765119890 minus 012

18 4395119890 minus 008 6828119890 minus 009 6784119890 minus 009 3983119890 minus 012

20 2944119890 minus 008 9406119890 minus 010 8820119890 minus 010 3812119890 minus 012

CPUtime(sec)

0019942 0025988 0027756 0029436

The modified KdV-Burgers equation describes various kindsof phenomena such as a mathematical model of turbulence[43] and the approximate theory of flow through a shockwavetraveling in viscous fluid [44] For this example the linearoperator 119865 and nonlinear operator 119866 are chosen as

119865 (119906) = 119906101584010158401015840

minus 11990610158401015840

119866 (119906) = minus61199061015840

1199062

(51)

Example 6 We consider the high nonlinear modified KdVequation

120597119906

120597119905=

1205973

119906

1205971199093+ (

120597119906

120597119909)

2

minus 1199062 (52)

subject to the initial condition

119906 (119909 0) =1

2+

119890minus119909

4

(53)

Table 9 Maximum errors 119864119873for the Fitzhugh-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 6326119890 minus 007 1311119890 minus 009 2886119890 minus 012 1131119890 minus 012

04 6721119890 minus 007 1467119890 minus 009 3310119890 minus 012 1564119890 minus 012

06 7140119890 minus 007 1602119890 minus 009 3617119890 minus 012 1936119890 minus 012

08 6730119890 minus 007 1496119890 minus 009 4707119890 minus 012 1196119890 minus 012

10 6660119890 minus 007 1487119890 minus 009 3675119890 minus 012 1264119890 minus 012

12 6449119890 minus 007 1366119890 minus 009 1897119890 minus 012 1727119890 minus 012

14 5690119890 minus 007 1083119890 minus 009 2972119890 minus 012 1200119890 minus 012

16 4931119890 minus 007 8010119890 minus 010 1519119890 minus 012 8590119890 minus 013

18 3986119890 minus 007 4658119890 minus 010 1068119890 minus 012 6790119890 minus 013

20 2904119890 minus 007 2968119890 minus 010 1592119890 minus 012 1770119890 minus 013

CPUtime(sec)

0041048 0049629 0055008 0053863

Table 10 Maximum errors 119864119873

for the Burgers-Huxley equationwhen 120574 = 05 120573 = 1 and119873

119905= 10

119905 119873119909

4 6 8 1002 2866119890 minus 006 1119119890 minus 008 3670119890 minus 011 1150119890 minus 012

04 3401119890 minus 006 1420119890 minus 008 5744119890 minus 011 1638119890 minus 012

06 3814119890 minus 006 1687119890 minus 008 7426119890 minus 011 1958119890 minus 012

08 3915119890 minus 006 1729119890 minus 008 8171119890 minus 011 7002119890 minus 013

10 3938119890 minus 006 1738119890 minus 008 8157119890 minus 011 1267119890 minus 012

12 3808119890 minus 006 1624119890 minus 008 7687119890 minus 011 1710119890 minus 012

14 3456119890 minus 006 1527119890 minus 008 6965119890 minus 011 5109119890 minus 013

16 3230119890 minus 006 1349119890 minus 008 5535119890 minus 011 8203119890 minus 013

18 2925119890 minus 006 1078119890 minus 008 3598119890 minus 011 8294119890 minus 013

20 2497119890 minus 006 7505119890 minus 009 2265119890 minus 011 9726119890 minus 014

CPUtime(sec)

0023822 0024901 002685 0032806

and exact solution

119906 (119909 119905) =1

119905 + 2+

119890minus(119909+119905)

(119905 + 2)2 (54)

For this example the linear operator 119865 and nonlinear opera-tor 119866 are chosen as

119865 (119906) = 119906101584010158401015840

119866 (119906) = (1199061015840

)2

minus 1199062

(55)

4 Results and Discussion

In this section we present the numerical solutions obtainedusing the BI-SQLM algorithm The number of collocationpoints in the space 119909 variable used to generate the results is119873119909

= 10 in all cases Similarly the number of collocationpoints in the time 119905 variable used is 119873

119905= 10 in all cases It

was found that sufficient accuracy was achieved using thesevalues in all numerical simulations

8 The Scientific World Journal

Table 11 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1002 2137119890 minus 007 3820119890 minus 010 4846119890 minus 013 9998119890 minus 013

04 2480119890 minus 007 4267119890 minus 010 5596119890 minus 013 8775119890 minus 013

06 2691119890 minus 007 4676119890 minus 010 6565119890 minus 013 2054119890 minus 012

08 2214119890 minus 007 3979119890 minus 010 8776119890 minus 013 1168119890 minus 012

10 2538119890 minus 007 4463119890 minus 010 9650119890 minus 013 8410119890 minus 013

12 2650119890 minus 007 4680119890 minus 010 7450119890 minus 013 5113119890 minus 013

14 2383119890 minus 007 4296119890 minus 010 7500119890 minus 013 1110119890 minus 012

16 2568119890 minus 007 4572119890 minus 010 9704119890 minus 013 2837119890 minus 013

18 2520119890 minus 007 4529119890 minus 010 7443119890 minus 013 5353119890 minus 013

20 2370119890 minus 007 4438119890 minus 010 2719119890 minus 013 8849119890 minus 013

CPUtime(sec)

0062066 0081646 0080718 010775

Table 12 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1002 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

04 8010119890 minus 005 3577119890 minus 007 3902119890 minus 008 1979119890 minus 010

06 7235119890 minus 005 2549119890 minus 007 2016119890 minus 008 4899119890 minus 010

08 6284119890 minus 005 1663119890 minus 007 1155119890 minus 007 2679119890 minus 010

10 1642119890 minus 005 1620119890 minus 007 1243119890 minus 007 2474119890 minus 010

12 2753119890 minus 005 1073119890 minus 007 1073119890 minus 007 1679119890 minus 010

14 3738119890 minus 006 8971119890 minus 008 8598119890 minus 008 4788119890 minus 011

16 1223119890 minus 005 2153119890 minus 008 2503119890 minus 008 2941119890 minus 011

18 5836119890 minus 006 2986119890 minus 008 9127119890 minus 009 5177119890 minus 011

20 9310119890 minus 006 6548119890 minus 008 7277119890 minus 008 1453119890 minus 009

CPUtime(sec)

0020609 0021241 0030617 0032816

In Tables 1 2 3 4 5 and 6 we give the maximumerrors between the exact and BI-SQLM results for the Fisherequation Burgers-Fisher equation Fitzhugh-Nagumo equa-tion Burgers-Huxley equation the modified KdV-Burgersequation and the modified KdV equation respectively at119905 isin [01 1] The results were computed in the space domain119909 isin [0 1] To give a sense of the computational efficiency ofthe method the computational time to generate the resultsis also given Tables 1ndash6 clearly show the accuracy of themethod The accuracy is seen to improve with an increasein the number of collocation points 119873

119909 It is remarkable to

note that accurate results with errors of order up to 10minus14

are obtained using very few collocation points in both the 119909

and 119905 variables 119873119905le 10 119873

119909le 10 This is a clear indication

that the BI-SQLM is powerful method that is appropriatein solving nonlinear evolution PDEs We remark also thatthe BI-SQLM is computationally fast as accurate results aregenerated in a fraction of a second in all the examplesconsidered in this work

0 05 1 15 206

062

064

066

068

07

072

Time (s)

The analytical solution versus the approximate solution ofFishers equation

ApproximateAnalytic

u(xt)

Figure 1 Fishers equation analytical solution graph

088

0885

089

0895

09

0905

091

0915

092

0925

093

The analytical solution versus the approximate solution of theBurgers-Fisher equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 2 Burger-Fishers equation analytical solution graph

In Tables 7 8 9 10 11 and 12 we give the maxi-mum errors of the BI-SQLM results for the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and themodified KdV equation respectively at selectedvalues of 119905 = 2 for different collocation points 119873

119905 in the

119905-variable The results in Tables 7ndash12 were computed on thespace domain 119909 isin [0 1] We note that the accuracy does notdetoriate when 119905 gt 1 for this method as is often the case withnumerical schemes such as finite differences

The Scientific World Journal 9

0 05 1 15 2Time (s)

ApproximateAnalytic

026

028

03

032

034

036

038

04

042

044

The analytical solution versus the approximate solution of theFitzhurg-Nagumo equation

u(xt)-a

xis

Figure 3 Fitzhugh-Nagumo equation analytical solution graph

025

035

045

055

06

05

04

03

The analytical solution and the approximate solution of theBurger-Huxley Equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 4 Burgers-Huxley equation analytical solution graph

Figures 1 2 3 4 5 and 6 show a comparison of theanalytical and approximate solutions of the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and the modified KdV equation respectively when 119905 =

2 The approximate solutions are in excellent agreement withthe analytical solutions and this demonstrates the accuracyof the algorithm presented in this study

In Figures 7 8 9 10 11 and 12 we present error analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation the

0 05 1 15 2Time (s)

ApproximateAnalytic

015

0155

016

0165

017

0175

018

0185

The analytical solution and the approximate solution of themodified KdV-Burger

u(xt)

Figure 5 Modified KdV-Burger equation analytical solution graph

0 05 1 15 2Time (s)

ApproximateAnalytic

0253

0254

0255

0256

0257

0258

0259

026

The analytical solution versus the approximate solution ofmodified KdV

u(xt)

Figure 6 Modified KdV equation analytical solution graph

modified KdV-Burgers equation and the modified KdVequation respectively when 119905 = 2

In Figures 13 14 15 16 17 and 18 convergence analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation themodified KdV-Burgers equation and the modified KdVequation respectively The figures present a variation of theerror norm at a fixed value of time (119905 = 1) with iterationsof the BI-SQLM scheme It can be seen that in almost allthe examples considered the iteration scheme takes about3 or 4 iterations to converge fully Beyond the point wherefull convergence is reached error norm levels off and does

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

Therefore the linearized equation can be expressed as

11990610158401015840

119903+1+ 12060111199031199061015840

119903+1+ 1206010119903119906119903+1

+ 119906119903+1

minus = 119877119903 (31)

Applying the spectral method both in 119909 and 119905 and initialcondition we get

D2U119903+1119894

+Φ1119903DU119903+1119894

+Φ0119903U119903+1119894

+ U119903+1119894

minus 2

119873119905minus1

sum

119895=0

119889119894119895U119903+1119895

= R119894

(32)

Equation (32) can be expressed as

[[[[

[

11986000

11986001

sdot sdot sdot 1198600119873119905minus1

11986010

11986011

sdot sdot sdot 1198601119873119905minus1

d

119860119873119905minus10

119860119873119905minus11

sdot sdot sdot 119860119873119905minus1119873119905minus1

]]]]

]

[[[[

[

U0

U1

U119873119905minus1

]]]]

]

=

[[[[

[

R0

R1

R119873119905minus1

]]]]

]

(33)

where

119860119894119894

= D2 +Φ(119894)1119903D +Φ

(119894)

0119903+ (1 minus 2119889

119894119894) I

119860119894119895

= minus 2119889119894119895I when 119894 = 119895

R119894= 119877119903+ 2119889119894119873119905

U119903119873119905

(34)

The boundary conditions are implemented in the first andlast row of the matrices 119860

119894119895and the column vectors R

119894for

119894 = 0 1 119873119905minus 1 and 119895 = 0 1 119873

119905minus 1 The procedure

for finding the variable coefficients 120601119894and matrices for the

remaining examples is similar

Example 2 We consider Fisherrsquos equation

120597119906

120597119905=

1205972

119906

1205971199092+ 120572119906 (1 minus 119906) (35)

subject to the initial condition

119906 (119909 0) =1

(1 + 119890radic1205726119909

)2 (36)

and exact solution [36]

119906 (119909 119905) =1

(1 + 119890radic1205726119909minus51205721199056

)2 (37)

where 120572 is a constant The Fisher equation represents areactive-diffusive system and is encountered in chemicalkinetics and population dynamics applications For thisexample the appropriate linear operator 119865 and nonlinearoperator 119866 are chosen as

119865 (119906) = 11990610158401015840

+ 120572119906 119866 (119906) = minus1205721199062

(38)

Table 1 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1001 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

02 3934119890 minus 008 3121119890 minus 011 1552119890 minus 012 1561119890 minus 012

03 5577119890 minus 008 4864119890 minus 011 1004119890 minus 012 1005119890 minus 012

04 6997119890 minus 008 6802119890 minus 011 7895119890 minus 013 8124119890 minus 013

05 8107119890 minus 008 7971119890 minus 011 1088119890 minus 012 1027119890 minus 012

06 8891119890 minus 008 8560119890 minus 011 8805119890 minus 013 7847119890 minus 013

07 9344119890 minus 008 8953119890 minus 011 6418119890 minus 013 6463119890 minus 013

08 9431119890 minus 008 8759119890 minus 011 6199119890 minus 013 6164119890 minus 013

09 9178119890 minus 008 8325119890 minus 011 3978119890 minus 013 3695119890 minus 013

10 8787119890 minus 008 7421119890 minus 011 7988119890 minus 014 5596119890 minus 014

CPUtime(sec)

0019942 0025988 0027756 0029436

Table 2Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 120574 = 120575 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 1142119890 minus 007 1369119890 minus 010 5891119890 minus 012 6143119890 minus 012

02 1178119890 minus 007 1373119890 minus 010 9570119890 minus 012 1013119890 minus 011

03 1186119890 minus 007 1479119890 minus 010 1489119890 minus 011 1512119890 minus 011

04 1069119890 minus 007 9450119890 minus 011 1703119890 minus 011 1702119890 minus 011

05 9030119890 minus 008 7944119890 minus 011 5283119890 minus 012 5736119890 minus 012

06 6963119890 minus 008 6618119890 minus 011 1639119890 minus 011 1626119890 minus 011

07 4638119890 minus 008 1579119890 minus 011 1362119890 minus 011 1364119890 minus 011

08 2457119890 minus 008 4030119890 minus 011 3934119890 minus 012 3852119890 minus 012

09 2028119890 minus 008 6006119890 minus 011 4466119890 minus 012 4727119890 minus 012

10 3147119890 minus 008 7708119890 minus 011 7757119890 minus 013 7261119890 minus 013

CPUTime(sec)

0010152 0015387 0019163 0021564

Example 3 Consider the Fitzhugh-Nagumo equation

120597119906

120597119905=

1205972

119906

1205971199092+ 119906 (119906 minus 120572) (1 minus 119906) (39)

with initial condition

119906 (119909 0) =1

2[1 minus coth(minus

119909

2radic2

)] (40)

This equation has the exact solution [37]

119906 (119909 119905) =1

2[1 minus coth(minus

119909

2radic2

+2120572 minus 1

4119905)] (41)

where 120572 is a parameter In this example the linear operator 119865and nonlinear operator 119866 are chosen as

119865 (119906) = 11990610158401015840

minus 120572119906 119866 (119906) = (1 + 120572) 1199062

minus 1199063

(42)

6 The Scientific World Journal

Table 3 Maximum errors 119864119873for the Fitzhug-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 5719119890 minus 007 1196119890 minus 009 2367119890 minus 012 9881119890 minus 014

02 6193119890 minus 007 1299119890 minus 009 2761119890 minus 012 3952119890 minus 014

03 6662119890 minus 007 1463119890 minus 009 3259119890 minus 012 8216119890 minus 014

04 6779119890 minus 007 1448119890 minus 009 3341119890 minus 012 8094119890 minus 014

05 6920119890 minus 007 1526119890 minus 009 3587119890 minus 012 5063119890 minus 014

06 7019119890 minus 007 1573119890 minus 009 3729119890 minus 012 3775119890 minus 014

07 6933119890 minus 007 1516119890 minus 009 3660119890 minus 012 8915119890 minus 014

08 6828119890 minus 007 81535119890 minus 009 3635119890 minus 012 7594119890 minus 014

09 6765119890 minus 007 1528119890 minus 009 3519119890 minus 012 3242119890 minus 013

10 6687119890 minus 007 1490119890 minus 009 3405119890 minus 012 1688119890 minus 013

CPUtime(sec)

0024281 0024901 0026810 0032389

Table 4Maximum errors119864119873for the Burger-Huxley equation when

120574 = 075 120573 = 1 and119873119905= 10

119905 119873119909

4 6 8 1001 2217119890 minus 006 8482119890 minus 009 2166119890 minus 011 7822119890 minus 014

02 2596119890 minus 006 9369119890 minus 009 2536119890 minus 011 1184119890 minus 013

03 2859119890 minus 006 1073119890 minus 008 3201119890 minus 011 1049119890 minus 013

04 3001119890 minus 006 1112119890 minus 008 3652119890 minus 011 9426119890 minus 014

05 3137119890 minus 006 1213119890 minus 008 4262119890 minus 011 1510119890 minus 013

06 3270119890 minus 006 1311119890 minus 008 4842119890 minus 011 2127119890 minus 013

07 3367119890 minus 006 1359119890 minus 008 5289119890 minus 011 1230119890 minus 013

08 3467119890 minus 006 1438119890 minus 008 5803119890 minus 011 1549119890 minus 013

09 3562119890 minus 006 1504119890 minus 008 6260119890 minus 011 3063119890 minus 013

10 3640119890 minus 006 1559119890 minus 008 6674119890 minus 011 2951119890 minus 013

CPUtime(sec)

0023822 0024901 002685 0032806

Example 4 Consider the Burgers-Huxley equation

120597119906

120597119905+ 120572119906120575

119906119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (119906120575

minus 120574) (43)

where 120572 120573 ge 0 are constant parameters 120575 is a positive integer(set to be 120575 = 1 in this study) and 120574 isin (0 1) The exactsolution subject to the initial condition

119906 (119909 0) =1

2minus

1

2tanh [

120573

119903 minus 120572119909] (44)

is reported in [38 39] as

119906 (119909 119905) =1

2minus

1

2tanh [

120573

119903 minus 120572(119909 minus 119888119905)] (45)

where

119903 = radic1205722+ 8120573 119888 =

(120572 minus 119903) (2120574 minus 1) + 2120572

4

(46)

Table 5 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1001 1803119890 minus 007 3419119890 minus 010 4449119890 minus 013 1572119890 minus 013

02 2614119890 minus 007 4347119890 minus 010 5049119890 minus 013 5992119890 minus 014

03 2717119890 minus 007 4677119890 minus 010 5532119890 minus 013 8128119890 minus 013

04 2009119890 minus 007 3663119890 minus 010 4771119890 minus 013 6158119890 minus 013

05 2580119890 minus 007 4410119890 minus 010 7518119890 minus 013 2555119890 minus 013

06 2653119890 minus 007 4606119890 minus 010 8738119890 minus 013 5756119890 minus 013

07 2248119890 minus 007 4039119890 minus 010 6210119890 minus 013 2393119890 minus 013

08 2572119890 minus 007 4476119890 minus 010 5432119890 minus 013 6812119890 minus 013

09 2436119890 minus 007 4351119890 minus 010 6111119890 minus 013 6287119890 minus 013

10 8275119890 minus 008 3721119890 minus 010 7569119890 minus 013 1087119890 minus 007

CPUtime(sec)

0015646 0021226 0030159 0035675

Table 6 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1001 7788119890 minus 005 3553119890 minus 007 7601119890 minus 010 2080119890 minus 010

02 1153119890 minus 004 4000119890 minus 007 5684119890 minus 010 1189119890 minus 010

03 1011119890 minus 004 3739119890 minus 007 4471119890 minus 010 4503119890 minus 010

04 3926119890 minus 005 1785119890 minus 007 6544119890 minus 010 4987119890 minus 010

05 6727119890 minus 005 2342119890 minus 007 2638119890 minus 010 1528119890 minus 010

06 6065119890 minus 005 2207119890 minus 007 4565119890 minus 010 4568119890 minus 010

07 2511119890 minus 005 1105119890 minus 007 4749119890 minus 010 3748119890 minus 010

08 4074119890 minus 005 1427119890 minus 007 1062119890 minus 010 1604119890 minus 010

09 2386119890 minus 005 1018119890 minus 007 2343119890 minus 010 8114119890 minus 011

10 1440119890 minus 006 7256119890 minus 008 1436119890 minus 009 1513119890 minus 011

CPUtime(sec)

0020609 0021241 0030617 0032816

The general solution (45) was reported in [40 41] In thisexample the linear operator 119865 and nonlinear operator 119866 arechosen as

119865 (119906) = 11990610158401015840

minus 120573120574119906

119866 (119906) = minus1205721199061199061015840

+ 120573 (1 + 120574) 1199062

minus 1205731199063

(47)

Example 5 We consider the modified KdV-Burgers equation

120597119906

120597119905=

1205973

119906

1205971199093minus

1205972

119906

1205971199092minus 61199062120597119906

120597119909

(48)

subject to the initial condition

119906 (119909 0) =1

6+

1

6tanh(

119909

6) (49)

and exact solution [42]

119906 (119909 119905) =1

6+

1

6tanh(

119909

6minus

119905

27) (50)

The Scientific World Journal 7

Table 7 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1002 1119119890 minus 011 7398119890 minus 013 8266119890 minus 013 3808119890 minus 014

04 3121119890 minus 011 1552119890 minus 012 7378119890 minus 013 3780119890 minus 014

06 4864119890 minus 011 1004119890 minus 012 3402119890 minus 012 7283119890 minus 014

08 6802119890 minus 011 7895119890 minus 013 1118119890 minus 012 3714119890 minus 014

10 7971119890 minus 011 1088119890 minus 012 1473119890 minus 012 1691119890 minus 013

12 8560119890 minus 011 8805119890 minus 013 2611119890 minus 012 3119119890 minus 013

14 8953119890 minus 011 6418119890 minus 013 6671119890 minus 012 1796119890 minus 013

16 8759119890 minus 011 6199119890 minus 013 1118119890 minus 011 1097119890 minus 013

18 8325119890 minus 011 3978119890 minus 013 7515119890 minus 013 6273119890 minus 014

20 7421119890 minus 011 7988119890 minus 014 3682119890 minus 012 2311119890 minus 013

CPUtime(sec)

0013542 0022967 0023792 0024758

Table 8Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 1223119890 minus 007 1400119890 minus 008 1402119890 minus 008 1094119890 minus 012

04 1145119890 minus 007 1919119890 minus 008 1918119890 minus 008 3919119890 minus 012

06 9192119890 minus 008 2082119890 minus 008 2085119890 minus 008 1953119890 minus 012

08 2293119890 minus 008 1793119890 minus 008 1793119890 minus 008 6340119890 minus 013

10 2395119890 minus 008 1337119890 minus 008 1339119890 minus 008 2381119890 minus 012

12 5778119890 minus 008 1954119890 minus 008 1930119890 minus 008 1005119890 minus 011

14 6045119890 minus 008 1620119890 minus 008 1620119890 minus 008 3535119890 minus 012

16 5244119890 minus 008 7218119890 minus 009 7345119890 minus 009 5765119890 minus 012

18 4395119890 minus 008 6828119890 minus 009 6784119890 minus 009 3983119890 minus 012

20 2944119890 minus 008 9406119890 minus 010 8820119890 minus 010 3812119890 minus 012

CPUtime(sec)

0019942 0025988 0027756 0029436

The modified KdV-Burgers equation describes various kindsof phenomena such as a mathematical model of turbulence[43] and the approximate theory of flow through a shockwavetraveling in viscous fluid [44] For this example the linearoperator 119865 and nonlinear operator 119866 are chosen as

119865 (119906) = 119906101584010158401015840

minus 11990610158401015840

119866 (119906) = minus61199061015840

1199062

(51)

Example 6 We consider the high nonlinear modified KdVequation

120597119906

120597119905=

1205973

119906

1205971199093+ (

120597119906

120597119909)

2

minus 1199062 (52)

subject to the initial condition

119906 (119909 0) =1

2+

119890minus119909

4

(53)

Table 9 Maximum errors 119864119873for the Fitzhugh-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 6326119890 minus 007 1311119890 minus 009 2886119890 minus 012 1131119890 minus 012

04 6721119890 minus 007 1467119890 minus 009 3310119890 minus 012 1564119890 minus 012

06 7140119890 minus 007 1602119890 minus 009 3617119890 minus 012 1936119890 minus 012

08 6730119890 minus 007 1496119890 minus 009 4707119890 minus 012 1196119890 minus 012

10 6660119890 minus 007 1487119890 minus 009 3675119890 minus 012 1264119890 minus 012

12 6449119890 minus 007 1366119890 minus 009 1897119890 minus 012 1727119890 minus 012

14 5690119890 minus 007 1083119890 minus 009 2972119890 minus 012 1200119890 minus 012

16 4931119890 minus 007 8010119890 minus 010 1519119890 minus 012 8590119890 minus 013

18 3986119890 minus 007 4658119890 minus 010 1068119890 minus 012 6790119890 minus 013

20 2904119890 minus 007 2968119890 minus 010 1592119890 minus 012 1770119890 minus 013

CPUtime(sec)

0041048 0049629 0055008 0053863

Table 10 Maximum errors 119864119873

for the Burgers-Huxley equationwhen 120574 = 05 120573 = 1 and119873

119905= 10

119905 119873119909

4 6 8 1002 2866119890 minus 006 1119119890 minus 008 3670119890 minus 011 1150119890 minus 012

04 3401119890 minus 006 1420119890 minus 008 5744119890 minus 011 1638119890 minus 012

06 3814119890 minus 006 1687119890 minus 008 7426119890 minus 011 1958119890 minus 012

08 3915119890 minus 006 1729119890 minus 008 8171119890 minus 011 7002119890 minus 013

10 3938119890 minus 006 1738119890 minus 008 8157119890 minus 011 1267119890 minus 012

12 3808119890 minus 006 1624119890 minus 008 7687119890 minus 011 1710119890 minus 012

14 3456119890 minus 006 1527119890 minus 008 6965119890 minus 011 5109119890 minus 013

16 3230119890 minus 006 1349119890 minus 008 5535119890 minus 011 8203119890 minus 013

18 2925119890 minus 006 1078119890 minus 008 3598119890 minus 011 8294119890 minus 013

20 2497119890 minus 006 7505119890 minus 009 2265119890 minus 011 9726119890 minus 014

CPUtime(sec)

0023822 0024901 002685 0032806

and exact solution

119906 (119909 119905) =1

119905 + 2+

119890minus(119909+119905)

(119905 + 2)2 (54)

For this example the linear operator 119865 and nonlinear opera-tor 119866 are chosen as

119865 (119906) = 119906101584010158401015840

119866 (119906) = (1199061015840

)2

minus 1199062

(55)

4 Results and Discussion

In this section we present the numerical solutions obtainedusing the BI-SQLM algorithm The number of collocationpoints in the space 119909 variable used to generate the results is119873119909

= 10 in all cases Similarly the number of collocationpoints in the time 119905 variable used is 119873

119905= 10 in all cases It

was found that sufficient accuracy was achieved using thesevalues in all numerical simulations

8 The Scientific World Journal

Table 11 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1002 2137119890 minus 007 3820119890 minus 010 4846119890 minus 013 9998119890 minus 013

04 2480119890 minus 007 4267119890 minus 010 5596119890 minus 013 8775119890 minus 013

06 2691119890 minus 007 4676119890 minus 010 6565119890 minus 013 2054119890 minus 012

08 2214119890 minus 007 3979119890 minus 010 8776119890 minus 013 1168119890 minus 012

10 2538119890 minus 007 4463119890 minus 010 9650119890 minus 013 8410119890 minus 013

12 2650119890 minus 007 4680119890 minus 010 7450119890 minus 013 5113119890 minus 013

14 2383119890 minus 007 4296119890 minus 010 7500119890 minus 013 1110119890 minus 012

16 2568119890 minus 007 4572119890 minus 010 9704119890 minus 013 2837119890 minus 013

18 2520119890 minus 007 4529119890 minus 010 7443119890 minus 013 5353119890 minus 013

20 2370119890 minus 007 4438119890 minus 010 2719119890 minus 013 8849119890 minus 013

CPUtime(sec)

0062066 0081646 0080718 010775

Table 12 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1002 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

04 8010119890 minus 005 3577119890 minus 007 3902119890 minus 008 1979119890 minus 010

06 7235119890 minus 005 2549119890 minus 007 2016119890 minus 008 4899119890 minus 010

08 6284119890 minus 005 1663119890 minus 007 1155119890 minus 007 2679119890 minus 010

10 1642119890 minus 005 1620119890 minus 007 1243119890 minus 007 2474119890 minus 010

12 2753119890 minus 005 1073119890 minus 007 1073119890 minus 007 1679119890 minus 010

14 3738119890 minus 006 8971119890 minus 008 8598119890 minus 008 4788119890 minus 011

16 1223119890 minus 005 2153119890 minus 008 2503119890 minus 008 2941119890 minus 011

18 5836119890 minus 006 2986119890 minus 008 9127119890 minus 009 5177119890 minus 011

20 9310119890 minus 006 6548119890 minus 008 7277119890 minus 008 1453119890 minus 009

CPUtime(sec)

0020609 0021241 0030617 0032816

In Tables 1 2 3 4 5 and 6 we give the maximumerrors between the exact and BI-SQLM results for the Fisherequation Burgers-Fisher equation Fitzhugh-Nagumo equa-tion Burgers-Huxley equation the modified KdV-Burgersequation and the modified KdV equation respectively at119905 isin [01 1] The results were computed in the space domain119909 isin [0 1] To give a sense of the computational efficiency ofthe method the computational time to generate the resultsis also given Tables 1ndash6 clearly show the accuracy of themethod The accuracy is seen to improve with an increasein the number of collocation points 119873

119909 It is remarkable to

note that accurate results with errors of order up to 10minus14

are obtained using very few collocation points in both the 119909

and 119905 variables 119873119905le 10 119873

119909le 10 This is a clear indication

that the BI-SQLM is powerful method that is appropriatein solving nonlinear evolution PDEs We remark also thatthe BI-SQLM is computationally fast as accurate results aregenerated in a fraction of a second in all the examplesconsidered in this work

0 05 1 15 206

062

064

066

068

07

072

Time (s)

The analytical solution versus the approximate solution ofFishers equation

ApproximateAnalytic

u(xt)

Figure 1 Fishers equation analytical solution graph

088

0885

089

0895

09

0905

091

0915

092

0925

093

The analytical solution versus the approximate solution of theBurgers-Fisher equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 2 Burger-Fishers equation analytical solution graph

In Tables 7 8 9 10 11 and 12 we give the maxi-mum errors of the BI-SQLM results for the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and themodified KdV equation respectively at selectedvalues of 119905 = 2 for different collocation points 119873

119905 in the

119905-variable The results in Tables 7ndash12 were computed on thespace domain 119909 isin [0 1] We note that the accuracy does notdetoriate when 119905 gt 1 for this method as is often the case withnumerical schemes such as finite differences

The Scientific World Journal 9

0 05 1 15 2Time (s)

ApproximateAnalytic

026

028

03

032

034

036

038

04

042

044

The analytical solution versus the approximate solution of theFitzhurg-Nagumo equation

u(xt)-a

xis

Figure 3 Fitzhugh-Nagumo equation analytical solution graph

025

035

045

055

06

05

04

03

The analytical solution and the approximate solution of theBurger-Huxley Equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 4 Burgers-Huxley equation analytical solution graph

Figures 1 2 3 4 5 and 6 show a comparison of theanalytical and approximate solutions of the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and the modified KdV equation respectively when 119905 =

2 The approximate solutions are in excellent agreement withthe analytical solutions and this demonstrates the accuracyof the algorithm presented in this study

In Figures 7 8 9 10 11 and 12 we present error analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation the

0 05 1 15 2Time (s)

ApproximateAnalytic

015

0155

016

0165

017

0175

018

0185

The analytical solution and the approximate solution of themodified KdV-Burger

u(xt)

Figure 5 Modified KdV-Burger equation analytical solution graph

0 05 1 15 2Time (s)

ApproximateAnalytic

0253

0254

0255

0256

0257

0258

0259

026

The analytical solution versus the approximate solution ofmodified KdV

u(xt)

Figure 6 Modified KdV equation analytical solution graph

modified KdV-Burgers equation and the modified KdVequation respectively when 119905 = 2

In Figures 13 14 15 16 17 and 18 convergence analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation themodified KdV-Burgers equation and the modified KdVequation respectively The figures present a variation of theerror norm at a fixed value of time (119905 = 1) with iterationsof the BI-SQLM scheme It can be seen that in almost allthe examples considered the iteration scheme takes about3 or 4 iterations to converge fully Beyond the point wherefull convergence is reached error norm levels off and does

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

Table 3 Maximum errors 119864119873for the Fitzhug-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1001 5719119890 minus 007 1196119890 minus 009 2367119890 minus 012 9881119890 minus 014

02 6193119890 minus 007 1299119890 minus 009 2761119890 minus 012 3952119890 minus 014

03 6662119890 minus 007 1463119890 minus 009 3259119890 minus 012 8216119890 minus 014

04 6779119890 minus 007 1448119890 minus 009 3341119890 minus 012 8094119890 minus 014

05 6920119890 minus 007 1526119890 minus 009 3587119890 minus 012 5063119890 minus 014

06 7019119890 minus 007 1573119890 minus 009 3729119890 minus 012 3775119890 minus 014

07 6933119890 minus 007 1516119890 minus 009 3660119890 minus 012 8915119890 minus 014

08 6828119890 minus 007 81535119890 minus 009 3635119890 minus 012 7594119890 minus 014

09 6765119890 minus 007 1528119890 minus 009 3519119890 minus 012 3242119890 minus 013

10 6687119890 minus 007 1490119890 minus 009 3405119890 minus 012 1688119890 minus 013

CPUtime(sec)

0024281 0024901 0026810 0032389

Table 4Maximum errors119864119873for the Burger-Huxley equation when

120574 = 075 120573 = 1 and119873119905= 10

119905 119873119909

4 6 8 1001 2217119890 minus 006 8482119890 minus 009 2166119890 minus 011 7822119890 minus 014

02 2596119890 minus 006 9369119890 minus 009 2536119890 minus 011 1184119890 minus 013

03 2859119890 minus 006 1073119890 minus 008 3201119890 minus 011 1049119890 minus 013

04 3001119890 minus 006 1112119890 minus 008 3652119890 minus 011 9426119890 minus 014

05 3137119890 minus 006 1213119890 minus 008 4262119890 minus 011 1510119890 minus 013

06 3270119890 minus 006 1311119890 minus 008 4842119890 minus 011 2127119890 minus 013

07 3367119890 minus 006 1359119890 minus 008 5289119890 minus 011 1230119890 minus 013

08 3467119890 minus 006 1438119890 minus 008 5803119890 minus 011 1549119890 minus 013

09 3562119890 minus 006 1504119890 minus 008 6260119890 minus 011 3063119890 minus 013

10 3640119890 minus 006 1559119890 minus 008 6674119890 minus 011 2951119890 minus 013

CPUtime(sec)

0023822 0024901 002685 0032806

Example 4 Consider the Burgers-Huxley equation

120597119906

120597119905+ 120572119906120575

119906119909=

1205972

119906

1205971199092+ 120573119906 (1 minus 119906

120575

) (119906120575

minus 120574) (43)

where 120572 120573 ge 0 are constant parameters 120575 is a positive integer(set to be 120575 = 1 in this study) and 120574 isin (0 1) The exactsolution subject to the initial condition

119906 (119909 0) =1

2minus

1

2tanh [

120573

119903 minus 120572119909] (44)

is reported in [38 39] as

119906 (119909 119905) =1

2minus

1

2tanh [

120573

119903 minus 120572(119909 minus 119888119905)] (45)

where

119903 = radic1205722+ 8120573 119888 =

(120572 minus 119903) (2120574 minus 1) + 2120572

4

(46)

Table 5 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1001 1803119890 minus 007 3419119890 minus 010 4449119890 minus 013 1572119890 minus 013

02 2614119890 minus 007 4347119890 minus 010 5049119890 minus 013 5992119890 minus 014

03 2717119890 minus 007 4677119890 minus 010 5532119890 minus 013 8128119890 minus 013

04 2009119890 minus 007 3663119890 minus 010 4771119890 minus 013 6158119890 minus 013

05 2580119890 minus 007 4410119890 minus 010 7518119890 minus 013 2555119890 minus 013

06 2653119890 minus 007 4606119890 minus 010 8738119890 minus 013 5756119890 minus 013

07 2248119890 minus 007 4039119890 minus 010 6210119890 minus 013 2393119890 minus 013

08 2572119890 minus 007 4476119890 minus 010 5432119890 minus 013 6812119890 minus 013

09 2436119890 minus 007 4351119890 minus 010 6111119890 minus 013 6287119890 minus 013

10 8275119890 minus 008 3721119890 minus 010 7569119890 minus 013 1087119890 minus 007

CPUtime(sec)

0015646 0021226 0030159 0035675

Table 6 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1001 7788119890 minus 005 3553119890 minus 007 7601119890 minus 010 2080119890 minus 010

02 1153119890 minus 004 4000119890 minus 007 5684119890 minus 010 1189119890 minus 010

03 1011119890 minus 004 3739119890 minus 007 4471119890 minus 010 4503119890 minus 010

04 3926119890 minus 005 1785119890 minus 007 6544119890 minus 010 4987119890 minus 010

05 6727119890 minus 005 2342119890 minus 007 2638119890 minus 010 1528119890 minus 010

06 6065119890 minus 005 2207119890 minus 007 4565119890 minus 010 4568119890 minus 010

07 2511119890 minus 005 1105119890 minus 007 4749119890 minus 010 3748119890 minus 010

08 4074119890 minus 005 1427119890 minus 007 1062119890 minus 010 1604119890 minus 010

09 2386119890 minus 005 1018119890 minus 007 2343119890 minus 010 8114119890 minus 011

10 1440119890 minus 006 7256119890 minus 008 1436119890 minus 009 1513119890 minus 011

CPUtime(sec)

0020609 0021241 0030617 0032816

The general solution (45) was reported in [40 41] In thisexample the linear operator 119865 and nonlinear operator 119866 arechosen as

119865 (119906) = 11990610158401015840

minus 120573120574119906

119866 (119906) = minus1205721199061199061015840

+ 120573 (1 + 120574) 1199062

minus 1205731199063

(47)

Example 5 We consider the modified KdV-Burgers equation

120597119906

120597119905=

1205973

119906

1205971199093minus

1205972

119906

1205971199092minus 61199062120597119906

120597119909

(48)

subject to the initial condition

119906 (119909 0) =1

6+

1

6tanh(

119909

6) (49)

and exact solution [42]

119906 (119909 119905) =1

6+

1

6tanh(

119909

6minus

119905

27) (50)

The Scientific World Journal 7

Table 7 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1002 1119119890 minus 011 7398119890 minus 013 8266119890 minus 013 3808119890 minus 014

04 3121119890 minus 011 1552119890 minus 012 7378119890 minus 013 3780119890 minus 014

06 4864119890 minus 011 1004119890 minus 012 3402119890 minus 012 7283119890 minus 014

08 6802119890 minus 011 7895119890 minus 013 1118119890 minus 012 3714119890 minus 014

10 7971119890 minus 011 1088119890 minus 012 1473119890 minus 012 1691119890 minus 013

12 8560119890 minus 011 8805119890 minus 013 2611119890 minus 012 3119119890 minus 013

14 8953119890 minus 011 6418119890 minus 013 6671119890 minus 012 1796119890 minus 013

16 8759119890 minus 011 6199119890 minus 013 1118119890 minus 011 1097119890 minus 013

18 8325119890 minus 011 3978119890 minus 013 7515119890 minus 013 6273119890 minus 014

20 7421119890 minus 011 7988119890 minus 014 3682119890 minus 012 2311119890 minus 013

CPUtime(sec)

0013542 0022967 0023792 0024758

Table 8Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 1223119890 minus 007 1400119890 minus 008 1402119890 minus 008 1094119890 minus 012

04 1145119890 minus 007 1919119890 minus 008 1918119890 minus 008 3919119890 minus 012

06 9192119890 minus 008 2082119890 minus 008 2085119890 minus 008 1953119890 minus 012

08 2293119890 minus 008 1793119890 minus 008 1793119890 minus 008 6340119890 minus 013

10 2395119890 minus 008 1337119890 minus 008 1339119890 minus 008 2381119890 minus 012

12 5778119890 minus 008 1954119890 minus 008 1930119890 minus 008 1005119890 minus 011

14 6045119890 minus 008 1620119890 minus 008 1620119890 minus 008 3535119890 minus 012

16 5244119890 minus 008 7218119890 minus 009 7345119890 minus 009 5765119890 minus 012

18 4395119890 minus 008 6828119890 minus 009 6784119890 minus 009 3983119890 minus 012

20 2944119890 minus 008 9406119890 minus 010 8820119890 minus 010 3812119890 minus 012

CPUtime(sec)

0019942 0025988 0027756 0029436

The modified KdV-Burgers equation describes various kindsof phenomena such as a mathematical model of turbulence[43] and the approximate theory of flow through a shockwavetraveling in viscous fluid [44] For this example the linearoperator 119865 and nonlinear operator 119866 are chosen as

119865 (119906) = 119906101584010158401015840

minus 11990610158401015840

119866 (119906) = minus61199061015840

1199062

(51)

Example 6 We consider the high nonlinear modified KdVequation

120597119906

120597119905=

1205973

119906

1205971199093+ (

120597119906

120597119909)

2

minus 1199062 (52)

subject to the initial condition

119906 (119909 0) =1

2+

119890minus119909

4

(53)

Table 9 Maximum errors 119864119873for the Fitzhugh-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 6326119890 minus 007 1311119890 minus 009 2886119890 minus 012 1131119890 minus 012

04 6721119890 minus 007 1467119890 minus 009 3310119890 minus 012 1564119890 minus 012

06 7140119890 minus 007 1602119890 minus 009 3617119890 minus 012 1936119890 minus 012

08 6730119890 minus 007 1496119890 minus 009 4707119890 minus 012 1196119890 minus 012

10 6660119890 minus 007 1487119890 minus 009 3675119890 minus 012 1264119890 minus 012

12 6449119890 minus 007 1366119890 minus 009 1897119890 minus 012 1727119890 minus 012

14 5690119890 minus 007 1083119890 minus 009 2972119890 minus 012 1200119890 minus 012

16 4931119890 minus 007 8010119890 minus 010 1519119890 minus 012 8590119890 minus 013

18 3986119890 minus 007 4658119890 minus 010 1068119890 minus 012 6790119890 minus 013

20 2904119890 minus 007 2968119890 minus 010 1592119890 minus 012 1770119890 minus 013

CPUtime(sec)

0041048 0049629 0055008 0053863

Table 10 Maximum errors 119864119873

for the Burgers-Huxley equationwhen 120574 = 05 120573 = 1 and119873

119905= 10

119905 119873119909

4 6 8 1002 2866119890 minus 006 1119119890 minus 008 3670119890 minus 011 1150119890 minus 012

04 3401119890 minus 006 1420119890 minus 008 5744119890 minus 011 1638119890 minus 012

06 3814119890 minus 006 1687119890 minus 008 7426119890 minus 011 1958119890 minus 012

08 3915119890 minus 006 1729119890 minus 008 8171119890 minus 011 7002119890 minus 013

10 3938119890 minus 006 1738119890 minus 008 8157119890 minus 011 1267119890 minus 012

12 3808119890 minus 006 1624119890 minus 008 7687119890 minus 011 1710119890 minus 012

14 3456119890 minus 006 1527119890 minus 008 6965119890 minus 011 5109119890 minus 013

16 3230119890 minus 006 1349119890 minus 008 5535119890 minus 011 8203119890 minus 013

18 2925119890 minus 006 1078119890 minus 008 3598119890 minus 011 8294119890 minus 013

20 2497119890 minus 006 7505119890 minus 009 2265119890 minus 011 9726119890 minus 014

CPUtime(sec)

0023822 0024901 002685 0032806

and exact solution

119906 (119909 119905) =1

119905 + 2+

119890minus(119909+119905)

(119905 + 2)2 (54)

For this example the linear operator 119865 and nonlinear opera-tor 119866 are chosen as

119865 (119906) = 119906101584010158401015840

119866 (119906) = (1199061015840

)2

minus 1199062

(55)

4 Results and Discussion

In this section we present the numerical solutions obtainedusing the BI-SQLM algorithm The number of collocationpoints in the space 119909 variable used to generate the results is119873119909

= 10 in all cases Similarly the number of collocationpoints in the time 119905 variable used is 119873

119905= 10 in all cases It

was found that sufficient accuracy was achieved using thesevalues in all numerical simulations

8 The Scientific World Journal

Table 11 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1002 2137119890 minus 007 3820119890 minus 010 4846119890 minus 013 9998119890 minus 013

04 2480119890 minus 007 4267119890 minus 010 5596119890 minus 013 8775119890 minus 013

06 2691119890 minus 007 4676119890 minus 010 6565119890 minus 013 2054119890 minus 012

08 2214119890 minus 007 3979119890 minus 010 8776119890 minus 013 1168119890 minus 012

10 2538119890 minus 007 4463119890 minus 010 9650119890 minus 013 8410119890 minus 013

12 2650119890 minus 007 4680119890 minus 010 7450119890 minus 013 5113119890 minus 013

14 2383119890 minus 007 4296119890 minus 010 7500119890 minus 013 1110119890 minus 012

16 2568119890 minus 007 4572119890 minus 010 9704119890 minus 013 2837119890 minus 013

18 2520119890 minus 007 4529119890 minus 010 7443119890 minus 013 5353119890 minus 013

20 2370119890 minus 007 4438119890 minus 010 2719119890 minus 013 8849119890 minus 013

CPUtime(sec)

0062066 0081646 0080718 010775

Table 12 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1002 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

04 8010119890 minus 005 3577119890 minus 007 3902119890 minus 008 1979119890 minus 010

06 7235119890 minus 005 2549119890 minus 007 2016119890 minus 008 4899119890 minus 010

08 6284119890 minus 005 1663119890 minus 007 1155119890 minus 007 2679119890 minus 010

10 1642119890 minus 005 1620119890 minus 007 1243119890 minus 007 2474119890 minus 010

12 2753119890 minus 005 1073119890 minus 007 1073119890 minus 007 1679119890 minus 010

14 3738119890 minus 006 8971119890 minus 008 8598119890 minus 008 4788119890 minus 011

16 1223119890 minus 005 2153119890 minus 008 2503119890 minus 008 2941119890 minus 011

18 5836119890 minus 006 2986119890 minus 008 9127119890 minus 009 5177119890 minus 011

20 9310119890 minus 006 6548119890 minus 008 7277119890 minus 008 1453119890 minus 009

CPUtime(sec)

0020609 0021241 0030617 0032816

In Tables 1 2 3 4 5 and 6 we give the maximumerrors between the exact and BI-SQLM results for the Fisherequation Burgers-Fisher equation Fitzhugh-Nagumo equa-tion Burgers-Huxley equation the modified KdV-Burgersequation and the modified KdV equation respectively at119905 isin [01 1] The results were computed in the space domain119909 isin [0 1] To give a sense of the computational efficiency ofthe method the computational time to generate the resultsis also given Tables 1ndash6 clearly show the accuracy of themethod The accuracy is seen to improve with an increasein the number of collocation points 119873

119909 It is remarkable to

note that accurate results with errors of order up to 10minus14

are obtained using very few collocation points in both the 119909

and 119905 variables 119873119905le 10 119873

119909le 10 This is a clear indication

that the BI-SQLM is powerful method that is appropriatein solving nonlinear evolution PDEs We remark also thatthe BI-SQLM is computationally fast as accurate results aregenerated in a fraction of a second in all the examplesconsidered in this work

0 05 1 15 206

062

064

066

068

07

072

Time (s)

The analytical solution versus the approximate solution ofFishers equation

ApproximateAnalytic

u(xt)

Figure 1 Fishers equation analytical solution graph

088

0885

089

0895

09

0905

091

0915

092

0925

093

The analytical solution versus the approximate solution of theBurgers-Fisher equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 2 Burger-Fishers equation analytical solution graph

In Tables 7 8 9 10 11 and 12 we give the maxi-mum errors of the BI-SQLM results for the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and themodified KdV equation respectively at selectedvalues of 119905 = 2 for different collocation points 119873

119905 in the

119905-variable The results in Tables 7ndash12 were computed on thespace domain 119909 isin [0 1] We note that the accuracy does notdetoriate when 119905 gt 1 for this method as is often the case withnumerical schemes such as finite differences

The Scientific World Journal 9

0 05 1 15 2Time (s)

ApproximateAnalytic

026

028

03

032

034

036

038

04

042

044

The analytical solution versus the approximate solution of theFitzhurg-Nagumo equation

u(xt)-a

xis

Figure 3 Fitzhugh-Nagumo equation analytical solution graph

025

035

045

055

06

05

04

03

The analytical solution and the approximate solution of theBurger-Huxley Equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 4 Burgers-Huxley equation analytical solution graph

Figures 1 2 3 4 5 and 6 show a comparison of theanalytical and approximate solutions of the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and the modified KdV equation respectively when 119905 =

2 The approximate solutions are in excellent agreement withthe analytical solutions and this demonstrates the accuracyof the algorithm presented in this study

In Figures 7 8 9 10 11 and 12 we present error analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation the

0 05 1 15 2Time (s)

ApproximateAnalytic

015

0155

016

0165

017

0175

018

0185

The analytical solution and the approximate solution of themodified KdV-Burger

u(xt)

Figure 5 Modified KdV-Burger equation analytical solution graph

0 05 1 15 2Time (s)

ApproximateAnalytic

0253

0254

0255

0256

0257

0258

0259

026

The analytical solution versus the approximate solution ofmodified KdV

u(xt)

Figure 6 Modified KdV equation analytical solution graph

modified KdV-Burgers equation and the modified KdVequation respectively when 119905 = 2

In Figures 13 14 15 16 17 and 18 convergence analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation themodified KdV-Burgers equation and the modified KdVequation respectively The figures present a variation of theerror norm at a fixed value of time (119905 = 1) with iterationsof the BI-SQLM scheme It can be seen that in almost allthe examples considered the iteration scheme takes about3 or 4 iterations to converge fully Beyond the point wherefull convergence is reached error norm levels off and does

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

Table 7 Maximum errors 119864119873for Fisher equation when 120572 = 1 using

119873119905= 10

119905 119873119909

4 6 8 1002 1119119890 minus 011 7398119890 minus 013 8266119890 minus 013 3808119890 minus 014

04 3121119890 minus 011 1552119890 minus 012 7378119890 minus 013 3780119890 minus 014

06 4864119890 minus 011 1004119890 minus 012 3402119890 minus 012 7283119890 minus 014

08 6802119890 minus 011 7895119890 minus 013 1118119890 minus 012 3714119890 minus 014

10 7971119890 minus 011 1088119890 minus 012 1473119890 minus 012 1691119890 minus 013

12 8560119890 minus 011 8805119890 minus 013 2611119890 minus 012 3119119890 minus 013

14 8953119890 minus 011 6418119890 minus 013 6671119890 minus 012 1796119890 minus 013

16 8759119890 minus 011 6199119890 minus 013 1118119890 minus 011 1097119890 minus 013

18 8325119890 minus 011 3978119890 minus 013 7515119890 minus 013 6273119890 minus 014

20 7421119890 minus 011 7988119890 minus 014 3682119890 minus 012 2311119890 minus 013

CPUtime(sec)

0013542 0022967 0023792 0024758

Table 8Maximum errors119864119873for the Burgers-Fisher equation when

120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 1223119890 minus 007 1400119890 minus 008 1402119890 minus 008 1094119890 minus 012

04 1145119890 minus 007 1919119890 minus 008 1918119890 minus 008 3919119890 minus 012

06 9192119890 minus 008 2082119890 minus 008 2085119890 minus 008 1953119890 minus 012

08 2293119890 minus 008 1793119890 minus 008 1793119890 minus 008 6340119890 minus 013

10 2395119890 minus 008 1337119890 minus 008 1339119890 minus 008 2381119890 minus 012

12 5778119890 minus 008 1954119890 minus 008 1930119890 minus 008 1005119890 minus 011

14 6045119890 minus 008 1620119890 minus 008 1620119890 minus 008 3535119890 minus 012

16 5244119890 minus 008 7218119890 minus 009 7345119890 minus 009 5765119890 minus 012

18 4395119890 minus 008 6828119890 minus 009 6784119890 minus 009 3983119890 minus 012

20 2944119890 minus 008 9406119890 minus 010 8820119890 minus 010 3812119890 minus 012

CPUtime(sec)

0019942 0025988 0027756 0029436

The modified KdV-Burgers equation describes various kindsof phenomena such as a mathematical model of turbulence[43] and the approximate theory of flow through a shockwavetraveling in viscous fluid [44] For this example the linearoperator 119865 and nonlinear operator 119866 are chosen as

119865 (119906) = 119906101584010158401015840

minus 11990610158401015840

119866 (119906) = minus61199061015840

1199062

(51)

Example 6 We consider the high nonlinear modified KdVequation

120597119906

120597119905=

1205973

119906

1205971199093+ (

120597119906

120597119909)

2

minus 1199062 (52)

subject to the initial condition

119906 (119909 0) =1

2+

119890minus119909

4

(53)

Table 9 Maximum errors 119864119873for the Fitzhugh-Nagumo equation

when 120572 = 1 using119873119905= 10

119905 119873119909

4 6 8 1002 6326119890 minus 007 1311119890 minus 009 2886119890 minus 012 1131119890 minus 012

04 6721119890 minus 007 1467119890 minus 009 3310119890 minus 012 1564119890 minus 012

06 7140119890 minus 007 1602119890 minus 009 3617119890 minus 012 1936119890 minus 012

08 6730119890 minus 007 1496119890 minus 009 4707119890 minus 012 1196119890 minus 012

10 6660119890 minus 007 1487119890 minus 009 3675119890 minus 012 1264119890 minus 012

12 6449119890 minus 007 1366119890 minus 009 1897119890 minus 012 1727119890 minus 012

14 5690119890 minus 007 1083119890 minus 009 2972119890 minus 012 1200119890 minus 012

16 4931119890 minus 007 8010119890 minus 010 1519119890 minus 012 8590119890 minus 013

18 3986119890 minus 007 4658119890 minus 010 1068119890 minus 012 6790119890 minus 013

20 2904119890 minus 007 2968119890 minus 010 1592119890 minus 012 1770119890 minus 013

CPUtime(sec)

0041048 0049629 0055008 0053863

Table 10 Maximum errors 119864119873

for the Burgers-Huxley equationwhen 120574 = 05 120573 = 1 and119873

119905= 10

119905 119873119909

4 6 8 1002 2866119890 minus 006 1119119890 minus 008 3670119890 minus 011 1150119890 minus 012

04 3401119890 minus 006 1420119890 minus 008 5744119890 minus 011 1638119890 minus 012

06 3814119890 minus 006 1687119890 minus 008 7426119890 minus 011 1958119890 minus 012

08 3915119890 minus 006 1729119890 minus 008 8171119890 minus 011 7002119890 minus 013

10 3938119890 minus 006 1738119890 minus 008 8157119890 minus 011 1267119890 minus 012

12 3808119890 minus 006 1624119890 minus 008 7687119890 minus 011 1710119890 minus 012

14 3456119890 minus 006 1527119890 minus 008 6965119890 minus 011 5109119890 minus 013

16 3230119890 minus 006 1349119890 minus 008 5535119890 minus 011 8203119890 minus 013

18 2925119890 minus 006 1078119890 minus 008 3598119890 minus 011 8294119890 minus 013

20 2497119890 minus 006 7505119890 minus 009 2265119890 minus 011 9726119890 minus 014

CPUtime(sec)

0023822 0024901 002685 0032806

and exact solution

119906 (119909 119905) =1

119905 + 2+

119890minus(119909+119905)

(119905 + 2)2 (54)

For this example the linear operator 119865 and nonlinear opera-tor 119866 are chosen as

119865 (119906) = 119906101584010158401015840

119866 (119906) = (1199061015840

)2

minus 1199062

(55)

4 Results and Discussion

In this section we present the numerical solutions obtainedusing the BI-SQLM algorithm The number of collocationpoints in the space 119909 variable used to generate the results is119873119909

= 10 in all cases Similarly the number of collocationpoints in the time 119905 variable used is 119873

119905= 10 in all cases It

was found that sufficient accuracy was achieved using thesevalues in all numerical simulations

8 The Scientific World Journal

Table 11 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1002 2137119890 minus 007 3820119890 minus 010 4846119890 minus 013 9998119890 minus 013

04 2480119890 minus 007 4267119890 minus 010 5596119890 minus 013 8775119890 minus 013

06 2691119890 minus 007 4676119890 minus 010 6565119890 minus 013 2054119890 minus 012

08 2214119890 minus 007 3979119890 minus 010 8776119890 minus 013 1168119890 minus 012

10 2538119890 minus 007 4463119890 minus 010 9650119890 minus 013 8410119890 minus 013

12 2650119890 minus 007 4680119890 minus 010 7450119890 minus 013 5113119890 minus 013

14 2383119890 minus 007 4296119890 minus 010 7500119890 minus 013 1110119890 minus 012

16 2568119890 minus 007 4572119890 minus 010 9704119890 minus 013 2837119890 minus 013

18 2520119890 minus 007 4529119890 minus 010 7443119890 minus 013 5353119890 minus 013

20 2370119890 minus 007 4438119890 minus 010 2719119890 minus 013 8849119890 minus 013

CPUtime(sec)

0062066 0081646 0080718 010775

Table 12 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1002 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

04 8010119890 minus 005 3577119890 minus 007 3902119890 minus 008 1979119890 minus 010

06 7235119890 minus 005 2549119890 minus 007 2016119890 minus 008 4899119890 minus 010

08 6284119890 minus 005 1663119890 minus 007 1155119890 minus 007 2679119890 minus 010

10 1642119890 minus 005 1620119890 minus 007 1243119890 minus 007 2474119890 minus 010

12 2753119890 minus 005 1073119890 minus 007 1073119890 minus 007 1679119890 minus 010

14 3738119890 minus 006 8971119890 minus 008 8598119890 minus 008 4788119890 minus 011

16 1223119890 minus 005 2153119890 minus 008 2503119890 minus 008 2941119890 minus 011

18 5836119890 minus 006 2986119890 minus 008 9127119890 minus 009 5177119890 minus 011

20 9310119890 minus 006 6548119890 minus 008 7277119890 minus 008 1453119890 minus 009

CPUtime(sec)

0020609 0021241 0030617 0032816

In Tables 1 2 3 4 5 and 6 we give the maximumerrors between the exact and BI-SQLM results for the Fisherequation Burgers-Fisher equation Fitzhugh-Nagumo equa-tion Burgers-Huxley equation the modified KdV-Burgersequation and the modified KdV equation respectively at119905 isin [01 1] The results were computed in the space domain119909 isin [0 1] To give a sense of the computational efficiency ofthe method the computational time to generate the resultsis also given Tables 1ndash6 clearly show the accuracy of themethod The accuracy is seen to improve with an increasein the number of collocation points 119873

119909 It is remarkable to

note that accurate results with errors of order up to 10minus14

are obtained using very few collocation points in both the 119909

and 119905 variables 119873119905le 10 119873

119909le 10 This is a clear indication

that the BI-SQLM is powerful method that is appropriatein solving nonlinear evolution PDEs We remark also thatthe BI-SQLM is computationally fast as accurate results aregenerated in a fraction of a second in all the examplesconsidered in this work

0 05 1 15 206

062

064

066

068

07

072

Time (s)

The analytical solution versus the approximate solution ofFishers equation

ApproximateAnalytic

u(xt)

Figure 1 Fishers equation analytical solution graph

088

0885

089

0895

09

0905

091

0915

092

0925

093

The analytical solution versus the approximate solution of theBurgers-Fisher equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 2 Burger-Fishers equation analytical solution graph

In Tables 7 8 9 10 11 and 12 we give the maxi-mum errors of the BI-SQLM results for the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and themodified KdV equation respectively at selectedvalues of 119905 = 2 for different collocation points 119873

119905 in the

119905-variable The results in Tables 7ndash12 were computed on thespace domain 119909 isin [0 1] We note that the accuracy does notdetoriate when 119905 gt 1 for this method as is often the case withnumerical schemes such as finite differences

The Scientific World Journal 9

0 05 1 15 2Time (s)

ApproximateAnalytic

026

028

03

032

034

036

038

04

042

044

The analytical solution versus the approximate solution of theFitzhurg-Nagumo equation

u(xt)-a

xis

Figure 3 Fitzhugh-Nagumo equation analytical solution graph

025

035

045

055

06

05

04

03

The analytical solution and the approximate solution of theBurger-Huxley Equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 4 Burgers-Huxley equation analytical solution graph

Figures 1 2 3 4 5 and 6 show a comparison of theanalytical and approximate solutions of the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and the modified KdV equation respectively when 119905 =

2 The approximate solutions are in excellent agreement withthe analytical solutions and this demonstrates the accuracyof the algorithm presented in this study

In Figures 7 8 9 10 11 and 12 we present error analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation the

0 05 1 15 2Time (s)

ApproximateAnalytic

015

0155

016

0165

017

0175

018

0185

The analytical solution and the approximate solution of themodified KdV-Burger

u(xt)

Figure 5 Modified KdV-Burger equation analytical solution graph

0 05 1 15 2Time (s)

ApproximateAnalytic

0253

0254

0255

0256

0257

0258

0259

026

The analytical solution versus the approximate solution ofmodified KdV

u(xt)

Figure 6 Modified KdV equation analytical solution graph

modified KdV-Burgers equation and the modified KdVequation respectively when 119905 = 2

In Figures 13 14 15 16 17 and 18 convergence analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation themodified KdV-Burgers equation and the modified KdVequation respectively The figures present a variation of theerror norm at a fixed value of time (119905 = 1) with iterationsof the BI-SQLM scheme It can be seen that in almost allthe examples considered the iteration scheme takes about3 or 4 iterations to converge fully Beyond the point wherefull convergence is reached error norm levels off and does

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

Table 11 Maximum errors 119864119873

for the modified KdV-Burgersequation with119873

119905= 10

119905 119873119909

4 6 8 1002 2137119890 minus 007 3820119890 minus 010 4846119890 minus 013 9998119890 minus 013

04 2480119890 minus 007 4267119890 minus 010 5596119890 minus 013 8775119890 minus 013

06 2691119890 minus 007 4676119890 minus 010 6565119890 minus 013 2054119890 minus 012

08 2214119890 minus 007 3979119890 minus 010 8776119890 minus 013 1168119890 minus 012

10 2538119890 minus 007 4463119890 minus 010 9650119890 minus 013 8410119890 minus 013

12 2650119890 minus 007 4680119890 minus 010 7450119890 minus 013 5113119890 minus 013

14 2383119890 minus 007 4296119890 minus 010 7500119890 minus 013 1110119890 minus 012

16 2568119890 minus 007 4572119890 minus 010 9704119890 minus 013 2837119890 minus 013

18 2520119890 minus 007 4529119890 minus 010 7443119890 minus 013 5353119890 minus 013

20 2370119890 minus 007 4438119890 minus 010 2719119890 minus 013 8849119890 minus 013

CPUtime(sec)

0062066 0081646 0080718 010775

Table 12 Maximum errors 119864119873for the highly nonlinear modified

KdV equation with119873119905= 10

119905 119873119909

4 6 8 1002 1986119890 minus 008 1119119890 minus 011 7398119890 minus 013 7171119890 minus 013

04 8010119890 minus 005 3577119890 minus 007 3902119890 minus 008 1979119890 minus 010

06 7235119890 minus 005 2549119890 minus 007 2016119890 minus 008 4899119890 minus 010

08 6284119890 minus 005 1663119890 minus 007 1155119890 minus 007 2679119890 minus 010

10 1642119890 minus 005 1620119890 minus 007 1243119890 minus 007 2474119890 minus 010

12 2753119890 minus 005 1073119890 minus 007 1073119890 minus 007 1679119890 minus 010

14 3738119890 minus 006 8971119890 minus 008 8598119890 minus 008 4788119890 minus 011

16 1223119890 minus 005 2153119890 minus 008 2503119890 minus 008 2941119890 minus 011

18 5836119890 minus 006 2986119890 minus 008 9127119890 minus 009 5177119890 minus 011

20 9310119890 minus 006 6548119890 minus 008 7277119890 minus 008 1453119890 minus 009

CPUtime(sec)

0020609 0021241 0030617 0032816

In Tables 1 2 3 4 5 and 6 we give the maximumerrors between the exact and BI-SQLM results for the Fisherequation Burgers-Fisher equation Fitzhugh-Nagumo equa-tion Burgers-Huxley equation the modified KdV-Burgersequation and the modified KdV equation respectively at119905 isin [01 1] The results were computed in the space domain119909 isin [0 1] To give a sense of the computational efficiency ofthe method the computational time to generate the resultsis also given Tables 1ndash6 clearly show the accuracy of themethod The accuracy is seen to improve with an increasein the number of collocation points 119873

119909 It is remarkable to

note that accurate results with errors of order up to 10minus14

are obtained using very few collocation points in both the 119909

and 119905 variables 119873119905le 10 119873

119909le 10 This is a clear indication

that the BI-SQLM is powerful method that is appropriatein solving nonlinear evolution PDEs We remark also thatthe BI-SQLM is computationally fast as accurate results aregenerated in a fraction of a second in all the examplesconsidered in this work

0 05 1 15 206

062

064

066

068

07

072

Time (s)

The analytical solution versus the approximate solution ofFishers equation

ApproximateAnalytic

u(xt)

Figure 1 Fishers equation analytical solution graph

088

0885

089

0895

09

0905

091

0915

092

0925

093

The analytical solution versus the approximate solution of theBurgers-Fisher equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 2 Burger-Fishers equation analytical solution graph

In Tables 7 8 9 10 11 and 12 we give the maxi-mum errors of the BI-SQLM results for the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and themodified KdV equation respectively at selectedvalues of 119905 = 2 for different collocation points 119873

119905 in the

119905-variable The results in Tables 7ndash12 were computed on thespace domain 119909 isin [0 1] We note that the accuracy does notdetoriate when 119905 gt 1 for this method as is often the case withnumerical schemes such as finite differences

The Scientific World Journal 9

0 05 1 15 2Time (s)

ApproximateAnalytic

026

028

03

032

034

036

038

04

042

044

The analytical solution versus the approximate solution of theFitzhurg-Nagumo equation

u(xt)-a

xis

Figure 3 Fitzhugh-Nagumo equation analytical solution graph

025

035

045

055

06

05

04

03

The analytical solution and the approximate solution of theBurger-Huxley Equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 4 Burgers-Huxley equation analytical solution graph

Figures 1 2 3 4 5 and 6 show a comparison of theanalytical and approximate solutions of the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and the modified KdV equation respectively when 119905 =

2 The approximate solutions are in excellent agreement withthe analytical solutions and this demonstrates the accuracyof the algorithm presented in this study

In Figures 7 8 9 10 11 and 12 we present error analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation the

0 05 1 15 2Time (s)

ApproximateAnalytic

015

0155

016

0165

017

0175

018

0185

The analytical solution and the approximate solution of themodified KdV-Burger

u(xt)

Figure 5 Modified KdV-Burger equation analytical solution graph

0 05 1 15 2Time (s)

ApproximateAnalytic

0253

0254

0255

0256

0257

0258

0259

026

The analytical solution versus the approximate solution ofmodified KdV

u(xt)

Figure 6 Modified KdV equation analytical solution graph

modified KdV-Burgers equation and the modified KdVequation respectively when 119905 = 2

In Figures 13 14 15 16 17 and 18 convergence analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation themodified KdV-Burgers equation and the modified KdVequation respectively The figures present a variation of theerror norm at a fixed value of time (119905 = 1) with iterationsof the BI-SQLM scheme It can be seen that in almost allthe examples considered the iteration scheme takes about3 or 4 iterations to converge fully Beyond the point wherefull convergence is reached error norm levels off and does

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 9

0 05 1 15 2Time (s)

ApproximateAnalytic

026

028

03

032

034

036

038

04

042

044

The analytical solution versus the approximate solution of theFitzhurg-Nagumo equation

u(xt)-a

xis

Figure 3 Fitzhugh-Nagumo equation analytical solution graph

025

035

045

055

06

05

04

03

The analytical solution and the approximate solution of theBurger-Huxley Equation

0 05 1 15 2Time (s)

ApproximateAnalytic

u(xt)

Figure 4 Burgers-Huxley equation analytical solution graph

Figures 1 2 3 4 5 and 6 show a comparison of theanalytical and approximate solutions of the Fisher equa-tion Burgers-Fisher equation Fitzhugh-Nagumo equationBurgers-Huxley equation the modified KdV-Burgers equa-tion and the modified KdV equation respectively when 119905 =

2 The approximate solutions are in excellent agreement withthe analytical solutions and this demonstrates the accuracyof the algorithm presented in this study

In Figures 7 8 9 10 11 and 12 we present error analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation the

0 05 1 15 2Time (s)

ApproximateAnalytic

015

0155

016

0165

017

0175

018

0185

The analytical solution and the approximate solution of themodified KdV-Burger

u(xt)

Figure 5 Modified KdV-Burger equation analytical solution graph

0 05 1 15 2Time (s)

ApproximateAnalytic

0253

0254

0255

0256

0257

0258

0259

026

The analytical solution versus the approximate solution ofmodified KdV

u(xt)

Figure 6 Modified KdV equation analytical solution graph

modified KdV-Burgers equation and the modified KdVequation respectively when 119905 = 2

In Figures 13 14 15 16 17 and 18 convergence analysisgraphs for the Fisher equation Burgers-Fisher equationFitzhugh-Nagumo equation Burgers-Huxley equation themodified KdV-Burgers equation and the modified KdVequation respectively The figures present a variation of theerror norm at a fixed value of time (119905 = 1) with iterationsof the BI-SQLM scheme It can be seen that in almost allthe examples considered the iteration scheme takes about3 or 4 iterations to converge fully Beyond the point wherefull convergence is reached error norm levels off and does

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 The Scientific World Journal

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

The error graph of the Fishers equation10

minus10

10minus11

10minus12

10minus13

Figure 7 Fishers equation error graph

The error graph of the Burgers-Fisher equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus10

10minus11

10minus12

10minus13

Figure 8 Burger-Fishers equation error graph

The error graph of the Fitzhurg-Nagumo equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus14

10minus12

10minus13

Figure 9 Fitzhugh-Nagumo equation error graph

The error graph of the Burger-Huxley equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

10minus14

Figure 10 Burgers-Huxley equation error graph

The error analysis graph of the modified KdV-Burger equation

0402 06 08 1 12 14 16 18 2Time (s)

Erro

r nor

m

10minus11

10minus12

10minus13

Figure 11 Modified KdV-Burger equation error graph

The error graph of modified KdV

Erro

r nor

m

10minus8

10minus9

10minus10

10minus11

0402 06 08 1 12 14 16 18 2Time (s)

Figure 12 Modified KdV equation error graph

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 11

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 13 Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus2

10minus4

10minus6

10minus8

10minus10

10minus12

10minus14

Eu

Figure 14 Burger-Fishers equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 15 Fitzhugh-Nagumo equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

Eu

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

10minus10

10minus11

Figure 16 Burgers-Huxley equation convergence graph

1 2 3 4 5 6 7 89 10Iterations

10minus8

10minus7

10minus9

10minus10

10minus11

10minus12

10minus13

Eu

Figure 17 Modified KdV-Burger equation convergence graph

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Eu

1 2 3 4 5 6 7 89 10Iterations

Figure 18 Modified KdV equation convergence graph

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 The Scientific World Journal

not improve with an increase in the number of iterationsThis plateau level gives an estimate of the maximum errorthat can be achieved when using the proposed method witha certain number of collocation points It is worth remarkingthat the accuracy of the method depends on the number ofcollocation points in both the 119909 and 119905 directions The resultsfrom Figures 13ndash18 clearly demonstrate that the BI-SQLM isaccurate

5 Conclusion

This paper has presented a new Chebyshev collocationspectral method for solving general nonlinear evolutionpartial differential equationsThe bivariate interpolated spec-tral quasilinearisation method (BI-SQLM) was developedby combining elements of the quasilinearisation methodand Chebyshev spectral collocation with bivariate LagrangeinterpolationThemain goal of the current studywas to assessthe accuracy robustness and effectiveness of the method insolving nonlinear partial differential equations

Numerical simulations were conducted on the modifiedKdV-Burger equation highly nonlinear modified KdV equa-tion the Fisher equation Burgers-Fisher equation Fitzhugh-Nagumo equation andBurgers-Huxley equation It is evidentfrom the study that the BI-SQLM gives accurate results ina computationally efficient manner Further evidence fromthis study is that the BI-SQLM gives solutions that areuniformly accurate and valid in large intervals of space andtime domains The apparent success of the method can beattributed to the use of the Chebyshev spectral collocationmethod with bivariate Lagrange interpolation in space andtime for differentiating This work contributes to the existingbody of literature on quasilinearisation tools for solvingcomplex nonlinear partial differential equations Furtherwork needs to be done to establish whether the BI-SQLM canbe equally successful in solving coupled systems of equations

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work was supported in part by the National ResearchFoundation of South Africa (Grant no 85596)

References

[1] G Adomian Stochastic Systems vol 169 of Mathematics inScience and Engineering Academic Press Orlando Fla USA1983

[2] G Adomian ldquoA review of the decomposition method inapplied mathematicsrdquo Journal of Mathematical Analysis andApplications vol 135 no 2 pp 501ndash544 1988

[3] L Bougoffa and R C Rach ldquoSolving nonlocal initial-boundaryvalue problems for linear and nonlinear parabolic and hyper-bolic partial differential equations by the Adomian decomposi-tion methodrdquo Applied Mathematics and Computation vol 225pp 50ndash61 2013

[4] S J Liao Advances in Homotopy Analysis Method WorldScientific Publishing Singapore 2014

[5] J He ldquoApplication of homotopy perturbation method to non-linear wave equationsrdquo Chaos Solitons and Fractals vol 26 no3 pp 695ndash700 2005

[6] S Abbasbandy ldquoThe application of homotopy analysis methodto solve a generalized Hirota-Satsuma coupled KdV equationrdquoPhysics Letters A General Atomic and Solid State Physics vol361 no 6 pp 478ndash483 2007

[7] L Song and H Zhang ldquoApplication of homotopy analy-sis method to fractional KdV-Burgers-KURamoto equationrdquoPhysics Letters A vol 367 no 1-2 pp 88ndash94 2007

[8] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[9] B R Duffy and E J Parkes ldquoTravelling solitary wave solutionsto a seventh-order generalized KdV equationrdquo Physics Letters Avol 214 no 5-6 pp 271ndash272 1996

[10] Z B Li ldquoExact solitary wave solutions of nonlinear evolutionequationsrdquo in Mathematics Mechanization and Application XS Gao and DMWang Eds Academic Press San Diego CalifUSA 2000

[11] U Lepik ldquoNumerical solution of evolution equations by theHaar wavelet methodrdquo Applied Mathematics and Computationvol 185 no 1 pp 695ndash704 2007

[12] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[13] G Hariharan K Kannan and K R Sharma ldquoHaar waveletmethod for solving Fishers equationrdquoAppliedMathematics andComputation vol 211 no 2 pp 284ndash292 2009

[14] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[15] C Chun ldquoSolitons and periodic solutions for the fifth-orderKdV equation with the Exp-function methodrdquo Physics LettersA vol 372 no 16 pp 2760ndash2766 2008

[16] X H Wu and J H He ldquoEXP-function method and its applica-tion to nonlinear equationsrdquo Chaos Solitons amp Fractals vol 38no 3 pp 903ndash910 2008

[17] FWWubs and E D deGoede ldquoAn explicit-implicitmethod fora class of time-dependent partial differential equationsrdquoAppliedNumerical Mathematics vol 9 no 2 pp 157ndash181 1992

[18] EM E Elbarbary andM El-Kady ldquoChebyshev finite differenceapproximation for the boundary value problemsrdquo AppliedMathematics and Computation vol 139 no 2-3 pp 513ndash5232003

[19] A C Vliegenthart ldquoOn finite-difference methods for theKorteweg-de Vries equationrdquo Journal of EngineeringMathemat-ics vol 5 pp 137ndash155 1971

[20] J Argyris and M Haase ldquoAn engineerrsquos guide to soliton phe-nomena application of the finite element methodrdquo ComputerMethods in AppliedMechanics and Engineering vol 61 no 1 pp71ndash122 1987

[21] G F Carey and Y Shen ldquoApproximations of the KdV equationby least squares finite elementsrdquo Computer Methods in AppliedMechanics and Engineering vol 93 no 1 pp 1ndash11 1991

[22] K Djidjeli W G Price P Temarel and E H Twizell ldquoAlinearized implicit pseudo-spectral method for certain non-linear water wave equationsrdquo Communications in Numerical

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 13

Methods in Engineering with Biomedical Applications vol 14 no10 pp 977ndash993 1998

[23] A H Khater R S Temsah and M M Hassan ldquoA Chebyshevspectral collocation method for solving Burgers-type equa-tionsrdquo Journal of Computational and Applied Mathematics vol222 no 2 pp 333ndash350 2008

[24] D Olmos and B D Shizgal ldquoA pseudospectral method ofsolution of Fishers equationrdquo Journal of Computational andApplied Mathematics vol 193 no 1 pp 219ndash242 2006

[25] M Javidi ldquoSpectral collocation method for the solution of thegeneralized Burger-Fisher equationrdquo Applied Mathematics andComputation vol 174 no 1 pp 345ndash352 2006

[26] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[27] M Dehghan and F Fakhar-Izadi ldquoPseudospectral methods forNagumoequationrdquo International Journal forNumericalMethodsin Biomedical Engineering vol 27 no 4 pp 553ndash561 2011

[28] T A Driscoll ldquoA composite Runge-Kutta method for thespectral solution of semilinear PDEsrdquo Journal of ComputationalPhysics vol 182 no 2 pp 357ndash367 2002

[29] M T Darvishi S Kheybari and F Khani ldquoA numerical solutionof Korteweg-de Vries equation by pseudospectral methodusing Darvishirsquos preconditioningsrdquo Applied Mathematics andComputation vol 182 no 1 pp 98ndash105 2006

[30] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[31] B A Jacobs and C Harley ldquoTwo hybrid methods for solv-ing two-dimensional linear time-fractional partial differentialequationsrdquo Abstract and Applied Analysis vol 2014 Article ID757204 10 pages 2014

[32] E Tohidi and A Kilicman ldquoAn efficient spectral approximationfor solving several types of parabolic pdes with nonlocalboundary conditionsrdquo Mathematical Problems in Engineeringvol 2014 Article ID 369029 6 pages 2014

[33] R E Bellman and R E Kalaba Quasilinearization and Non-linear Boundary-Value Problems vol 3 ofModern Analytic andComputional Methods in Science and Mathematics AmericanElsevier New York NY USA 1965

[34] L N Trefethen Spectral Methods in MATLAB SIAM Philadel-phia Pa USA 2000

[35] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[36] A Wazwaz and A Gorguis ldquoAn analytic study of Fisherrsquosequation by using Adomian decomposition methodrdquo AppliedMathematics and Computation vol 154 no 3 pp 609ndash6202004

[37] H Li and Y Guo ldquoNew exact solutions to the FitzHugh-Nagumo equationrdquoAppliedMathematics and Computation vol180 no 2 pp 524ndash528 2006

[38] E Fan ldquoTraveling wave solutions for nonlinear equationsusing symbolic computationrdquo Computers amp Mathematics withApplications vol 43 no 6-7 pp 671ndash680 2002

[39] Y N Kyrychko M V Bartuccelli and K B Blyuss ldquoPersistenceof travelling wave solutions of a fourth order diffusion systemrdquoJournal of Computational and AppliedMathematics vol 176 no2 pp 433ndash443 2005

[40] I Hashim M S M Noorani and M R Said Al-HadidildquoSolving the generalized Burgers-Huxley equation using theAdomian decompositionmethodrdquoMathematical andComputerModelling vol 43 no 11-12 pp 1404ndash1411 2006

[41] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[42] M A Helal and M S Mehanna ldquoA comparison between twodifferent methods for solving KdV-Burgers equationrdquo ChaosSolitons and Fractals vol 28 no 2 pp 320ndash326 2006

[43] J M Burgers ldquoA mathematical model illustrating the theory ofturbulencerdquo in Advances in Applied Mechanics vol 1 pp 171ndash199 1948

[44] J D Cole ldquoOn a quasi-linear parabolic equation occurring inaerodynamicsrdquo Quarterly of Applied Mathematics vol 9 pp225ndash236 1951

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of