Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 715756 9 pageshttpdxdoiorg1011552013715756

Research ArticleNew Spectral Second Kind Chebyshev Wavelets Algorithmfor Solving Linear and Nonlinear Second-Order DifferentialEquations Involving Singular and Bratu Type Equations

W M Abd-Elhameed12 E H Doha2 and Y H Youssri2

1 Department of Mathematics Faculty of Science King Abdulaziz University Jeddah Saudi Arabia2Department of Mathematics Faculty of Science Cairo University Giza 12613 Egypt

Correspondence should be addressed to Y H Youssri youssriscicuedueg

Received 27 April 2013 Accepted 31 October 2013

Academic Editor Turgut Ozis

Copyright copy 2013 W M Abd-Elhameed et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A new spectral algorithm based on shifted second kind Chebyshev wavelets operational matrices of derivatives is introduced andused for solving linear and nonlinear second-order two-point boundary value problems The main idea for obtaining spectralnumerical solutions for these equations is essentially developed by reducing the linear or nonlinear equations with their initialandor boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficientsConvergence analysis and some efficient specific illustrative examples including singular and Bratu type equations are considered todemonstrate the validity and the applicability of themethod Numerical results obtained are compared favorably with the analyticalknown solutions

1 Introduction

Spectral methods are one of the principal methods of dis-cretization for the numerical solution of differential equa-tions The main advantage of these methods lies in theiraccuracy for a given number of unknowns (see eg [1ndash3]) For smooth problems in simple geometries they offerexponential rates of convergencespectral accuracy In con-trast finite difference and finite-element methods yield onlyalgebraic convergence ratesThe threemost widely used spec-tral versions are the Galerkin collocation and tau methodsCollocation methods have become increasingly popular forsolving differential equations also they are very useful inproviding highly accurate solutions to nonlinear differentialequations (see eg [4 5])

The subject of wavelets has recently drawn a great deal ofattention frommathematical scientists in various disciplinesIt is creating a common link between mathematicians physi-cists and electrical engineers Wavelets theory is a relativelynew and an emerging area in mathematical research It hasbeen applied to a wide range of engineering disciplines

particularly wavelets are very successfully used in signal anal-ysis for wave form representation and segmentations timefrequency analysis and fast algorithms for easy implementa-tion Wavelets permit the accurate representation of a varietyof functions and operators Moreover wavelets establish aconnection with fast numerical algorithms (see [6 7])

The subject of nonlinear differential equations is a well-established part of mathematics and its systematic devel-opment goes back to the early days of the development ofcalculusMany recent advances inmathematics paralleled bya renewed and flourishing interaction between mathematicssciences and engineering have again shown that manyphenomena in applied sciences modelled by differentialequations will yield some mathematical explanation of thesephenomena [8ndash10]

Even-order differential equations have been extensivelydiscussed by a large number of authors due to their greatimportance in various applications in many fields For exam-ple in the sequence of papers [1 2 11] the Galerkinmethod isapplied for handling these problems They constructed suit-able basis functions which satisfy the boundary conditions of

2 Abstract and Applied Analysis

the given differential equation For this purpose they usedcompact combinations of various orthogonal polynomialsThe suggested algorithms in these papers are suitable forhandling one- and two-dimensional linear high even-orderboundary value problems

In this paper we aim to give some algorithms for handlingboth linear and nonlinear second-order boundary valueproblems based on introducing a new matrix of derivativesthen applying Petrov-Galerkin method on linear equationsand collocation method on nonlinear equations High accu-rate spectral wavelets solutions are achieved with a smallnumber of retained modes compared with the usual spectralmethods also we can handle singular differential equationswith discontinuous variable coefficients

Among the second-order boundary value problems is theone-dimensional Bratu problem which has a long historyBratursquos own paper appeared in 1914 [12] generalizations aresometimes called the Liouville-Gelfand or Liouville-Gelfand-Bratu problem in honor of Gelrsquofand [13] and the nineteenthcentury work of the great Frenchmathematician Liouville Inrecent years it has been a popular testbed for numerical andperturbation methods [14ndash16]

Simplification of the solid fuel ignition model in thermalcombustion theory yields an elliptic nonlinear partial dif-ferential equation namely the Bratu problem Also due toits use in a large variety of applications many authors havecontributed to the study of such problem Some applicationsof Bratu problem are the model of thermal reaction processthe Chandrasekhar model of the expansion of the Universechemical reaction theory nanotechnology and radiative heattransfer (see [17ndash21])

The application of Legendre wavelets for solving differen-tial and integral equations is thoroughly considered by manyauthors (see [22ndash26]) Also Chebyshev wavelets are used forsolving some differential fractional and integral equations(see [5 27ndash31])

In [32 33] a cosine and sine (CAS) wavelets operationalmatrix of fractional order integration has been derived andused to solve integrodifferential equations of fractional order

One approach for solving differential equations is basedon converting the differential equations into integral equa-tions through integration approximating various signalsinvolved in the equation by truncated orthogonal series andusing the operational matrix of integration to eliminate theintegral operations

Special attentions have been given to applications of blockpulse functions [34] Legendre polynomials [35] Chebyshevpolynomials [36] Haar wavelets [37] Legendre wavelets [2425 38] and Chebyshev wavelets [27] Another approach isbased on using operational matrix of derivatives in order toreduce the problem into solving a system of algebraic equa-tions (see [23])

The main aim of this paper is to develop a new spec-tral algorithm for solving second-order two-point boundaryvalue problems based on shifted second kind Chebyshevwavelets operational matrix of derivatives The methodreduces the differential equation with its initial andorboundary conditions to a system of algebraic equations in theunknown expansion coefficients Large systems of algebraic

equations may lead to greater computational complexityand large storage requirements However the second kindChebyshev wavelets are structurally sparse this reduces dras-tically the computational complexity of solving the resultingalgebraic system

The structure of the paper is as follows In Section 2we give some relevant properties of second kind Chebyshevpolynomials and their shifted forms In Section 3 we developa new shifted second kind Chebyshev wavelets operationalmatrices of derivatives (SCWOMD) also we ascertain theconvergence analysis of the proposed algorithm As an appli-cation of SCWOMD numerical solutions of second-orderlinear and nonlinear initial value problems or two-pointboundary value problems are implemented and presentedin Section 4 In Section 5 some numerical examples arepresented to show the efficiency and the applicability of thepresented algorithm Some concluding remarks are given inSection 6

2 Some Properties of Second Kind ChebyshevPolynomials and Their Shifted Forms

In the present section we discuss some relevant propertiesof the second kind Chebyshev polynomials and their shiftedforms

21 Second Kind Chebyshev Polynomials It is well knownthat the second kind Chebyshev polynomials are defined on[minus1 1] by

119880119899(119909) =

sin (119899 + 1) 120579sin 120579

119909 = cos 120579 (1)

These polynomials are orthogonal on [minus1 1] that is

int1

minus1

radic1 minus 1199092119880119898(119909)119880119899(119909) 119889119909 =

0 119898 = 119899120587

2 119898 = 119899

(2)

The following properties of second kind Chebyshev poly-nomials (see eg [39]) are of fundamental importance inthe sequel They are eigenfunctions of the following singularSturm-Liouville equation

(1 minus 1199092

)1198632

120601119896(119909) minus 3119909119863120601

119896(119909) + 119896 (119896 + 2) 120601

119896(119909) = 0 (3)

where 119863 equiv 119889119889119909 and may be generated by using the recur-rence relation

119880119896+1(119909) = 2119909119880

119896(119909) minus 119880

119896minus1(119909) 119896 = 1 2 3 (4)

starting from 1198800(119909) = 1 and 119880

1(119909) = 2119909 or from Rodrigues

formula

119880119899(119909) =

(minus2)119899

(119899 + 1)

(2119899 + 1)radic1 minus 1199092119863119899

[(1 minus 1199092

)119899+(12)

] (5)

The following theorem is needed hereafter

Abstract and Applied Analysis 3

Theorem 1 Thefirst derivative of second kind Chebyshev poly-nomials is given by

119863119880119899(119909) = 2

119899minus1

sum119896=0

(119896+119899) odd

(119896 + 1)119880119896(119909) (6)

For a proof of Theorem 1 see [39]

22 Shifted Second Kind Chebyshev Polynomials The shiftedsecond kind Chebyshev polynomials are defined on [0 1] by119880lowast

119899(119909) = 119880

119899(2119909 minus 1) All results of second kind Chebyshev

polynomials can be easily transformed to give the corre-sponding results for their shifted forms The orthogonalityrelation with respect to the weight function radic119909 minus 1199092 is givenby

int1

0

radic119909 minus 1199092119880lowast

119899(119909)119880lowast

119898(119909) 119889119909 =

0 119898 = 119899120587

8 119898 = 119899

(7)

The first derivative of 119880lowast119899(119909) is given in the following corol-

lary

Corollary 2 The first derivative of the shifted second kindChebyshev polynomial is given by

119863119880lowast

119899(119909) = 4

119899minus1

sum119896=0

(119896+119899)odd

(119896 + 1)119880lowast

119896(119909) (8)

3 Shifted Second Kind Chebyshev OperationalMatrix of Derivatives

Wavelets constitute of a family of functions constructed fromdilation and translation of single function called the motherwavelet When the dilation parameter 119886 and the translationparameter 119887 vary continuously then we have the followingfamily of continuous wavelets

120595119886119887(119905) = |119886|

minus12

120595(119905 minus 119887

119886) 119886 119887 isin R 119886 = 0 (9)

Second kind Chebyshev wavelets 120595119899119898(119905) = 120595(119896 119899119898 119905) have

four arguments 119896 119899 can assume any positive integer 119898 isthe order of second kind Chebyshev polynomials and 119905 is thenormalized time They are defined on the interval [0 1] by

120595119899119898(119905) =

2(119896+3)2

radic120587119880lowast

119898(2119896

119905 minus 119899) 119905 isin [119899

2119896119899 + 1

2119896]

0 otherwise

119898 = 0 1 119872 119899 = 0 1 2119896

minus 1

(10)

31 Function Approximation A function 119891(119905) defined over[0 1] may be expanded in terms of second kind Chebyshevwavelets as

119891 (119905) =

infin

sum119899=0

infin

sum119898=0

119888119899119898120595119899119898(119905) (11)

where

119888119899119898= (119891 (119905) 120595

119899119898(119905))120596= int1

0

120596 (119905) 119891 (119905) 120595119899119898(119905) 119889119905 (12)

and 119908(119905) = radic119905 minus 1199052 If the infinite series is truncated then119891(119905) can be approximated as

119891 (119905) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119888119899119898120595119899119898(119905) = 119862

119879

Ψ (119905) (13)

where 119862 and Ψ(119905) are 2119896(119872 + 1) times 1matrices defined by

119862 = [11988800 11988801 119888

0119872 119888

2119896minus1119872

1198882119896minus11 119888

2119896minus1119872

]119879

(14)

Ψ (119905) = [12059500 12059501 120595

0119872

1205952119896minus1119872

1205952119896minus11 120595

2119896minus1119872

]119879

(15)

32 Convergence Analysis We state and prove a theoremascertaining that the second kind Chebyshev wavelet expan-sion of a function 119891(119909) with bounded second derivativeconverges uniformly to 119891(119909)

Theorem 3 A function 119891(119909) isin 1198712120596[0 1] with |11989110158401015840(119909)| ⩽ 119871

can be expanded as an infinite sum of Chebyshev wavelets andthe series converges uniformly to119891(119909) Explicitly the expansioncoefficients in (12) satisfy the following inequality

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

8radic2120587 119871

(119899 + 1)52

(119898 + 1)2

forall119898 gt 1 119899 ⩾ 0 (16)

Proof From (12) it follows that

119888119899119898=2(119896+3)2

radic120587int(119899+1)2

119896

1198992119896

119891 (119909)119880lowast

119898(2119896

119909 minus 119899)120596 (2119896

119909 minus 119899) 119889119909

(17)

If we set 2119896119909 minus 119899 = cos 120579 in (17) then we get

119888119899119898=2(minus119896+3)2

radic120587

times int120587

0

119891(cos 120579 + 1198992119896

) sin (119898 + 1) 120579 sin 120579 119889120579

=2(minus119896+1)2

radic120587

times int120587

0

119891(cos 120579 + 1198992119896

) [cos119898120579 minus cos (119898 + 2) 120579] 119889120579

(18)

which gives after integration by parts two times

119888119899119898=

2

251198962radic2120587int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579 (19)

4 Abstract and Applied Analysis

where

120574119898(120579) =

sin 120579119898(sin (119898 minus 1) 120579119898 minus 1

minussin (119898 + 1) 120579119898 + 1

)

minussin 120579119898 + 2

(sin (119898 + 1) 120579119898 + 1

minussin (119898 + 3) 120579119898 + 3

)

(20)

Therefore we have

10038161003816100381610038161198881198991198981003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816

1

2(5119896minus1)2radic120587int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

=1

2(5119896minus1)2radic120587

10038161003816100381610038161003816100381610038161003816int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

⩽119871

2(5119896minus1)2radic120587int120587

0

1003816100381610038161003816120574119898 (120579)1003816100381610038161003816 119889120579

⩽119871radic120587

2(5119896minus1)2

times [1

119898(1

119898 minus 1+

1

119898 + 1)

+1

119898 + 2(1

119898 + 1+

1

119898 + 3)]

=119871radic120587

2(5119896minus5)2 (1198982 + 2119898 minus 3)

lt2119871radic120587

2(5119896minus5)2(119898 + 1)2

(21)

Since 119899 ⩽ 2119896 minus 1 we have

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

8radic2120587119871

(119899 + 1)52

(119898 + 1)2

(22)

and this completes the proof of Theorem 3

A shifted second kind Chebyshev wavelets operationalmatrix of the first derivative is stated andproved in the follow-ing theorem

Theorem 4 Let Ψ(119905) be the second kind Chebyshev waveletsvector defined in (15) Then the first derivative of the vectorΨ(119905) can be expressed as

119889Ψ (119905)

119889119905= DΨ (119905) (23)

where D is 2119896(119872 + 1) square operational matrix of derivativeand is defined by

D =(

(

119865 0 0

0 119865 0

0 0 119865

)

)

(24)

in which 119865 is an (119872 + 1) square matrix whose (119903 119904)th elementis defined by

119865119903119904= 2119896+2

119904 119903 ⩾ 2 119903 gt 119904 (119903 + 119904) odd0 otherwise

(25)

Proof If we make use of the shifted second kind Chebyshevpolynomials then the 119903th element of the vector Ψ(119905) definedin (15) can be written in the following form

Ψ119903(119905) = 120595

119899119898(119905) =

2(119896+3)2

radic120587119880lowast

119898(2119896

119905 minus 119899) 120594[1198992119896(119899+1)2

119896]

119894 = 1 2 2119896

(119872 + 1)

(26)

where 119903 = 119899(119872 + 1) + (119898 + 1) 119898 = 0 1 119872 119899 =0 1 (2

119896

minus 1) and

120594[1198992119896(119899+1)2

119896]=

1 119905 isin [119899

2119896119899 + 1

2119896]

0 otherwise(27)

If we differentiate (26) with respect to 119905 then we get

119889Ψ119903(119905)

119889119905=2(2119896+3)2

radic120587

119889

119889119905[119880lowast

119898(2119896

119905 minus 119899)] 120594[1198992119896(119899+1)2

119896] (28)

It is to be noted here that the RHS of (28) is zero outside theinterval [1198992119896 (119899 + 1)2119896] hence its second kind Chebyshevwavelets expansion only has those elements in Ψ(119905) that arenonzero in the interval [1198992119896 (119899 + 1)2119896] that is Ψ

119894(119905) 119894 =

119899(119872 + 1) + 1 119899(119872 + 1) + 2 119899(119872 + 1) + 119872 + 1 and as aconsequence this enables one to expand (119889Ψ

119903(119905)119889119905) in terms

of shifted second kind Chebyshev wavelets Ψ119894(119905) 119894 = 119899(119872 +

1) + 1 119899(119872 + 1) + 2 119899(119872 + 1) +119872 + 1 in the form

119889Ψ119903(119905)

119889119905=

(119899+1)(119872+1)

sum119894=119899(119872+1)+1

119886119894Ψ119894(119905) (29)

and accordingly (28) implies that the operational matrixD isa block matrix defined as in (24) Moreover we have

119889119880lowast

0(119905)

119889119905= 0 (30)

and this implies that119889

119889119905Ψ119903(119905) = 0

for 119903 = 1 (119872 + 1) + 1 2 (119872 + 1) + 1

(2119896

minus 1) (119872 + 1) + 1

(31)

Consequently the elements of the first row of matrix 119865defined in (24) are zeros Now andwith the aid of Corollary 2the first derivative of Ψ

119903(119905)may be expressed in the form

119889Ψ119903(119905)

119889119905

=2(2119896+7)2

radic120587

119898minus1

sum119895=0

(119895+119898) odd

(119895 + 1)119880lowast

119895(2119896

119905 minus 119899) 120594[1198992119896(119899+1)2

119896](32)

Abstract and Applied Analysis 5

Expanding this equation in terms of second kind Chebyshevwavelets basis we get

119889Ψ119903(119905)

119889119905= 2119896+2

119898minus1

sum119904=0

(119904+119898) odd

119904Ψ119899(119872+1)+119904

(119905) (33)

so if 119865119903119904

are chosen as

119865119903119904=

2119896+2

119904 119903 = 2 119872 + 1 119904 = 1 119903 minus 1

(119903 + 119904) odd0 otherwise

(34)

then (23) is obtained and the proof of the theorem iscompleted

Corollary 5 The operational matrix for the 119899th derivative canbe obtained from

119889119899

Ψ (119905)

119889119905119899= D119899Ψ (119905) 119899 = 1 2 (35)

whereD119899 is the 119899th power of matrix D

4 Second-Order Two-Point BoundaryValue Problems

In this section we are interested in solving linear and nonlin-ear two-point boundary value problems subject to homoge-nous or nonhomogenous initial or boundary conditionsbased on the wavelets operational matrices constructed inSection 3

41 Linear Second-Order Two-Point Boundary Value Prob-lems Consider the linear second-order differential equation

11991010158401015840

(119909) + 1198911(119909) 1199101015840

(119909) + 1198912(119909) 119910 (119909) = 119892 (119909) 119909 isin [0 1]

(36)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (37)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (38)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (39)

If we approximate 119910(119909)1198911(119909)1198912(119909) and 119892(119909) in terms of the

second kind Chebyshev wavelets basis then one can write

119910 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119888119899119898120595119899119898(119909) = 119862

119879

Ψ (119909)

1198911(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

1Ψ (119909)

1198912(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

2Ψ (119905)

119892 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119892119899119898120595119899119898(119909) = 119866

119879

Ψ (119909)

(40)

where 119862119879 1198651198791 1198651198792 and 119866119879 are defined similarly as in (14)

Relations (23) and (35) enable one to approximate 1199101015840(119909) and11991010158401015840

(119909) as

1199101015840

(119909) ≃ 119862119879DΨ (119909) 119910

10158401015840

(119909) ≃ 119862119879D2Ψ (119909) (41)

Now substitution of relations (40) and (41) into (36) enablesus to define the residual 119877(119909) of this equation as

119877 (119909) = 119862119879D2Ψ (119909) + 119865119879

1Ψ (119909) (Ψ (119909))

119879D119879119862

+ 119865119879

2Ψ (119909) (Ψ (119909))

119879

119862 minus 119866119879

Ψ (119909) (42)

and application of the Petrov-Galerkin method (see [40])yields the following (2119896(119872 + 1) minus 2) linear equations in theunknown expansion coefficients 119888

119899119898 namely

int1

0

radic119909 minus 1199092Ψ119895(119909) 119877 (119909) 119889119909 = 0

119895 = 1 2 2119896

(119872 + 1) minus 2

(43)

Moreover the initial conditions (37) the boundary condi-tions (38) and the mixed boundary conditions (39) leadrespectively to the following equations

119862119879

Ψ (0) = 120572 119862119879DΨ (0) = 120573 (44)

119862119879

Ψ (0) = 120572 119862119879

Ψ (1) = 120573 (45)

1198861119862119879

Ψ (0) + 1198862119862119879DΨ (0) = 120572

1198871119862119879

Ψ (1) + 1198872119862119879DΨ (1) = 120573

(46)

Thus (43) with the two equations of (44) or (45) or (46)generates 2119896(119872 + 1) a set of linear equations which can besolved for the unknown components of the vector 119862 andhence an approximate spectral wavelets solution to119910(119909) givenin (40) can be obtained

42 Nonlinear Second-Order Two-Point Boundary Value Prob-lems Consider the nonlinear differential equation

11991010158401015840

(119909) = 119865 (119909 119910 (119909) 1199101015840

(119909)) (47)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (48)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (49)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (50)

6 Abstract and Applied Analysis

If we follow the same procedure of Section 41 approximate119910(119909) as in (40) and make use of (23) and (35) then we obtain

119862119879D2Ψ (119909) = 119865 (119909 119910 (119909) 119862119879DΨ (119909)) (51)

To find an approximate solution to 119910(119909) we compute (51)at the first 2119896(119872+1)minus2 roots of119880lowast

2119896(119872+1)

(119909)These equationswith (44) or (45) or (46) generate 2119896(119872 + 1) nonlinearequations in the expansion coefficients 119888

119899119898 which can be

solved with the aid of Newtonrsquos iterative method

5 Numerical Results and Discussions

In this section the presented algorithms in Section 4 areapplied to solve linear and nonlinear second-order initialand boundary value problems as well as Bratursquos equationsSome examples subject to different initial boundary andmixed boundary conditions are considered to illustrate theefficiency and the applicability of the proposed algorithmsAll computations are performed by using Mathematica 8

Example 1 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) + (1 minus 119909) 1199101015840

(119909) + 2119910 (119909) = (1 + 2119909 minus 1199092

) sin119909

0 lt 119909 lt 1 119910 (0) = 1 119910 (1) = 0

(52)

with the exact solution 119910(119909) = (1 minus 119909) cos119909In Table 1 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 2 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) minus sin1199091199101015840 (119909) + 119910 (119909) = sin2119909 0 lt 119909 lt 1 (53)

subject to the mixed boundary conditions

119910 (0) + 1199101015840

(0) = 1 sin (1) 119910 (1) + cos (1) 1199101015840 (1) = 0(54)

with the exact solution 119910(119909) = cos119909In Table 2 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 3 Consider the following nonlinear initial valueproblem (see [41])

411991010158401015840

(119909) minus 2(1199101015840

(119909))2

+ 119910 (119909) = 0

0 lt 119909 lt 1 119910 (0) = minus1 1199101015840

(0) = 0

(55)

with the exact solution 119910(119909) = (11990928) minus 1We solve (55) using the algorithmdescribed in Section 42

for the case corresponds to 119872 = 2 119896 = 0 to obtain anapproximate solution of119910(119909) First if wemake use of (24) and

Table 1 Maximum absolute error 119864 for Example 1

119896 119872 119864 119896 119872 119864

3 228 sdot 10minus3

2 923 sdot 10minus6

0 4 632 sdot 10minus5

1 3 862 sdot 10minus8

5 623 sdot 10minus6

4 107 sdot 10minus9

Table 2 Maximum absolute error 119864 for Example 2

119896 119872 119864 119896 119872 119864

3 252 sdot 10minus5

2 490 sdot 10minus8

0 4 211 sdot 10minus6

1 3 255 sdot 10minus9

5 901 sdot 10minus8

4 350 sdot 10minus10

(35) then the two operational matrices D and D2 are givenrespectively by

D = (0 0 0

4 0 0

0 8 0

) D2 = (0 0 0

0 0 0

32 0 0

) (56)

Moreover Ψ(119909) can be evaluated to give

Ψ (119909) = radic2

120587(

2

8119909 minus 4

321199092

minus 32119909 + 6

) (57)

If we set

119862 = (11988800 11988801 11988802)119879

= radic120587

2(1198880 1198881 1198882)119879

(58)

then (55) takes the form

4119862119879D2Ψ (119909) minus 2(119862119879DΨ (119909))

2

+ 119862119879

Ψ (119909) = 0 (59)

which is equivalent to

1198880minus 21198881+ 131119888

2minus 641198882

1+ 512119888

11198882minus 1024119888

2

2

+ 4 (1198881minus 41198882minus 256119888

11198882+ 1024119888

2

2) 119909

+ 16 (1198882minus 256119888

2

2) 1199092

= 0

(60)

We only need to satisfy this equation at the first root of119880lowast3(119909)

that is at 1199091= (2 minus radic2)4 to get

1198880minus 641198882

1+ radic2119888

1(2561198882minus 1) + (129 minus 512119888

2) 1198882= 0 (61)

Furthermore the use of initial conditions in (55) leads to thetwo equations

21198880minus 41198881+ 61198882= minus1

1198881minus 41198882= 0

(62)

The solution of the nonlinear system of (61)-(62) gives

1198880=minus123

256 119888

1=1

64 119888

2=1

256 (63)

Abstract and Applied Analysis 7

and consequently

119910 (119909) = (minus123

2561

641

256)(

2

8119909 minus 4

321199092

minus 32119909 + 6

)

=1199092

8minus 1

(64)

which is the exact solution

Example 4 Consider the following nonlinear second-orderboundary value problem

11991010158401015840

(119909) + (1199101015840

(119909))2

minus 119910 (119909) = sinh2119909

119910 (0) = 1 119910 (1) = cosh 1(65)

with the exact solution 119910(119909) = cosh 119909In Table 3 we list the maximum absolute error 119864 for

various values of 119896 and119872

Example 5 Consider the following linear singular initialvalue problem (see [23])

11991010158401015840

(119909) + 119891 (119909) 1199101015840

(119909) + 119910 (119909) = 119892 (119909)

119910 (0) = 0 1199101015840

(0) = 1

(66)

where

119891 (119909) =

minus4119909 +4

3 0 ⩽ 119909 lt

1

4

119909 minus1

3

1

4⩽ 119909 ⩽ 1

(67)

and 119892(119909) is chosen such that the exact solution of (66) is119910(119909) = 119890

119909 sin119909(1 + 1199092) We apply the algorithm describedin Section 41 with 119896 = 2 In Table 4 we list the maximumabsolute error for various values of119872 while in Figure 1 theexact solution is comparedwith the three numerical solutionscorresponding to 119896 = 2119872 = 2 119896 = 2119872 = 4 119896 = 2 and119872 = 6 and also in Table 5 we give a comparison between theerror obtained in [23] and the present method 119896 = 2119872 = 8

Example 6 Consider the following Bratu equation (see [17ndash20])

11991010158401015840

(119909) + 120582119890119910(119909)

= 0 119910 (0) = 119910 (1) = 0 0 ⩽ 119909 ⩽ 1 (68)

with the analytical solution

119910 (119909) = minus2 ln [cosh ((1205794) (2119909 minus 1))cosh (1205794)

] (69)

where 120579 is the solution of the nonlinear equation 120579 =

radic2120582 cosh 120579The presented algorithm in Section 42 is appliedto numerically solve (68) for the three cases correspondingto 120582 = 1 2 and 351 which yield 120579 = 151716 235755 and466781 respectively In Table 6 the maximum absolute error119864 is listed for 119896 = 0 and various values of119872 while in Table 7we give a comparison between the best errors obtained by

Table 3 Maximum absolute error 119864 for Example 4

119896 119872 119864 119896 119872 119864

03 569 sdot 10

minus4

1 2 112 sdot 10minus6

4 135 sdot 10minus5

3 197 sdot 10minus8

Table 4 Maximum absolute error 119864 for Example 5

119872 119864 119872 119864

2 298 sdot 10minus4 6 185 sdot 10

minus10

4 170 sdot 10minus5 8 912 sdot 10

minus11

Table 5 Comparison between different errors for Example 5

119909 02 04 06 08

LegendreWavelets [23] 430 sdot 10

minus12

124 sdot 10minus11

297 sdot 10minus12

435 sdot 10minus12

SCWOMD 647 sdot 10minus13

121 sdot 10minus12

174 sdot 10minus12

232 sdot 10minus12

Table 6 Maximum absolute error 119864 for Example 6

119872 120582 119864 120582 119864 120582 119864

10

1

2162 sdot 10minus7

2

2758 sdot 10minus5

351

7658 sdot 10minus4

12 4453 sdot 10minus9

8082 sdot 10minus6

8665 sdot 10minus5

14 8461 sdot 10minus11

1461 sdot 10minus9

2384 sdot 10minus8

16 1546 sdot 10minus12

5342 sdot 10minus10

6587 sdot 10minus9

18 2782 sdot 10minus14

1644 sdot 10minus11

2387 sdot 10minus10

20 7216 sdot 10minus16

4024 sdot 10minus12

1058 sdot 10minus10

Table 7 Comparison between the best errors for Example 6 for 119896 =0

SCWOMDError in [17] Error in [18] Error in [19] Error in [20]722 sdot 10

minus16

102 sdot 10minus6

889 sdot 10minus6

135 sdot 10minus5

301 sdot 10minus3

our method and some other numerical methods used tosolve Example 6 This table shows that our method is moreaccurate if compared with the methods developed in [17ndash20] In addition Figure 2 illustrates a comparison betweendifferent solutions obtained by our algorithm in case of 120582 = 1119896 = 0 and119872 = 1 2 3

Remark 6 It is worth noting here that the obtained numericalresults in the previous six solved examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecompared favorably with the known analytical solutions

6 Concluding Remarks

In this paper an algorithm for obtaining a numerical spectralsolution for second-order linear and nonlinear boundaryvalue problems is discussed The derivation of this algorithmis essentially based on constructing the shifted second kindChebyshev wavelets operational matrix of differentiationOne of the main advantages of the presented algorithm isits availability for application on both linear and nonlin-ear second-order boundary value problems including some

8 Abstract and Applied Analysis

Exactk = 2M = 2

k = 2M = 4

k = 2M = 6

x

00 02 04 06 08 10002

005

010

020

050

100

y(x)

Figure 1 Numerical and exact solutions of Example 5

00 02 04 06 08 100010

0100

0050

0020

0030

0015

0150

0070

Exactk = 0M = 1

k = 0M = 2

k = 0M = 3

x

y(x)

Figure 2 Different solutions of Example 6

singular equations and also Bratu type equations Anotheradvantage of the developed algorithm is that high accurateapproximate solutions are achieved using a small number ofthe second kindChebyshev waveletsThe obtained numericalresults are compared favorably with the analytical ones

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support Also they would like to thank the referee forhis valuable comments and suggestions which improved thepaper into its present form

References

[1] EHDoha andWMAbd-Elhameed ldquoEfficient spectral-Galer-kin algorithms for direct solution of second-order equationsusing ultraspherical polynomialsrdquo SIAM Journal on ScientificComputing vol 24 no 2 pp 548ndash571 2002

[2] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[4] E H Doha W M Abd-Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[5] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourth kindsrdquo Abstract and Applied Analysis vol2013 Article ID 542839 9 pages 2013

[6] A Constantmldes Applied Numerical Methods With PersonalComputers McGraw-Hill New York NY USA 1987

[7] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[8] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013

[9] N Daoudi-Merzagui F Derrab and A Boucherif ldquoSubhar-monic solutions of nonautonomous second order differentialequations with singular nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 903281 20 pages 2012

[10] M Turkyilmazoglu ldquoEffective computation of exact and ana-lytic approximate solutions to singular nonlinear equationsof Lane-Emden-Fowler typerdquo Applied Mathematical Modellingvol 37 pp 7539ndash7548 2013

[11] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[12] G Bratu ldquoSur les equations integrales non lineairesrdquo Bulletin dela Societe Mathematique de France vol 42 pp 113ndash142 1914

[13] I M Gelrsquofand ldquoSome problems in the theory of quasilinearequationsrdquoAmericanMathematical Society Translations vol 29pp 295ndash381 1963

[14] A Serghini Mounim and B M de Dormale ldquoFrom the fittingtechniques to accurate schemes for the Liouville-Bratu-Gelfandproblemrdquo Numerical Methods for Partial Differential Equationsvol 22 no 4 pp 761ndash775 2006

[15] M I Syam and A Hamdan ldquoAn efficient method for solvingBratu equationsrdquo Applied Mathematics and Computation vol176 no 2 pp 704ndash713 2006

[16] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[17] S Abbasbandy M S Hashemi and C-S Liu ldquoThe Lie-groupshooting method for solving the Bratu equationrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no11 pp 4238ndash4249 2011

[18] H Caglar N Caglar M Ozer A Valarıstos and A N Anag-nostopoulos ldquoB-spline method for solving Bratursquos problemrdquoInternational Journal of Computer Mathematics vol 87 no 8pp 1885ndash1891 2010

Abstract and Applied Analysis 9

[19] S A Khuri ldquoA new approach to Bratursquos problemrdquo AppliedMathematics and Computation vol 147 no 1 pp 131ndash136 2004

[20] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvingboundary value problemsrdquo Journal of Computational Physicsvol 159 no 2 pp 125ndash138 2000

[21] J P Boyd ldquoOne-point pseudospectral collocation for the one-dimensional Bratu equationrdquoAppliedMathematics and Compu-tation vol 217 no 12 pp 5553ndash5565 2011

[22] H Jafari S A Yousefi M A Firoozjaee S Momani and CM Khalique ldquoApplication of Legendre wavelets for solvingfractional differential equationsrdquo Computers amp Mathematicswith Applications vol 62 no 3 pp 1038ndash1045 2011

[23] F Mohammadi and M M Hosseini ldquoA new Legendre waveletoperational matrix of derivative and its applications in solvingthe singular ordinary differential equationsrdquo Journal of theFranklin Institute vol 348 no 8 pp 1787ndash1796 2011

[24] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[25] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[26] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[27] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[28] M Dehghan and A Saadatmandi ldquoChebyshev finite differencemethod for Fredholm integro-differential equationrdquo Interna-tional Journal of Computer Mathematics vol 85 no 1 pp 123ndash130 2008

[29] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[30] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[31] Y Li ldquoSolving a nonlinear fractional differential equation usingChebyshev waveletsrdquoCommunications in Nonlinear Science andNumerical Simulation vol 15 no 9 pp 2284ndash2292 2010

[32] H Saeedi M Mohseni Moghadam N Mollahasani and G NChuev ldquoA CAS wavelet method for solving nonlinear Fredholmintegro-differential equations of fractional orderrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no3 pp 1154ndash1163 2011

[33] H Saeedi and M M Moghadam ldquoNumerical solution ofnonlinear Volterra integro-differential equations of arbitraryorder by CAS waveletsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 3 pp 1216ndash1226 2011

[34] C F Cheng Y T Tsay and T T Wu ldquoWalsh operational matri-ces for fractional calculus and their application to distributedsystemsrdquo Journal of the Franklin Institute vol 303 no 3 pp 267ndash284 1977

[35] R Y Chang and M L Wang ldquoShifted Legendre direct methodfor variational problemsrdquo Journal of Optimization Theory andApplications vol 39 no 2 pp 299ndash307 1983

[36] I R Horng and J H Chou ldquoShifted Chebyshev direct methodfor solving variational problemsrdquo International Journal of Sys-tems Science vol 16 no 7 pp 855ndash861 1985

[37] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996

[38] F Khellat and S A Yousefi ldquoThe linear Legendre motherwavelets operational matrix of integration and its applicationrdquoJournal of the Franklin Institute vol 343 no 2 pp 181ndash190 2006

[39] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp HallCRC Boca Raton Fla USA 2003

[40] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[41] M R Eslahchi M Dehghan and S Ahmadi ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

Submit your manuscripts athttpwwwhindawicom

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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

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

2 Abstract and Applied Analysis

the given differential equation For this purpose they usedcompact combinations of various orthogonal polynomialsThe suggested algorithms in these papers are suitable forhandling one- and two-dimensional linear high even-orderboundary value problems

In this paper we aim to give some algorithms for handlingboth linear and nonlinear second-order boundary valueproblems based on introducing a new matrix of derivativesthen applying Petrov-Galerkin method on linear equationsand collocation method on nonlinear equations High accu-rate spectral wavelets solutions are achieved with a smallnumber of retained modes compared with the usual spectralmethods also we can handle singular differential equationswith discontinuous variable coefficients

Among the second-order boundary value problems is theone-dimensional Bratu problem which has a long historyBratursquos own paper appeared in 1914 [12] generalizations aresometimes called the Liouville-Gelfand or Liouville-Gelfand-Bratu problem in honor of Gelrsquofand [13] and the nineteenthcentury work of the great Frenchmathematician Liouville Inrecent years it has been a popular testbed for numerical andperturbation methods [14ndash16]

Simplification of the solid fuel ignition model in thermalcombustion theory yields an elliptic nonlinear partial dif-ferential equation namely the Bratu problem Also due toits use in a large variety of applications many authors havecontributed to the study of such problem Some applicationsof Bratu problem are the model of thermal reaction processthe Chandrasekhar model of the expansion of the Universechemical reaction theory nanotechnology and radiative heattransfer (see [17ndash21])

The application of Legendre wavelets for solving differen-tial and integral equations is thoroughly considered by manyauthors (see [22ndash26]) Also Chebyshev wavelets are used forsolving some differential fractional and integral equations(see [5 27ndash31])

In [32 33] a cosine and sine (CAS) wavelets operationalmatrix of fractional order integration has been derived andused to solve integrodifferential equations of fractional order

One approach for solving differential equations is basedon converting the differential equations into integral equa-tions through integration approximating various signalsinvolved in the equation by truncated orthogonal series andusing the operational matrix of integration to eliminate theintegral operations

Special attentions have been given to applications of blockpulse functions [34] Legendre polynomials [35] Chebyshevpolynomials [36] Haar wavelets [37] Legendre wavelets [2425 38] and Chebyshev wavelets [27] Another approach isbased on using operational matrix of derivatives in order toreduce the problem into solving a system of algebraic equa-tions (see [23])

The main aim of this paper is to develop a new spec-tral algorithm for solving second-order two-point boundaryvalue problems based on shifted second kind Chebyshevwavelets operational matrix of derivatives The methodreduces the differential equation with its initial andorboundary conditions to a system of algebraic equations in theunknown expansion coefficients Large systems of algebraic

equations may lead to greater computational complexityand large storage requirements However the second kindChebyshev wavelets are structurally sparse this reduces dras-tically the computational complexity of solving the resultingalgebraic system

The structure of the paper is as follows In Section 2we give some relevant properties of second kind Chebyshevpolynomials and their shifted forms In Section 3 we developa new shifted second kind Chebyshev wavelets operationalmatrices of derivatives (SCWOMD) also we ascertain theconvergence analysis of the proposed algorithm As an appli-cation of SCWOMD numerical solutions of second-orderlinear and nonlinear initial value problems or two-pointboundary value problems are implemented and presentedin Section 4 In Section 5 some numerical examples arepresented to show the efficiency and the applicability of thepresented algorithm Some concluding remarks are given inSection 6

2 Some Properties of Second Kind ChebyshevPolynomials and Their Shifted Forms

In the present section we discuss some relevant propertiesof the second kind Chebyshev polynomials and their shiftedforms

21 Second Kind Chebyshev Polynomials It is well knownthat the second kind Chebyshev polynomials are defined on[minus1 1] by

119880119899(119909) =

sin (119899 + 1) 120579sin 120579

119909 = cos 120579 (1)

These polynomials are orthogonal on [minus1 1] that is

int1

minus1

radic1 minus 1199092119880119898(119909)119880119899(119909) 119889119909 =

0 119898 = 119899120587

2 119898 = 119899

(2)

The following properties of second kind Chebyshev poly-nomials (see eg [39]) are of fundamental importance inthe sequel They are eigenfunctions of the following singularSturm-Liouville equation

(1 minus 1199092

)1198632

120601119896(119909) minus 3119909119863120601

119896(119909) + 119896 (119896 + 2) 120601

119896(119909) = 0 (3)

where 119863 equiv 119889119889119909 and may be generated by using the recur-rence relation

119880119896+1(119909) = 2119909119880

119896(119909) minus 119880

119896minus1(119909) 119896 = 1 2 3 (4)

starting from 1198800(119909) = 1 and 119880

1(119909) = 2119909 or from Rodrigues

formula

119880119899(119909) =

(minus2)119899

(119899 + 1)

(2119899 + 1)radic1 minus 1199092119863119899

[(1 minus 1199092

)119899+(12)

] (5)

The following theorem is needed hereafter

Abstract and Applied Analysis 3

Theorem 1 Thefirst derivative of second kind Chebyshev poly-nomials is given by

119863119880119899(119909) = 2

119899minus1

sum119896=0

(119896+119899) odd

(119896 + 1)119880119896(119909) (6)

For a proof of Theorem 1 see [39]

22 Shifted Second Kind Chebyshev Polynomials The shiftedsecond kind Chebyshev polynomials are defined on [0 1] by119880lowast

119899(119909) = 119880

119899(2119909 minus 1) All results of second kind Chebyshev

polynomials can be easily transformed to give the corre-sponding results for their shifted forms The orthogonalityrelation with respect to the weight function radic119909 minus 1199092 is givenby

int1

0

radic119909 minus 1199092119880lowast

119899(119909)119880lowast

119898(119909) 119889119909 =

0 119898 = 119899120587

8 119898 = 119899

(7)

The first derivative of 119880lowast119899(119909) is given in the following corol-

lary

Corollary 2 The first derivative of the shifted second kindChebyshev polynomial is given by

119863119880lowast

119899(119909) = 4

119899minus1

sum119896=0

(119896+119899)odd

(119896 + 1)119880lowast

119896(119909) (8)

3 Shifted Second Kind Chebyshev OperationalMatrix of Derivatives

Wavelets constitute of a family of functions constructed fromdilation and translation of single function called the motherwavelet When the dilation parameter 119886 and the translationparameter 119887 vary continuously then we have the followingfamily of continuous wavelets

120595119886119887(119905) = |119886|

minus12

120595(119905 minus 119887

119886) 119886 119887 isin R 119886 = 0 (9)

Second kind Chebyshev wavelets 120595119899119898(119905) = 120595(119896 119899119898 119905) have

four arguments 119896 119899 can assume any positive integer 119898 isthe order of second kind Chebyshev polynomials and 119905 is thenormalized time They are defined on the interval [0 1] by

120595119899119898(119905) =

2(119896+3)2

radic120587119880lowast

119898(2119896

119905 minus 119899) 119905 isin [119899

2119896119899 + 1

2119896]

0 otherwise

119898 = 0 1 119872 119899 = 0 1 2119896

minus 1

(10)

31 Function Approximation A function 119891(119905) defined over[0 1] may be expanded in terms of second kind Chebyshevwavelets as

119891 (119905) =

infin

sum119899=0

infin

sum119898=0

119888119899119898120595119899119898(119905) (11)

where

119888119899119898= (119891 (119905) 120595

119899119898(119905))120596= int1

0

120596 (119905) 119891 (119905) 120595119899119898(119905) 119889119905 (12)

and 119908(119905) = radic119905 minus 1199052 If the infinite series is truncated then119891(119905) can be approximated as

119891 (119905) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119888119899119898120595119899119898(119905) = 119862

119879

Ψ (119905) (13)

where 119862 and Ψ(119905) are 2119896(119872 + 1) times 1matrices defined by

119862 = [11988800 11988801 119888

0119872 119888

2119896minus1119872

1198882119896minus11 119888

2119896minus1119872

]119879

(14)

Ψ (119905) = [12059500 12059501 120595

0119872

1205952119896minus1119872

1205952119896minus11 120595

2119896minus1119872

]119879

(15)

32 Convergence Analysis We state and prove a theoremascertaining that the second kind Chebyshev wavelet expan-sion of a function 119891(119909) with bounded second derivativeconverges uniformly to 119891(119909)

Theorem 3 A function 119891(119909) isin 1198712120596[0 1] with |11989110158401015840(119909)| ⩽ 119871

can be expanded as an infinite sum of Chebyshev wavelets andthe series converges uniformly to119891(119909) Explicitly the expansioncoefficients in (12) satisfy the following inequality

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

8radic2120587 119871

(119899 + 1)52

(119898 + 1)2

forall119898 gt 1 119899 ⩾ 0 (16)

Proof From (12) it follows that

119888119899119898=2(119896+3)2

radic120587int(119899+1)2

119896

1198992119896

119891 (119909)119880lowast

119898(2119896

119909 minus 119899)120596 (2119896

119909 minus 119899) 119889119909

(17)

If we set 2119896119909 minus 119899 = cos 120579 in (17) then we get

119888119899119898=2(minus119896+3)2

radic120587

times int120587

0

119891(cos 120579 + 1198992119896

) sin (119898 + 1) 120579 sin 120579 119889120579

=2(minus119896+1)2

radic120587

times int120587

0

119891(cos 120579 + 1198992119896

) [cos119898120579 minus cos (119898 + 2) 120579] 119889120579

(18)

which gives after integration by parts two times

119888119899119898=

2

251198962radic2120587int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579 (19)

4 Abstract and Applied Analysis

where

120574119898(120579) =

sin 120579119898(sin (119898 minus 1) 120579119898 minus 1

minussin (119898 + 1) 120579119898 + 1

)

minussin 120579119898 + 2

(sin (119898 + 1) 120579119898 + 1

minussin (119898 + 3) 120579119898 + 3

)

(20)

Therefore we have

10038161003816100381610038161198881198991198981003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816

1

2(5119896minus1)2radic120587int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

=1

2(5119896minus1)2radic120587

10038161003816100381610038161003816100381610038161003816int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

⩽119871

2(5119896minus1)2radic120587int120587

0

1003816100381610038161003816120574119898 (120579)1003816100381610038161003816 119889120579

⩽119871radic120587

2(5119896minus1)2

times [1

119898(1

119898 minus 1+

1

119898 + 1)

+1

119898 + 2(1

119898 + 1+

1

119898 + 3)]

=119871radic120587

2(5119896minus5)2 (1198982 + 2119898 minus 3)

lt2119871radic120587

2(5119896minus5)2(119898 + 1)2

(21)

Since 119899 ⩽ 2119896 minus 1 we have

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

8radic2120587119871

(119899 + 1)52

(119898 + 1)2

(22)

and this completes the proof of Theorem 3

A shifted second kind Chebyshev wavelets operationalmatrix of the first derivative is stated andproved in the follow-ing theorem

Theorem 4 Let Ψ(119905) be the second kind Chebyshev waveletsvector defined in (15) Then the first derivative of the vectorΨ(119905) can be expressed as

119889Ψ (119905)

119889119905= DΨ (119905) (23)

where D is 2119896(119872 + 1) square operational matrix of derivativeand is defined by

D =(

(

119865 0 0

0 119865 0

0 0 119865

)

)

(24)

in which 119865 is an (119872 + 1) square matrix whose (119903 119904)th elementis defined by

119865119903119904= 2119896+2

119904 119903 ⩾ 2 119903 gt 119904 (119903 + 119904) odd0 otherwise

(25)

Proof If we make use of the shifted second kind Chebyshevpolynomials then the 119903th element of the vector Ψ(119905) definedin (15) can be written in the following form

Ψ119903(119905) = 120595

119899119898(119905) =

2(119896+3)2

radic120587119880lowast

119898(2119896

119905 minus 119899) 120594[1198992119896(119899+1)2

119896]

119894 = 1 2 2119896

(119872 + 1)

(26)

where 119903 = 119899(119872 + 1) + (119898 + 1) 119898 = 0 1 119872 119899 =0 1 (2

119896

minus 1) and

120594[1198992119896(119899+1)2

119896]=

1 119905 isin [119899

2119896119899 + 1

2119896]

0 otherwise(27)

If we differentiate (26) with respect to 119905 then we get

119889Ψ119903(119905)

119889119905=2(2119896+3)2

radic120587

119889

119889119905[119880lowast

119898(2119896

119905 minus 119899)] 120594[1198992119896(119899+1)2

119896] (28)

It is to be noted here that the RHS of (28) is zero outside theinterval [1198992119896 (119899 + 1)2119896] hence its second kind Chebyshevwavelets expansion only has those elements in Ψ(119905) that arenonzero in the interval [1198992119896 (119899 + 1)2119896] that is Ψ

119894(119905) 119894 =

119899(119872 + 1) + 1 119899(119872 + 1) + 2 119899(119872 + 1) + 119872 + 1 and as aconsequence this enables one to expand (119889Ψ

119903(119905)119889119905) in terms

of shifted second kind Chebyshev wavelets Ψ119894(119905) 119894 = 119899(119872 +

1) + 1 119899(119872 + 1) + 2 119899(119872 + 1) +119872 + 1 in the form

119889Ψ119903(119905)

119889119905=

(119899+1)(119872+1)

sum119894=119899(119872+1)+1

119886119894Ψ119894(119905) (29)

and accordingly (28) implies that the operational matrixD isa block matrix defined as in (24) Moreover we have

119889119880lowast

0(119905)

119889119905= 0 (30)

and this implies that119889

119889119905Ψ119903(119905) = 0

for 119903 = 1 (119872 + 1) + 1 2 (119872 + 1) + 1

(2119896

minus 1) (119872 + 1) + 1

(31)

Consequently the elements of the first row of matrix 119865defined in (24) are zeros Now andwith the aid of Corollary 2the first derivative of Ψ

119903(119905)may be expressed in the form

119889Ψ119903(119905)

119889119905

=2(2119896+7)2

radic120587

119898minus1

sum119895=0

(119895+119898) odd

(119895 + 1)119880lowast

119895(2119896

119905 minus 119899) 120594[1198992119896(119899+1)2

119896](32)

Abstract and Applied Analysis 5

Expanding this equation in terms of second kind Chebyshevwavelets basis we get

119889Ψ119903(119905)

119889119905= 2119896+2

119898minus1

sum119904=0

(119904+119898) odd

119904Ψ119899(119872+1)+119904

(119905) (33)

so if 119865119903119904

are chosen as

119865119903119904=

2119896+2

119904 119903 = 2 119872 + 1 119904 = 1 119903 minus 1

(119903 + 119904) odd0 otherwise

(34)

then (23) is obtained and the proof of the theorem iscompleted

Corollary 5 The operational matrix for the 119899th derivative canbe obtained from

119889119899

Ψ (119905)

119889119905119899= D119899Ψ (119905) 119899 = 1 2 (35)

whereD119899 is the 119899th power of matrix D

4 Second-Order Two-Point BoundaryValue Problems

In this section we are interested in solving linear and nonlin-ear two-point boundary value problems subject to homoge-nous or nonhomogenous initial or boundary conditionsbased on the wavelets operational matrices constructed inSection 3

41 Linear Second-Order Two-Point Boundary Value Prob-lems Consider the linear second-order differential equation

11991010158401015840

(119909) + 1198911(119909) 1199101015840

(119909) + 1198912(119909) 119910 (119909) = 119892 (119909) 119909 isin [0 1]

(36)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (37)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (38)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (39)

If we approximate 119910(119909)1198911(119909)1198912(119909) and 119892(119909) in terms of the

second kind Chebyshev wavelets basis then one can write

119910 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119888119899119898120595119899119898(119909) = 119862

119879

Ψ (119909)

1198911(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

1Ψ (119909)

1198912(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

2Ψ (119905)

119892 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119892119899119898120595119899119898(119909) = 119866

119879

Ψ (119909)

(40)

where 119862119879 1198651198791 1198651198792 and 119866119879 are defined similarly as in (14)

Relations (23) and (35) enable one to approximate 1199101015840(119909) and11991010158401015840

(119909) as

1199101015840

(119909) ≃ 119862119879DΨ (119909) 119910

10158401015840

(119909) ≃ 119862119879D2Ψ (119909) (41)

Now substitution of relations (40) and (41) into (36) enablesus to define the residual 119877(119909) of this equation as

119877 (119909) = 119862119879D2Ψ (119909) + 119865119879

1Ψ (119909) (Ψ (119909))

119879D119879119862

+ 119865119879

2Ψ (119909) (Ψ (119909))

119879

119862 minus 119866119879

Ψ (119909) (42)

and application of the Petrov-Galerkin method (see [40])yields the following (2119896(119872 + 1) minus 2) linear equations in theunknown expansion coefficients 119888

119899119898 namely

int1

0

radic119909 minus 1199092Ψ119895(119909) 119877 (119909) 119889119909 = 0

119895 = 1 2 2119896

(119872 + 1) minus 2

(43)

Moreover the initial conditions (37) the boundary condi-tions (38) and the mixed boundary conditions (39) leadrespectively to the following equations

119862119879

Ψ (0) = 120572 119862119879DΨ (0) = 120573 (44)

119862119879

Ψ (0) = 120572 119862119879

Ψ (1) = 120573 (45)

1198861119862119879

Ψ (0) + 1198862119862119879DΨ (0) = 120572

1198871119862119879

Ψ (1) + 1198872119862119879DΨ (1) = 120573

(46)

Thus (43) with the two equations of (44) or (45) or (46)generates 2119896(119872 + 1) a set of linear equations which can besolved for the unknown components of the vector 119862 andhence an approximate spectral wavelets solution to119910(119909) givenin (40) can be obtained

42 Nonlinear Second-Order Two-Point Boundary Value Prob-lems Consider the nonlinear differential equation

11991010158401015840

(119909) = 119865 (119909 119910 (119909) 1199101015840

(119909)) (47)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (48)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (49)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (50)

6 Abstract and Applied Analysis

If we follow the same procedure of Section 41 approximate119910(119909) as in (40) and make use of (23) and (35) then we obtain

119862119879D2Ψ (119909) = 119865 (119909 119910 (119909) 119862119879DΨ (119909)) (51)

To find an approximate solution to 119910(119909) we compute (51)at the first 2119896(119872+1)minus2 roots of119880lowast

2119896(119872+1)

(119909)These equationswith (44) or (45) or (46) generate 2119896(119872 + 1) nonlinearequations in the expansion coefficients 119888

119899119898 which can be

solved with the aid of Newtonrsquos iterative method

5 Numerical Results and Discussions

In this section the presented algorithms in Section 4 areapplied to solve linear and nonlinear second-order initialand boundary value problems as well as Bratursquos equationsSome examples subject to different initial boundary andmixed boundary conditions are considered to illustrate theefficiency and the applicability of the proposed algorithmsAll computations are performed by using Mathematica 8

Example 1 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) + (1 minus 119909) 1199101015840

(119909) + 2119910 (119909) = (1 + 2119909 minus 1199092

) sin119909

0 lt 119909 lt 1 119910 (0) = 1 119910 (1) = 0

(52)

with the exact solution 119910(119909) = (1 minus 119909) cos119909In Table 1 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 2 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) minus sin1199091199101015840 (119909) + 119910 (119909) = sin2119909 0 lt 119909 lt 1 (53)

subject to the mixed boundary conditions

119910 (0) + 1199101015840

(0) = 1 sin (1) 119910 (1) + cos (1) 1199101015840 (1) = 0(54)

with the exact solution 119910(119909) = cos119909In Table 2 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 3 Consider the following nonlinear initial valueproblem (see [41])

411991010158401015840

(119909) minus 2(1199101015840

(119909))2

+ 119910 (119909) = 0

0 lt 119909 lt 1 119910 (0) = minus1 1199101015840

(0) = 0

(55)

with the exact solution 119910(119909) = (11990928) minus 1We solve (55) using the algorithmdescribed in Section 42

for the case corresponds to 119872 = 2 119896 = 0 to obtain anapproximate solution of119910(119909) First if wemake use of (24) and

Table 1 Maximum absolute error 119864 for Example 1

119896 119872 119864 119896 119872 119864

3 228 sdot 10minus3

2 923 sdot 10minus6

0 4 632 sdot 10minus5

1 3 862 sdot 10minus8

5 623 sdot 10minus6

4 107 sdot 10minus9

Table 2 Maximum absolute error 119864 for Example 2

119896 119872 119864 119896 119872 119864

3 252 sdot 10minus5

2 490 sdot 10minus8

0 4 211 sdot 10minus6

1 3 255 sdot 10minus9

5 901 sdot 10minus8

4 350 sdot 10minus10

(35) then the two operational matrices D and D2 are givenrespectively by

D = (0 0 0

4 0 0

0 8 0

) D2 = (0 0 0

0 0 0

32 0 0

) (56)

Moreover Ψ(119909) can be evaluated to give

Ψ (119909) = radic2

120587(

2

8119909 minus 4

321199092

minus 32119909 + 6

) (57)

If we set

119862 = (11988800 11988801 11988802)119879

= radic120587

2(1198880 1198881 1198882)119879

(58)

then (55) takes the form

4119862119879D2Ψ (119909) minus 2(119862119879DΨ (119909))

2

+ 119862119879

Ψ (119909) = 0 (59)

which is equivalent to

1198880minus 21198881+ 131119888

2minus 641198882

1+ 512119888

11198882minus 1024119888

2

2

+ 4 (1198881minus 41198882minus 256119888

11198882+ 1024119888

2

2) 119909

+ 16 (1198882minus 256119888

2

2) 1199092

= 0

(60)

We only need to satisfy this equation at the first root of119880lowast3(119909)

that is at 1199091= (2 minus radic2)4 to get

1198880minus 641198882

1+ radic2119888

1(2561198882minus 1) + (129 minus 512119888

2) 1198882= 0 (61)

Furthermore the use of initial conditions in (55) leads to thetwo equations

21198880minus 41198881+ 61198882= minus1

1198881minus 41198882= 0

(62)

The solution of the nonlinear system of (61)-(62) gives

1198880=minus123

256 119888

1=1

64 119888

2=1

256 (63)

Abstract and Applied Analysis 7

and consequently

119910 (119909) = (minus123

2561

641

256)(

2

8119909 minus 4

321199092

minus 32119909 + 6

)

=1199092

8minus 1

(64)

which is the exact solution

Example 4 Consider the following nonlinear second-orderboundary value problem

11991010158401015840

(119909) + (1199101015840

(119909))2

minus 119910 (119909) = sinh2119909

119910 (0) = 1 119910 (1) = cosh 1(65)

with the exact solution 119910(119909) = cosh 119909In Table 3 we list the maximum absolute error 119864 for

various values of 119896 and119872

Example 5 Consider the following linear singular initialvalue problem (see [23])

11991010158401015840

(119909) + 119891 (119909) 1199101015840

(119909) + 119910 (119909) = 119892 (119909)

119910 (0) = 0 1199101015840

(0) = 1

(66)

where

119891 (119909) =

minus4119909 +4

3 0 ⩽ 119909 lt

1

4

119909 minus1

3

1

4⩽ 119909 ⩽ 1

(67)

and 119892(119909) is chosen such that the exact solution of (66) is119910(119909) = 119890

119909 sin119909(1 + 1199092) We apply the algorithm describedin Section 41 with 119896 = 2 In Table 4 we list the maximumabsolute error for various values of119872 while in Figure 1 theexact solution is comparedwith the three numerical solutionscorresponding to 119896 = 2119872 = 2 119896 = 2119872 = 4 119896 = 2 and119872 = 6 and also in Table 5 we give a comparison between theerror obtained in [23] and the present method 119896 = 2119872 = 8

Example 6 Consider the following Bratu equation (see [17ndash20])

11991010158401015840

(119909) + 120582119890119910(119909)

= 0 119910 (0) = 119910 (1) = 0 0 ⩽ 119909 ⩽ 1 (68)

with the analytical solution

119910 (119909) = minus2 ln [cosh ((1205794) (2119909 minus 1))cosh (1205794)

] (69)

where 120579 is the solution of the nonlinear equation 120579 =

radic2120582 cosh 120579The presented algorithm in Section 42 is appliedto numerically solve (68) for the three cases correspondingto 120582 = 1 2 and 351 which yield 120579 = 151716 235755 and466781 respectively In Table 6 the maximum absolute error119864 is listed for 119896 = 0 and various values of119872 while in Table 7we give a comparison between the best errors obtained by

Table 3 Maximum absolute error 119864 for Example 4

119896 119872 119864 119896 119872 119864

03 569 sdot 10

minus4

1 2 112 sdot 10minus6

4 135 sdot 10minus5

3 197 sdot 10minus8

Table 4 Maximum absolute error 119864 for Example 5

119872 119864 119872 119864

2 298 sdot 10minus4 6 185 sdot 10

minus10

4 170 sdot 10minus5 8 912 sdot 10

minus11

Table 5 Comparison between different errors for Example 5

119909 02 04 06 08

LegendreWavelets [23] 430 sdot 10

minus12

124 sdot 10minus11

297 sdot 10minus12

435 sdot 10minus12

SCWOMD 647 sdot 10minus13

121 sdot 10minus12

174 sdot 10minus12

232 sdot 10minus12

Table 6 Maximum absolute error 119864 for Example 6

119872 120582 119864 120582 119864 120582 119864

10

1

2162 sdot 10minus7

2

2758 sdot 10minus5

351

7658 sdot 10minus4

12 4453 sdot 10minus9

8082 sdot 10minus6

8665 sdot 10minus5

14 8461 sdot 10minus11

1461 sdot 10minus9

2384 sdot 10minus8

16 1546 sdot 10minus12

5342 sdot 10minus10

6587 sdot 10minus9

18 2782 sdot 10minus14

1644 sdot 10minus11

2387 sdot 10minus10

20 7216 sdot 10minus16

4024 sdot 10minus12

1058 sdot 10minus10

Table 7 Comparison between the best errors for Example 6 for 119896 =0

SCWOMDError in [17] Error in [18] Error in [19] Error in [20]722 sdot 10

minus16

102 sdot 10minus6

889 sdot 10minus6

135 sdot 10minus5

301 sdot 10minus3

our method and some other numerical methods used tosolve Example 6 This table shows that our method is moreaccurate if compared with the methods developed in [17ndash20] In addition Figure 2 illustrates a comparison betweendifferent solutions obtained by our algorithm in case of 120582 = 1119896 = 0 and119872 = 1 2 3

Remark 6 It is worth noting here that the obtained numericalresults in the previous six solved examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecompared favorably with the known analytical solutions

6 Concluding Remarks

In this paper an algorithm for obtaining a numerical spectralsolution for second-order linear and nonlinear boundaryvalue problems is discussed The derivation of this algorithmis essentially based on constructing the shifted second kindChebyshev wavelets operational matrix of differentiationOne of the main advantages of the presented algorithm isits availability for application on both linear and nonlin-ear second-order boundary value problems including some

8 Abstract and Applied Analysis

Exactk = 2M = 2

k = 2M = 4

k = 2M = 6

x

00 02 04 06 08 10002

005

010

020

050

100

y(x)

Figure 1 Numerical and exact solutions of Example 5

00 02 04 06 08 100010

0100

0050

0020

0030

0015

0150

0070

Exactk = 0M = 1

k = 0M = 2

k = 0M = 3

x

y(x)

Figure 2 Different solutions of Example 6

singular equations and also Bratu type equations Anotheradvantage of the developed algorithm is that high accurateapproximate solutions are achieved using a small number ofthe second kindChebyshev waveletsThe obtained numericalresults are compared favorably with the analytical ones

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support Also they would like to thank the referee forhis valuable comments and suggestions which improved thepaper into its present form

References

[1] EHDoha andWMAbd-Elhameed ldquoEfficient spectral-Galer-kin algorithms for direct solution of second-order equationsusing ultraspherical polynomialsrdquo SIAM Journal on ScientificComputing vol 24 no 2 pp 548ndash571 2002

[2] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[4] E H Doha W M Abd-Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[5] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourth kindsrdquo Abstract and Applied Analysis vol2013 Article ID 542839 9 pages 2013

[6] A Constantmldes Applied Numerical Methods With PersonalComputers McGraw-Hill New York NY USA 1987

[7] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[8] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013

[9] N Daoudi-Merzagui F Derrab and A Boucherif ldquoSubhar-monic solutions of nonautonomous second order differentialequations with singular nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 903281 20 pages 2012

[10] M Turkyilmazoglu ldquoEffective computation of exact and ana-lytic approximate solutions to singular nonlinear equationsof Lane-Emden-Fowler typerdquo Applied Mathematical Modellingvol 37 pp 7539ndash7548 2013

[11] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[12] G Bratu ldquoSur les equations integrales non lineairesrdquo Bulletin dela Societe Mathematique de France vol 42 pp 113ndash142 1914

[13] I M Gelrsquofand ldquoSome problems in the theory of quasilinearequationsrdquoAmericanMathematical Society Translations vol 29pp 295ndash381 1963

[14] A Serghini Mounim and B M de Dormale ldquoFrom the fittingtechniques to accurate schemes for the Liouville-Bratu-Gelfandproblemrdquo Numerical Methods for Partial Differential Equationsvol 22 no 4 pp 761ndash775 2006

[15] M I Syam and A Hamdan ldquoAn efficient method for solvingBratu equationsrdquo Applied Mathematics and Computation vol176 no 2 pp 704ndash713 2006

[16] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[17] S Abbasbandy M S Hashemi and C-S Liu ldquoThe Lie-groupshooting method for solving the Bratu equationrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no11 pp 4238ndash4249 2011

[18] H Caglar N Caglar M Ozer A Valarıstos and A N Anag-nostopoulos ldquoB-spline method for solving Bratursquos problemrdquoInternational Journal of Computer Mathematics vol 87 no 8pp 1885ndash1891 2010

Abstract and Applied Analysis 9

[19] S A Khuri ldquoA new approach to Bratursquos problemrdquo AppliedMathematics and Computation vol 147 no 1 pp 131ndash136 2004

[20] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvingboundary value problemsrdquo Journal of Computational Physicsvol 159 no 2 pp 125ndash138 2000

[21] J P Boyd ldquoOne-point pseudospectral collocation for the one-dimensional Bratu equationrdquoAppliedMathematics and Compu-tation vol 217 no 12 pp 5553ndash5565 2011

[22] H Jafari S A Yousefi M A Firoozjaee S Momani and CM Khalique ldquoApplication of Legendre wavelets for solvingfractional differential equationsrdquo Computers amp Mathematicswith Applications vol 62 no 3 pp 1038ndash1045 2011

[23] F Mohammadi and M M Hosseini ldquoA new Legendre waveletoperational matrix of derivative and its applications in solvingthe singular ordinary differential equationsrdquo Journal of theFranklin Institute vol 348 no 8 pp 1787ndash1796 2011

[24] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[25] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[26] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[27] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[28] M Dehghan and A Saadatmandi ldquoChebyshev finite differencemethod for Fredholm integro-differential equationrdquo Interna-tional Journal of Computer Mathematics vol 85 no 1 pp 123ndash130 2008

[29] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[30] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[31] Y Li ldquoSolving a nonlinear fractional differential equation usingChebyshev waveletsrdquoCommunications in Nonlinear Science andNumerical Simulation vol 15 no 9 pp 2284ndash2292 2010

[32] H Saeedi M Mohseni Moghadam N Mollahasani and G NChuev ldquoA CAS wavelet method for solving nonlinear Fredholmintegro-differential equations of fractional orderrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no3 pp 1154ndash1163 2011

[33] H Saeedi and M M Moghadam ldquoNumerical solution ofnonlinear Volterra integro-differential equations of arbitraryorder by CAS waveletsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 3 pp 1216ndash1226 2011

[34] C F Cheng Y T Tsay and T T Wu ldquoWalsh operational matri-ces for fractional calculus and their application to distributedsystemsrdquo Journal of the Franklin Institute vol 303 no 3 pp 267ndash284 1977

[35] R Y Chang and M L Wang ldquoShifted Legendre direct methodfor variational problemsrdquo Journal of Optimization Theory andApplications vol 39 no 2 pp 299ndash307 1983

[36] I R Horng and J H Chou ldquoShifted Chebyshev direct methodfor solving variational problemsrdquo International Journal of Sys-tems Science vol 16 no 7 pp 855ndash861 1985

[37] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996

[38] F Khellat and S A Yousefi ldquoThe linear Legendre motherwavelets operational matrix of integration and its applicationrdquoJournal of the Franklin Institute vol 343 no 2 pp 181ndash190 2006

[39] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp HallCRC Boca Raton Fla USA 2003

[40] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[41] M R Eslahchi M Dehghan and S Ahmadi ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Theorem 1 Thefirst derivative of second kind Chebyshev poly-nomials is given by

119863119880119899(119909) = 2

119899minus1

sum119896=0

(119896+119899) odd

(119896 + 1)119880119896(119909) (6)

For a proof of Theorem 1 see [39]

22 Shifted Second Kind Chebyshev Polynomials The shiftedsecond kind Chebyshev polynomials are defined on [0 1] by119880lowast

119899(119909) = 119880

119899(2119909 minus 1) All results of second kind Chebyshev

polynomials can be easily transformed to give the corre-sponding results for their shifted forms The orthogonalityrelation with respect to the weight function radic119909 minus 1199092 is givenby

int1

0

radic119909 minus 1199092119880lowast

119899(119909)119880lowast

119898(119909) 119889119909 =

0 119898 = 119899120587

8 119898 = 119899

(7)

The first derivative of 119880lowast119899(119909) is given in the following corol-

lary

Corollary 2 The first derivative of the shifted second kindChebyshev polynomial is given by

119863119880lowast

119899(119909) = 4

119899minus1

sum119896=0

(119896+119899)odd

(119896 + 1)119880lowast

119896(119909) (8)

3 Shifted Second Kind Chebyshev OperationalMatrix of Derivatives

Wavelets constitute of a family of functions constructed fromdilation and translation of single function called the motherwavelet When the dilation parameter 119886 and the translationparameter 119887 vary continuously then we have the followingfamily of continuous wavelets

120595119886119887(119905) = |119886|

minus12

120595(119905 minus 119887

119886) 119886 119887 isin R 119886 = 0 (9)

Second kind Chebyshev wavelets 120595119899119898(119905) = 120595(119896 119899119898 119905) have

four arguments 119896 119899 can assume any positive integer 119898 isthe order of second kind Chebyshev polynomials and 119905 is thenormalized time They are defined on the interval [0 1] by

120595119899119898(119905) =

2(119896+3)2

radic120587119880lowast

119898(2119896

119905 minus 119899) 119905 isin [119899

2119896119899 + 1

2119896]

0 otherwise

119898 = 0 1 119872 119899 = 0 1 2119896

minus 1

(10)

31 Function Approximation A function 119891(119905) defined over[0 1] may be expanded in terms of second kind Chebyshevwavelets as

119891 (119905) =

infin

sum119899=0

infin

sum119898=0

119888119899119898120595119899119898(119905) (11)

where

119888119899119898= (119891 (119905) 120595

119899119898(119905))120596= int1

0

120596 (119905) 119891 (119905) 120595119899119898(119905) 119889119905 (12)

and 119908(119905) = radic119905 minus 1199052 If the infinite series is truncated then119891(119905) can be approximated as

119891 (119905) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119888119899119898120595119899119898(119905) = 119862

119879

Ψ (119905) (13)

where 119862 and Ψ(119905) are 2119896(119872 + 1) times 1matrices defined by

119862 = [11988800 11988801 119888

0119872 119888

2119896minus1119872

1198882119896minus11 119888

2119896minus1119872

]119879

(14)

Ψ (119905) = [12059500 12059501 120595

0119872

1205952119896minus1119872

1205952119896minus11 120595

2119896minus1119872

]119879

(15)

32 Convergence Analysis We state and prove a theoremascertaining that the second kind Chebyshev wavelet expan-sion of a function 119891(119909) with bounded second derivativeconverges uniformly to 119891(119909)

Theorem 3 A function 119891(119909) isin 1198712120596[0 1] with |11989110158401015840(119909)| ⩽ 119871

can be expanded as an infinite sum of Chebyshev wavelets andthe series converges uniformly to119891(119909) Explicitly the expansioncoefficients in (12) satisfy the following inequality

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

8radic2120587 119871

(119899 + 1)52

(119898 + 1)2

forall119898 gt 1 119899 ⩾ 0 (16)

Proof From (12) it follows that

119888119899119898=2(119896+3)2

radic120587int(119899+1)2

119896

1198992119896

119891 (119909)119880lowast

119898(2119896

119909 minus 119899)120596 (2119896

119909 minus 119899) 119889119909

(17)

If we set 2119896119909 minus 119899 = cos 120579 in (17) then we get

119888119899119898=2(minus119896+3)2

radic120587

times int120587

0

119891(cos 120579 + 1198992119896

) sin (119898 + 1) 120579 sin 120579 119889120579

=2(minus119896+1)2

radic120587

times int120587

0

119891(cos 120579 + 1198992119896

) [cos119898120579 minus cos (119898 + 2) 120579] 119889120579

(18)

which gives after integration by parts two times

119888119899119898=

2

251198962radic2120587int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579 (19)

4 Abstract and Applied Analysis

where

120574119898(120579) =

sin 120579119898(sin (119898 minus 1) 120579119898 minus 1

minussin (119898 + 1) 120579119898 + 1

)

minussin 120579119898 + 2

(sin (119898 + 1) 120579119898 + 1

minussin (119898 + 3) 120579119898 + 3

)

(20)

Therefore we have

10038161003816100381610038161198881198991198981003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816

1

2(5119896minus1)2radic120587int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

=1

2(5119896minus1)2radic120587

10038161003816100381610038161003816100381610038161003816int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

⩽119871

2(5119896minus1)2radic120587int120587

0

1003816100381610038161003816120574119898 (120579)1003816100381610038161003816 119889120579

⩽119871radic120587

2(5119896minus1)2

times [1

119898(1

119898 minus 1+

1

119898 + 1)

+1

119898 + 2(1

119898 + 1+

1

119898 + 3)]

=119871radic120587

2(5119896minus5)2 (1198982 + 2119898 minus 3)

lt2119871radic120587

2(5119896minus5)2(119898 + 1)2

(21)

Since 119899 ⩽ 2119896 minus 1 we have

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

8radic2120587119871

(119899 + 1)52

(119898 + 1)2

(22)

and this completes the proof of Theorem 3

A shifted second kind Chebyshev wavelets operationalmatrix of the first derivative is stated andproved in the follow-ing theorem

Theorem 4 Let Ψ(119905) be the second kind Chebyshev waveletsvector defined in (15) Then the first derivative of the vectorΨ(119905) can be expressed as

119889Ψ (119905)

119889119905= DΨ (119905) (23)

where D is 2119896(119872 + 1) square operational matrix of derivativeand is defined by

D =(

(

119865 0 0

0 119865 0

0 0 119865

)

)

(24)

in which 119865 is an (119872 + 1) square matrix whose (119903 119904)th elementis defined by

119865119903119904= 2119896+2

119904 119903 ⩾ 2 119903 gt 119904 (119903 + 119904) odd0 otherwise

(25)

Proof If we make use of the shifted second kind Chebyshevpolynomials then the 119903th element of the vector Ψ(119905) definedin (15) can be written in the following form

Ψ119903(119905) = 120595

119899119898(119905) =

2(119896+3)2

radic120587119880lowast

119898(2119896

119905 minus 119899) 120594[1198992119896(119899+1)2

119896]

119894 = 1 2 2119896

(119872 + 1)

(26)

where 119903 = 119899(119872 + 1) + (119898 + 1) 119898 = 0 1 119872 119899 =0 1 (2

119896

minus 1) and

120594[1198992119896(119899+1)2

119896]=

1 119905 isin [119899

2119896119899 + 1

2119896]

0 otherwise(27)

If we differentiate (26) with respect to 119905 then we get

119889Ψ119903(119905)

119889119905=2(2119896+3)2

radic120587

119889

119889119905[119880lowast

119898(2119896

119905 minus 119899)] 120594[1198992119896(119899+1)2

119896] (28)

It is to be noted here that the RHS of (28) is zero outside theinterval [1198992119896 (119899 + 1)2119896] hence its second kind Chebyshevwavelets expansion only has those elements in Ψ(119905) that arenonzero in the interval [1198992119896 (119899 + 1)2119896] that is Ψ

119894(119905) 119894 =

119899(119872 + 1) + 1 119899(119872 + 1) + 2 119899(119872 + 1) + 119872 + 1 and as aconsequence this enables one to expand (119889Ψ

119903(119905)119889119905) in terms

of shifted second kind Chebyshev wavelets Ψ119894(119905) 119894 = 119899(119872 +

1) + 1 119899(119872 + 1) + 2 119899(119872 + 1) +119872 + 1 in the form

119889Ψ119903(119905)

119889119905=

(119899+1)(119872+1)

sum119894=119899(119872+1)+1

119886119894Ψ119894(119905) (29)

and accordingly (28) implies that the operational matrixD isa block matrix defined as in (24) Moreover we have

119889119880lowast

0(119905)

119889119905= 0 (30)

and this implies that119889

119889119905Ψ119903(119905) = 0

for 119903 = 1 (119872 + 1) + 1 2 (119872 + 1) + 1

(2119896

minus 1) (119872 + 1) + 1

(31)

Consequently the elements of the first row of matrix 119865defined in (24) are zeros Now andwith the aid of Corollary 2the first derivative of Ψ

119903(119905)may be expressed in the form

119889Ψ119903(119905)

119889119905

=2(2119896+7)2

radic120587

119898minus1

sum119895=0

(119895+119898) odd

(119895 + 1)119880lowast

119895(2119896

119905 minus 119899) 120594[1198992119896(119899+1)2

119896](32)

Abstract and Applied Analysis 5

Expanding this equation in terms of second kind Chebyshevwavelets basis we get

119889Ψ119903(119905)

119889119905= 2119896+2

119898minus1

sum119904=0

(119904+119898) odd

119904Ψ119899(119872+1)+119904

(119905) (33)

so if 119865119903119904

are chosen as

119865119903119904=

2119896+2

119904 119903 = 2 119872 + 1 119904 = 1 119903 minus 1

(119903 + 119904) odd0 otherwise

(34)

then (23) is obtained and the proof of the theorem iscompleted

Corollary 5 The operational matrix for the 119899th derivative canbe obtained from

119889119899

Ψ (119905)

119889119905119899= D119899Ψ (119905) 119899 = 1 2 (35)

whereD119899 is the 119899th power of matrix D

4 Second-Order Two-Point BoundaryValue Problems

In this section we are interested in solving linear and nonlin-ear two-point boundary value problems subject to homoge-nous or nonhomogenous initial or boundary conditionsbased on the wavelets operational matrices constructed inSection 3

41 Linear Second-Order Two-Point Boundary Value Prob-lems Consider the linear second-order differential equation

11991010158401015840

(119909) + 1198911(119909) 1199101015840

(119909) + 1198912(119909) 119910 (119909) = 119892 (119909) 119909 isin [0 1]

(36)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (37)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (38)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (39)

If we approximate 119910(119909)1198911(119909)1198912(119909) and 119892(119909) in terms of the

second kind Chebyshev wavelets basis then one can write

119910 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119888119899119898120595119899119898(119909) = 119862

119879

Ψ (119909)

1198911(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

1Ψ (119909)

1198912(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

2Ψ (119905)

119892 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119892119899119898120595119899119898(119909) = 119866

119879

Ψ (119909)

(40)

where 119862119879 1198651198791 1198651198792 and 119866119879 are defined similarly as in (14)

Relations (23) and (35) enable one to approximate 1199101015840(119909) and11991010158401015840

(119909) as

1199101015840

(119909) ≃ 119862119879DΨ (119909) 119910

10158401015840

(119909) ≃ 119862119879D2Ψ (119909) (41)

Now substitution of relations (40) and (41) into (36) enablesus to define the residual 119877(119909) of this equation as

119877 (119909) = 119862119879D2Ψ (119909) + 119865119879

1Ψ (119909) (Ψ (119909))

119879D119879119862

+ 119865119879

2Ψ (119909) (Ψ (119909))

119879

119862 minus 119866119879

Ψ (119909) (42)

and application of the Petrov-Galerkin method (see [40])yields the following (2119896(119872 + 1) minus 2) linear equations in theunknown expansion coefficients 119888

119899119898 namely

int1

0

radic119909 minus 1199092Ψ119895(119909) 119877 (119909) 119889119909 = 0

119895 = 1 2 2119896

(119872 + 1) minus 2

(43)

Moreover the initial conditions (37) the boundary condi-tions (38) and the mixed boundary conditions (39) leadrespectively to the following equations

119862119879

Ψ (0) = 120572 119862119879DΨ (0) = 120573 (44)

119862119879

Ψ (0) = 120572 119862119879

Ψ (1) = 120573 (45)

1198861119862119879

Ψ (0) + 1198862119862119879DΨ (0) = 120572

1198871119862119879

Ψ (1) + 1198872119862119879DΨ (1) = 120573

(46)

Thus (43) with the two equations of (44) or (45) or (46)generates 2119896(119872 + 1) a set of linear equations which can besolved for the unknown components of the vector 119862 andhence an approximate spectral wavelets solution to119910(119909) givenin (40) can be obtained

42 Nonlinear Second-Order Two-Point Boundary Value Prob-lems Consider the nonlinear differential equation

11991010158401015840

(119909) = 119865 (119909 119910 (119909) 1199101015840

(119909)) (47)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (48)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (49)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (50)

6 Abstract and Applied Analysis

If we follow the same procedure of Section 41 approximate119910(119909) as in (40) and make use of (23) and (35) then we obtain

119862119879D2Ψ (119909) = 119865 (119909 119910 (119909) 119862119879DΨ (119909)) (51)

To find an approximate solution to 119910(119909) we compute (51)at the first 2119896(119872+1)minus2 roots of119880lowast

2119896(119872+1)

(119909)These equationswith (44) or (45) or (46) generate 2119896(119872 + 1) nonlinearequations in the expansion coefficients 119888

119899119898 which can be

solved with the aid of Newtonrsquos iterative method

5 Numerical Results and Discussions

In this section the presented algorithms in Section 4 areapplied to solve linear and nonlinear second-order initialand boundary value problems as well as Bratursquos equationsSome examples subject to different initial boundary andmixed boundary conditions are considered to illustrate theefficiency and the applicability of the proposed algorithmsAll computations are performed by using Mathematica 8

Example 1 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) + (1 minus 119909) 1199101015840

(119909) + 2119910 (119909) = (1 + 2119909 minus 1199092

) sin119909

0 lt 119909 lt 1 119910 (0) = 1 119910 (1) = 0

(52)

with the exact solution 119910(119909) = (1 minus 119909) cos119909In Table 1 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 2 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) minus sin1199091199101015840 (119909) + 119910 (119909) = sin2119909 0 lt 119909 lt 1 (53)

subject to the mixed boundary conditions

119910 (0) + 1199101015840

(0) = 1 sin (1) 119910 (1) + cos (1) 1199101015840 (1) = 0(54)

with the exact solution 119910(119909) = cos119909In Table 2 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 3 Consider the following nonlinear initial valueproblem (see [41])

411991010158401015840

(119909) minus 2(1199101015840

(119909))2

+ 119910 (119909) = 0

0 lt 119909 lt 1 119910 (0) = minus1 1199101015840

(0) = 0

(55)

with the exact solution 119910(119909) = (11990928) minus 1We solve (55) using the algorithmdescribed in Section 42

for the case corresponds to 119872 = 2 119896 = 0 to obtain anapproximate solution of119910(119909) First if wemake use of (24) and

Table 1 Maximum absolute error 119864 for Example 1

119896 119872 119864 119896 119872 119864

3 228 sdot 10minus3

2 923 sdot 10minus6

0 4 632 sdot 10minus5

1 3 862 sdot 10minus8

5 623 sdot 10minus6

4 107 sdot 10minus9

Table 2 Maximum absolute error 119864 for Example 2

119896 119872 119864 119896 119872 119864

3 252 sdot 10minus5

2 490 sdot 10minus8

0 4 211 sdot 10minus6

1 3 255 sdot 10minus9

5 901 sdot 10minus8

4 350 sdot 10minus10

(35) then the two operational matrices D and D2 are givenrespectively by

D = (0 0 0

4 0 0

0 8 0

) D2 = (0 0 0

0 0 0

32 0 0

) (56)

Moreover Ψ(119909) can be evaluated to give

Ψ (119909) = radic2

120587(

2

8119909 minus 4

321199092

minus 32119909 + 6

) (57)

If we set

119862 = (11988800 11988801 11988802)119879

= radic120587

2(1198880 1198881 1198882)119879

(58)

then (55) takes the form

4119862119879D2Ψ (119909) minus 2(119862119879DΨ (119909))

2

+ 119862119879

Ψ (119909) = 0 (59)

which is equivalent to

1198880minus 21198881+ 131119888

2minus 641198882

1+ 512119888

11198882minus 1024119888

2

2

+ 4 (1198881minus 41198882minus 256119888

11198882+ 1024119888

2

2) 119909

+ 16 (1198882minus 256119888

2

2) 1199092

= 0

(60)

We only need to satisfy this equation at the first root of119880lowast3(119909)

that is at 1199091= (2 minus radic2)4 to get

1198880minus 641198882

1+ radic2119888

1(2561198882minus 1) + (129 minus 512119888

2) 1198882= 0 (61)

Furthermore the use of initial conditions in (55) leads to thetwo equations

21198880minus 41198881+ 61198882= minus1

1198881minus 41198882= 0

(62)

The solution of the nonlinear system of (61)-(62) gives

1198880=minus123

256 119888

1=1

64 119888

2=1

256 (63)

Abstract and Applied Analysis 7

and consequently

119910 (119909) = (minus123

2561

641

256)(

2

8119909 minus 4

321199092

minus 32119909 + 6

)

=1199092

8minus 1

(64)

which is the exact solution

Example 4 Consider the following nonlinear second-orderboundary value problem

11991010158401015840

(119909) + (1199101015840

(119909))2

minus 119910 (119909) = sinh2119909

119910 (0) = 1 119910 (1) = cosh 1(65)

with the exact solution 119910(119909) = cosh 119909In Table 3 we list the maximum absolute error 119864 for

various values of 119896 and119872

Example 5 Consider the following linear singular initialvalue problem (see [23])

11991010158401015840

(119909) + 119891 (119909) 1199101015840

(119909) + 119910 (119909) = 119892 (119909)

119910 (0) = 0 1199101015840

(0) = 1

(66)

where

119891 (119909) =

minus4119909 +4

3 0 ⩽ 119909 lt

1

4

119909 minus1

3

1

4⩽ 119909 ⩽ 1

(67)

and 119892(119909) is chosen such that the exact solution of (66) is119910(119909) = 119890

119909 sin119909(1 + 1199092) We apply the algorithm describedin Section 41 with 119896 = 2 In Table 4 we list the maximumabsolute error for various values of119872 while in Figure 1 theexact solution is comparedwith the three numerical solutionscorresponding to 119896 = 2119872 = 2 119896 = 2119872 = 4 119896 = 2 and119872 = 6 and also in Table 5 we give a comparison between theerror obtained in [23] and the present method 119896 = 2119872 = 8

Example 6 Consider the following Bratu equation (see [17ndash20])

11991010158401015840

(119909) + 120582119890119910(119909)

= 0 119910 (0) = 119910 (1) = 0 0 ⩽ 119909 ⩽ 1 (68)

with the analytical solution

119910 (119909) = minus2 ln [cosh ((1205794) (2119909 minus 1))cosh (1205794)

] (69)

where 120579 is the solution of the nonlinear equation 120579 =

radic2120582 cosh 120579The presented algorithm in Section 42 is appliedto numerically solve (68) for the three cases correspondingto 120582 = 1 2 and 351 which yield 120579 = 151716 235755 and466781 respectively In Table 6 the maximum absolute error119864 is listed for 119896 = 0 and various values of119872 while in Table 7we give a comparison between the best errors obtained by

Table 3 Maximum absolute error 119864 for Example 4

119896 119872 119864 119896 119872 119864

03 569 sdot 10

minus4

1 2 112 sdot 10minus6

4 135 sdot 10minus5

3 197 sdot 10minus8

Table 4 Maximum absolute error 119864 for Example 5

119872 119864 119872 119864

2 298 sdot 10minus4 6 185 sdot 10

minus10

4 170 sdot 10minus5 8 912 sdot 10

minus11

Table 5 Comparison between different errors for Example 5

119909 02 04 06 08

LegendreWavelets [23] 430 sdot 10

minus12

124 sdot 10minus11

297 sdot 10minus12

435 sdot 10minus12

SCWOMD 647 sdot 10minus13

121 sdot 10minus12

174 sdot 10minus12

232 sdot 10minus12

Table 6 Maximum absolute error 119864 for Example 6

119872 120582 119864 120582 119864 120582 119864

10

1

2162 sdot 10minus7

2

2758 sdot 10minus5

351

7658 sdot 10minus4

12 4453 sdot 10minus9

8082 sdot 10minus6

8665 sdot 10minus5

14 8461 sdot 10minus11

1461 sdot 10minus9

2384 sdot 10minus8

16 1546 sdot 10minus12

5342 sdot 10minus10

6587 sdot 10minus9

18 2782 sdot 10minus14

1644 sdot 10minus11

2387 sdot 10minus10

20 7216 sdot 10minus16

4024 sdot 10minus12

1058 sdot 10minus10

Table 7 Comparison between the best errors for Example 6 for 119896 =0

SCWOMDError in [17] Error in [18] Error in [19] Error in [20]722 sdot 10

minus16

102 sdot 10minus6

889 sdot 10minus6

135 sdot 10minus5

301 sdot 10minus3

our method and some other numerical methods used tosolve Example 6 This table shows that our method is moreaccurate if compared with the methods developed in [17ndash20] In addition Figure 2 illustrates a comparison betweendifferent solutions obtained by our algorithm in case of 120582 = 1119896 = 0 and119872 = 1 2 3

Remark 6 It is worth noting here that the obtained numericalresults in the previous six solved examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecompared favorably with the known analytical solutions

6 Concluding Remarks

In this paper an algorithm for obtaining a numerical spectralsolution for second-order linear and nonlinear boundaryvalue problems is discussed The derivation of this algorithmis essentially based on constructing the shifted second kindChebyshev wavelets operational matrix of differentiationOne of the main advantages of the presented algorithm isits availability for application on both linear and nonlin-ear second-order boundary value problems including some

8 Abstract and Applied Analysis

Exactk = 2M = 2

k = 2M = 4

k = 2M = 6

x

00 02 04 06 08 10002

005

010

020

050

100

y(x)

Figure 1 Numerical and exact solutions of Example 5

00 02 04 06 08 100010

0100

0050

0020

0030

0015

0150

0070

Exactk = 0M = 1

k = 0M = 2

k = 0M = 3

x

y(x)

Figure 2 Different solutions of Example 6

singular equations and also Bratu type equations Anotheradvantage of the developed algorithm is that high accurateapproximate solutions are achieved using a small number ofthe second kindChebyshev waveletsThe obtained numericalresults are compared favorably with the analytical ones

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support Also they would like to thank the referee forhis valuable comments and suggestions which improved thepaper into its present form

References

[1] EHDoha andWMAbd-Elhameed ldquoEfficient spectral-Galer-kin algorithms for direct solution of second-order equationsusing ultraspherical polynomialsrdquo SIAM Journal on ScientificComputing vol 24 no 2 pp 548ndash571 2002

[2] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[4] E H Doha W M Abd-Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[5] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourth kindsrdquo Abstract and Applied Analysis vol2013 Article ID 542839 9 pages 2013

[6] A Constantmldes Applied Numerical Methods With PersonalComputers McGraw-Hill New York NY USA 1987

[7] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[8] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013

[9] N Daoudi-Merzagui F Derrab and A Boucherif ldquoSubhar-monic solutions of nonautonomous second order differentialequations with singular nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 903281 20 pages 2012

[10] M Turkyilmazoglu ldquoEffective computation of exact and ana-lytic approximate solutions to singular nonlinear equationsof Lane-Emden-Fowler typerdquo Applied Mathematical Modellingvol 37 pp 7539ndash7548 2013

[11] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[12] G Bratu ldquoSur les equations integrales non lineairesrdquo Bulletin dela Societe Mathematique de France vol 42 pp 113ndash142 1914

[13] I M Gelrsquofand ldquoSome problems in the theory of quasilinearequationsrdquoAmericanMathematical Society Translations vol 29pp 295ndash381 1963

[14] A Serghini Mounim and B M de Dormale ldquoFrom the fittingtechniques to accurate schemes for the Liouville-Bratu-Gelfandproblemrdquo Numerical Methods for Partial Differential Equationsvol 22 no 4 pp 761ndash775 2006

[15] M I Syam and A Hamdan ldquoAn efficient method for solvingBratu equationsrdquo Applied Mathematics and Computation vol176 no 2 pp 704ndash713 2006

[16] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[17] S Abbasbandy M S Hashemi and C-S Liu ldquoThe Lie-groupshooting method for solving the Bratu equationrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no11 pp 4238ndash4249 2011

[18] H Caglar N Caglar M Ozer A Valarıstos and A N Anag-nostopoulos ldquoB-spline method for solving Bratursquos problemrdquoInternational Journal of Computer Mathematics vol 87 no 8pp 1885ndash1891 2010

Abstract and Applied Analysis 9

[19] S A Khuri ldquoA new approach to Bratursquos problemrdquo AppliedMathematics and Computation vol 147 no 1 pp 131ndash136 2004

[20] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvingboundary value problemsrdquo Journal of Computational Physicsvol 159 no 2 pp 125ndash138 2000

[21] J P Boyd ldquoOne-point pseudospectral collocation for the one-dimensional Bratu equationrdquoAppliedMathematics and Compu-tation vol 217 no 12 pp 5553ndash5565 2011

[22] H Jafari S A Yousefi M A Firoozjaee S Momani and CM Khalique ldquoApplication of Legendre wavelets for solvingfractional differential equationsrdquo Computers amp Mathematicswith Applications vol 62 no 3 pp 1038ndash1045 2011

[23] F Mohammadi and M M Hosseini ldquoA new Legendre waveletoperational matrix of derivative and its applications in solvingthe singular ordinary differential equationsrdquo Journal of theFranklin Institute vol 348 no 8 pp 1787ndash1796 2011

[24] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[25] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[26] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[27] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[28] M Dehghan and A Saadatmandi ldquoChebyshev finite differencemethod for Fredholm integro-differential equationrdquo Interna-tional Journal of Computer Mathematics vol 85 no 1 pp 123ndash130 2008

[29] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[30] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[31] Y Li ldquoSolving a nonlinear fractional differential equation usingChebyshev waveletsrdquoCommunications in Nonlinear Science andNumerical Simulation vol 15 no 9 pp 2284ndash2292 2010

[32] H Saeedi M Mohseni Moghadam N Mollahasani and G NChuev ldquoA CAS wavelet method for solving nonlinear Fredholmintegro-differential equations of fractional orderrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no3 pp 1154ndash1163 2011

[33] H Saeedi and M M Moghadam ldquoNumerical solution ofnonlinear Volterra integro-differential equations of arbitraryorder by CAS waveletsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 3 pp 1216ndash1226 2011

[34] C F Cheng Y T Tsay and T T Wu ldquoWalsh operational matri-ces for fractional calculus and their application to distributedsystemsrdquo Journal of the Franklin Institute vol 303 no 3 pp 267ndash284 1977

[35] R Y Chang and M L Wang ldquoShifted Legendre direct methodfor variational problemsrdquo Journal of Optimization Theory andApplications vol 39 no 2 pp 299ndash307 1983

[36] I R Horng and J H Chou ldquoShifted Chebyshev direct methodfor solving variational problemsrdquo International Journal of Sys-tems Science vol 16 no 7 pp 855ndash861 1985

[37] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996

[38] F Khellat and S A Yousefi ldquoThe linear Legendre motherwavelets operational matrix of integration and its applicationrdquoJournal of the Franklin Institute vol 343 no 2 pp 181ndash190 2006

[39] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp HallCRC Boca Raton Fla USA 2003

[40] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[41] M R Eslahchi M Dehghan and S Ahmadi ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

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 Abstract and Applied Analysis

where

120574119898(120579) =

sin 120579119898(sin (119898 minus 1) 120579119898 minus 1

minussin (119898 + 1) 120579119898 + 1

)

minussin 120579119898 + 2

(sin (119898 + 1) 120579119898 + 1

minussin (119898 + 3) 120579119898 + 3

)

(20)

Therefore we have

10038161003816100381610038161198881198991198981003816100381610038161003816 =10038161003816100381610038161003816100381610038161003816

1

2(5119896minus1)2radic120587int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

=1

2(5119896minus1)2radic120587

10038161003816100381610038161003816100381610038161003816int120587

0

11989110158401015840

(cos 120579 + 1198992119896

)120574119898(120579) 119889120579

10038161003816100381610038161003816100381610038161003816

⩽119871

2(5119896minus1)2radic120587int120587

0

1003816100381610038161003816120574119898 (120579)1003816100381610038161003816 119889120579

⩽119871radic120587

2(5119896minus1)2

times [1

119898(1

119898 minus 1+

1

119898 + 1)

+1

119898 + 2(1

119898 + 1+

1

119898 + 3)]

=119871radic120587

2(5119896minus5)2 (1198982 + 2119898 minus 3)

lt2119871radic120587

2(5119896minus5)2(119898 + 1)2

(21)

Since 119899 ⩽ 2119896 minus 1 we have

10038161003816100381610038161198881198991198981003816100381610038161003816 lt

8radic2120587119871

(119899 + 1)52

(119898 + 1)2

(22)

and this completes the proof of Theorem 3

A shifted second kind Chebyshev wavelets operationalmatrix of the first derivative is stated andproved in the follow-ing theorem

Theorem 4 Let Ψ(119905) be the second kind Chebyshev waveletsvector defined in (15) Then the first derivative of the vectorΨ(119905) can be expressed as

119889Ψ (119905)

119889119905= DΨ (119905) (23)

where D is 2119896(119872 + 1) square operational matrix of derivativeand is defined by

D =(

(

119865 0 0

0 119865 0

0 0 119865

)

)

(24)

in which 119865 is an (119872 + 1) square matrix whose (119903 119904)th elementis defined by

119865119903119904= 2119896+2

119904 119903 ⩾ 2 119903 gt 119904 (119903 + 119904) odd0 otherwise

(25)

Proof If we make use of the shifted second kind Chebyshevpolynomials then the 119903th element of the vector Ψ(119905) definedin (15) can be written in the following form

Ψ119903(119905) = 120595

119899119898(119905) =

2(119896+3)2

radic120587119880lowast

119898(2119896

119905 minus 119899) 120594[1198992119896(119899+1)2

119896]

119894 = 1 2 2119896

(119872 + 1)

(26)

where 119903 = 119899(119872 + 1) + (119898 + 1) 119898 = 0 1 119872 119899 =0 1 (2

119896

minus 1) and

120594[1198992119896(119899+1)2

119896]=

1 119905 isin [119899

2119896119899 + 1

2119896]

0 otherwise(27)

If we differentiate (26) with respect to 119905 then we get

119889Ψ119903(119905)

119889119905=2(2119896+3)2

radic120587

119889

119889119905[119880lowast

119898(2119896

119905 minus 119899)] 120594[1198992119896(119899+1)2

119896] (28)

It is to be noted here that the RHS of (28) is zero outside theinterval [1198992119896 (119899 + 1)2119896] hence its second kind Chebyshevwavelets expansion only has those elements in Ψ(119905) that arenonzero in the interval [1198992119896 (119899 + 1)2119896] that is Ψ

119894(119905) 119894 =

119899(119872 + 1) + 1 119899(119872 + 1) + 2 119899(119872 + 1) + 119872 + 1 and as aconsequence this enables one to expand (119889Ψ

119903(119905)119889119905) in terms

of shifted second kind Chebyshev wavelets Ψ119894(119905) 119894 = 119899(119872 +

1) + 1 119899(119872 + 1) + 2 119899(119872 + 1) +119872 + 1 in the form

119889Ψ119903(119905)

119889119905=

(119899+1)(119872+1)

sum119894=119899(119872+1)+1

119886119894Ψ119894(119905) (29)

and accordingly (28) implies that the operational matrixD isa block matrix defined as in (24) Moreover we have

119889119880lowast

0(119905)

119889119905= 0 (30)

and this implies that119889

119889119905Ψ119903(119905) = 0

for 119903 = 1 (119872 + 1) + 1 2 (119872 + 1) + 1

(2119896

minus 1) (119872 + 1) + 1

(31)

Consequently the elements of the first row of matrix 119865defined in (24) are zeros Now andwith the aid of Corollary 2the first derivative of Ψ

119903(119905)may be expressed in the form

119889Ψ119903(119905)

119889119905

=2(2119896+7)2

radic120587

119898minus1

sum119895=0

(119895+119898) odd

(119895 + 1)119880lowast

119895(2119896

119905 minus 119899) 120594[1198992119896(119899+1)2

119896](32)

Abstract and Applied Analysis 5

Expanding this equation in terms of second kind Chebyshevwavelets basis we get

119889Ψ119903(119905)

119889119905= 2119896+2

119898minus1

sum119904=0

(119904+119898) odd

119904Ψ119899(119872+1)+119904

(119905) (33)

so if 119865119903119904

are chosen as

119865119903119904=

2119896+2

119904 119903 = 2 119872 + 1 119904 = 1 119903 minus 1

(119903 + 119904) odd0 otherwise

(34)

then (23) is obtained and the proof of the theorem iscompleted

Corollary 5 The operational matrix for the 119899th derivative canbe obtained from

119889119899

Ψ (119905)

119889119905119899= D119899Ψ (119905) 119899 = 1 2 (35)

whereD119899 is the 119899th power of matrix D

4 Second-Order Two-Point BoundaryValue Problems

In this section we are interested in solving linear and nonlin-ear two-point boundary value problems subject to homoge-nous or nonhomogenous initial or boundary conditionsbased on the wavelets operational matrices constructed inSection 3

41 Linear Second-Order Two-Point Boundary Value Prob-lems Consider the linear second-order differential equation

11991010158401015840

(119909) + 1198911(119909) 1199101015840

(119909) + 1198912(119909) 119910 (119909) = 119892 (119909) 119909 isin [0 1]

(36)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (37)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (38)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (39)

If we approximate 119910(119909)1198911(119909)1198912(119909) and 119892(119909) in terms of the

second kind Chebyshev wavelets basis then one can write

119910 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119888119899119898120595119899119898(119909) = 119862

119879

Ψ (119909)

1198911(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

1Ψ (119909)

1198912(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

2Ψ (119905)

119892 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119892119899119898120595119899119898(119909) = 119866

119879

Ψ (119909)

(40)

where 119862119879 1198651198791 1198651198792 and 119866119879 are defined similarly as in (14)

Relations (23) and (35) enable one to approximate 1199101015840(119909) and11991010158401015840

(119909) as

1199101015840

(119909) ≃ 119862119879DΨ (119909) 119910

10158401015840

(119909) ≃ 119862119879D2Ψ (119909) (41)

Now substitution of relations (40) and (41) into (36) enablesus to define the residual 119877(119909) of this equation as

119877 (119909) = 119862119879D2Ψ (119909) + 119865119879

1Ψ (119909) (Ψ (119909))

119879D119879119862

+ 119865119879

2Ψ (119909) (Ψ (119909))

119879

119862 minus 119866119879

Ψ (119909) (42)

and application of the Petrov-Galerkin method (see [40])yields the following (2119896(119872 + 1) minus 2) linear equations in theunknown expansion coefficients 119888

119899119898 namely

int1

0

radic119909 minus 1199092Ψ119895(119909) 119877 (119909) 119889119909 = 0

119895 = 1 2 2119896

(119872 + 1) minus 2

(43)

Moreover the initial conditions (37) the boundary condi-tions (38) and the mixed boundary conditions (39) leadrespectively to the following equations

119862119879

Ψ (0) = 120572 119862119879DΨ (0) = 120573 (44)

119862119879

Ψ (0) = 120572 119862119879

Ψ (1) = 120573 (45)

1198861119862119879

Ψ (0) + 1198862119862119879DΨ (0) = 120572

1198871119862119879

Ψ (1) + 1198872119862119879DΨ (1) = 120573

(46)

Thus (43) with the two equations of (44) or (45) or (46)generates 2119896(119872 + 1) a set of linear equations which can besolved for the unknown components of the vector 119862 andhence an approximate spectral wavelets solution to119910(119909) givenin (40) can be obtained

42 Nonlinear Second-Order Two-Point Boundary Value Prob-lems Consider the nonlinear differential equation

11991010158401015840

(119909) = 119865 (119909 119910 (119909) 1199101015840

(119909)) (47)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (48)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (49)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (50)

6 Abstract and Applied Analysis

If we follow the same procedure of Section 41 approximate119910(119909) as in (40) and make use of (23) and (35) then we obtain

119862119879D2Ψ (119909) = 119865 (119909 119910 (119909) 119862119879DΨ (119909)) (51)

To find an approximate solution to 119910(119909) we compute (51)at the first 2119896(119872+1)minus2 roots of119880lowast

2119896(119872+1)

(119909)These equationswith (44) or (45) or (46) generate 2119896(119872 + 1) nonlinearequations in the expansion coefficients 119888

119899119898 which can be

solved with the aid of Newtonrsquos iterative method

5 Numerical Results and Discussions

In this section the presented algorithms in Section 4 areapplied to solve linear and nonlinear second-order initialand boundary value problems as well as Bratursquos equationsSome examples subject to different initial boundary andmixed boundary conditions are considered to illustrate theefficiency and the applicability of the proposed algorithmsAll computations are performed by using Mathematica 8

Example 1 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) + (1 minus 119909) 1199101015840

(119909) + 2119910 (119909) = (1 + 2119909 minus 1199092

) sin119909

0 lt 119909 lt 1 119910 (0) = 1 119910 (1) = 0

(52)

with the exact solution 119910(119909) = (1 minus 119909) cos119909In Table 1 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 2 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) minus sin1199091199101015840 (119909) + 119910 (119909) = sin2119909 0 lt 119909 lt 1 (53)

subject to the mixed boundary conditions

119910 (0) + 1199101015840

(0) = 1 sin (1) 119910 (1) + cos (1) 1199101015840 (1) = 0(54)

with the exact solution 119910(119909) = cos119909In Table 2 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 3 Consider the following nonlinear initial valueproblem (see [41])

411991010158401015840

(119909) minus 2(1199101015840

(119909))2

+ 119910 (119909) = 0

0 lt 119909 lt 1 119910 (0) = minus1 1199101015840

(0) = 0

(55)

with the exact solution 119910(119909) = (11990928) minus 1We solve (55) using the algorithmdescribed in Section 42

for the case corresponds to 119872 = 2 119896 = 0 to obtain anapproximate solution of119910(119909) First if wemake use of (24) and

Table 1 Maximum absolute error 119864 for Example 1

119896 119872 119864 119896 119872 119864

3 228 sdot 10minus3

2 923 sdot 10minus6

0 4 632 sdot 10minus5

1 3 862 sdot 10minus8

5 623 sdot 10minus6

4 107 sdot 10minus9

Table 2 Maximum absolute error 119864 for Example 2

119896 119872 119864 119896 119872 119864

3 252 sdot 10minus5

2 490 sdot 10minus8

0 4 211 sdot 10minus6

1 3 255 sdot 10minus9

5 901 sdot 10minus8

4 350 sdot 10minus10

(35) then the two operational matrices D and D2 are givenrespectively by

D = (0 0 0

4 0 0

0 8 0

) D2 = (0 0 0

0 0 0

32 0 0

) (56)

Moreover Ψ(119909) can be evaluated to give

Ψ (119909) = radic2

120587(

2

8119909 minus 4

321199092

minus 32119909 + 6

) (57)

If we set

119862 = (11988800 11988801 11988802)119879

= radic120587

2(1198880 1198881 1198882)119879

(58)

then (55) takes the form

4119862119879D2Ψ (119909) minus 2(119862119879DΨ (119909))

2

+ 119862119879

Ψ (119909) = 0 (59)

which is equivalent to

1198880minus 21198881+ 131119888

2minus 641198882

1+ 512119888

11198882minus 1024119888

2

2

+ 4 (1198881minus 41198882minus 256119888

11198882+ 1024119888

2

2) 119909

+ 16 (1198882minus 256119888

2

2) 1199092

= 0

(60)

We only need to satisfy this equation at the first root of119880lowast3(119909)

that is at 1199091= (2 minus radic2)4 to get

1198880minus 641198882

1+ radic2119888

1(2561198882minus 1) + (129 minus 512119888

2) 1198882= 0 (61)

Furthermore the use of initial conditions in (55) leads to thetwo equations

21198880minus 41198881+ 61198882= minus1

1198881minus 41198882= 0

(62)

The solution of the nonlinear system of (61)-(62) gives

1198880=minus123

256 119888

1=1

64 119888

2=1

256 (63)

Abstract and Applied Analysis 7

and consequently

119910 (119909) = (minus123

2561

641

256)(

2

8119909 minus 4

321199092

minus 32119909 + 6

)

=1199092

8minus 1

(64)

which is the exact solution

Example 4 Consider the following nonlinear second-orderboundary value problem

11991010158401015840

(119909) + (1199101015840

(119909))2

minus 119910 (119909) = sinh2119909

119910 (0) = 1 119910 (1) = cosh 1(65)

with the exact solution 119910(119909) = cosh 119909In Table 3 we list the maximum absolute error 119864 for

various values of 119896 and119872

Example 5 Consider the following linear singular initialvalue problem (see [23])

11991010158401015840

(119909) + 119891 (119909) 1199101015840

(119909) + 119910 (119909) = 119892 (119909)

119910 (0) = 0 1199101015840

(0) = 1

(66)

where

119891 (119909) =

minus4119909 +4

3 0 ⩽ 119909 lt

1

4

119909 minus1

3

1

4⩽ 119909 ⩽ 1

(67)

and 119892(119909) is chosen such that the exact solution of (66) is119910(119909) = 119890

119909 sin119909(1 + 1199092) We apply the algorithm describedin Section 41 with 119896 = 2 In Table 4 we list the maximumabsolute error for various values of119872 while in Figure 1 theexact solution is comparedwith the three numerical solutionscorresponding to 119896 = 2119872 = 2 119896 = 2119872 = 4 119896 = 2 and119872 = 6 and also in Table 5 we give a comparison between theerror obtained in [23] and the present method 119896 = 2119872 = 8

Example 6 Consider the following Bratu equation (see [17ndash20])

11991010158401015840

(119909) + 120582119890119910(119909)

= 0 119910 (0) = 119910 (1) = 0 0 ⩽ 119909 ⩽ 1 (68)

with the analytical solution

119910 (119909) = minus2 ln [cosh ((1205794) (2119909 minus 1))cosh (1205794)

] (69)

where 120579 is the solution of the nonlinear equation 120579 =

radic2120582 cosh 120579The presented algorithm in Section 42 is appliedto numerically solve (68) for the three cases correspondingto 120582 = 1 2 and 351 which yield 120579 = 151716 235755 and466781 respectively In Table 6 the maximum absolute error119864 is listed for 119896 = 0 and various values of119872 while in Table 7we give a comparison between the best errors obtained by

Table 3 Maximum absolute error 119864 for Example 4

119896 119872 119864 119896 119872 119864

03 569 sdot 10

minus4

1 2 112 sdot 10minus6

4 135 sdot 10minus5

3 197 sdot 10minus8

Table 4 Maximum absolute error 119864 for Example 5

119872 119864 119872 119864

2 298 sdot 10minus4 6 185 sdot 10

minus10

4 170 sdot 10minus5 8 912 sdot 10

minus11

Table 5 Comparison between different errors for Example 5

119909 02 04 06 08

LegendreWavelets [23] 430 sdot 10

minus12

124 sdot 10minus11

297 sdot 10minus12

435 sdot 10minus12

SCWOMD 647 sdot 10minus13

121 sdot 10minus12

174 sdot 10minus12

232 sdot 10minus12

Table 6 Maximum absolute error 119864 for Example 6

119872 120582 119864 120582 119864 120582 119864

10

1

2162 sdot 10minus7

2

2758 sdot 10minus5

351

7658 sdot 10minus4

12 4453 sdot 10minus9

8082 sdot 10minus6

8665 sdot 10minus5

14 8461 sdot 10minus11

1461 sdot 10minus9

2384 sdot 10minus8

16 1546 sdot 10minus12

5342 sdot 10minus10

6587 sdot 10minus9

18 2782 sdot 10minus14

1644 sdot 10minus11

2387 sdot 10minus10

20 7216 sdot 10minus16

4024 sdot 10minus12

1058 sdot 10minus10

Table 7 Comparison between the best errors for Example 6 for 119896 =0

SCWOMDError in [17] Error in [18] Error in [19] Error in [20]722 sdot 10

minus16

102 sdot 10minus6

889 sdot 10minus6

135 sdot 10minus5

301 sdot 10minus3

our method and some other numerical methods used tosolve Example 6 This table shows that our method is moreaccurate if compared with the methods developed in [17ndash20] In addition Figure 2 illustrates a comparison betweendifferent solutions obtained by our algorithm in case of 120582 = 1119896 = 0 and119872 = 1 2 3

Remark 6 It is worth noting here that the obtained numericalresults in the previous six solved examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecompared favorably with the known analytical solutions

6 Concluding Remarks

In this paper an algorithm for obtaining a numerical spectralsolution for second-order linear and nonlinear boundaryvalue problems is discussed The derivation of this algorithmis essentially based on constructing the shifted second kindChebyshev wavelets operational matrix of differentiationOne of the main advantages of the presented algorithm isits availability for application on both linear and nonlin-ear second-order boundary value problems including some

8 Abstract and Applied Analysis

Exactk = 2M = 2

k = 2M = 4

k = 2M = 6

x

00 02 04 06 08 10002

005

010

020

050

100

y(x)

Figure 1 Numerical and exact solutions of Example 5

00 02 04 06 08 100010

0100

0050

0020

0030

0015

0150

0070

Exactk = 0M = 1

k = 0M = 2

k = 0M = 3

x

y(x)

Figure 2 Different solutions of Example 6

singular equations and also Bratu type equations Anotheradvantage of the developed algorithm is that high accurateapproximate solutions are achieved using a small number ofthe second kindChebyshev waveletsThe obtained numericalresults are compared favorably with the analytical ones

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support Also they would like to thank the referee forhis valuable comments and suggestions which improved thepaper into its present form

References

[1] EHDoha andWMAbd-Elhameed ldquoEfficient spectral-Galer-kin algorithms for direct solution of second-order equationsusing ultraspherical polynomialsrdquo SIAM Journal on ScientificComputing vol 24 no 2 pp 548ndash571 2002

[2] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[4] E H Doha W M Abd-Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[5] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourth kindsrdquo Abstract and Applied Analysis vol2013 Article ID 542839 9 pages 2013

[6] A Constantmldes Applied Numerical Methods With PersonalComputers McGraw-Hill New York NY USA 1987

[7] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[8] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013

[9] N Daoudi-Merzagui F Derrab and A Boucherif ldquoSubhar-monic solutions of nonautonomous second order differentialequations with singular nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 903281 20 pages 2012

[10] M Turkyilmazoglu ldquoEffective computation of exact and ana-lytic approximate solutions to singular nonlinear equationsof Lane-Emden-Fowler typerdquo Applied Mathematical Modellingvol 37 pp 7539ndash7548 2013

[11] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[12] G Bratu ldquoSur les equations integrales non lineairesrdquo Bulletin dela Societe Mathematique de France vol 42 pp 113ndash142 1914

[13] I M Gelrsquofand ldquoSome problems in the theory of quasilinearequationsrdquoAmericanMathematical Society Translations vol 29pp 295ndash381 1963

[14] A Serghini Mounim and B M de Dormale ldquoFrom the fittingtechniques to accurate schemes for the Liouville-Bratu-Gelfandproblemrdquo Numerical Methods for Partial Differential Equationsvol 22 no 4 pp 761ndash775 2006

[15] M I Syam and A Hamdan ldquoAn efficient method for solvingBratu equationsrdquo Applied Mathematics and Computation vol176 no 2 pp 704ndash713 2006

[16] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[17] S Abbasbandy M S Hashemi and C-S Liu ldquoThe Lie-groupshooting method for solving the Bratu equationrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no11 pp 4238ndash4249 2011

[18] H Caglar N Caglar M Ozer A Valarıstos and A N Anag-nostopoulos ldquoB-spline method for solving Bratursquos problemrdquoInternational Journal of Computer Mathematics vol 87 no 8pp 1885ndash1891 2010

Abstract and Applied Analysis 9

[19] S A Khuri ldquoA new approach to Bratursquos problemrdquo AppliedMathematics and Computation vol 147 no 1 pp 131ndash136 2004

[20] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvingboundary value problemsrdquo Journal of Computational Physicsvol 159 no 2 pp 125ndash138 2000

[21] J P Boyd ldquoOne-point pseudospectral collocation for the one-dimensional Bratu equationrdquoAppliedMathematics and Compu-tation vol 217 no 12 pp 5553ndash5565 2011

[22] H Jafari S A Yousefi M A Firoozjaee S Momani and CM Khalique ldquoApplication of Legendre wavelets for solvingfractional differential equationsrdquo Computers amp Mathematicswith Applications vol 62 no 3 pp 1038ndash1045 2011

[23] F Mohammadi and M M Hosseini ldquoA new Legendre waveletoperational matrix of derivative and its applications in solvingthe singular ordinary differential equationsrdquo Journal of theFranklin Institute vol 348 no 8 pp 1787ndash1796 2011

[24] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[25] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[26] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[27] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[28] M Dehghan and A Saadatmandi ldquoChebyshev finite differencemethod for Fredholm integro-differential equationrdquo Interna-tional Journal of Computer Mathematics vol 85 no 1 pp 123ndash130 2008

[29] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[30] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[31] Y Li ldquoSolving a nonlinear fractional differential equation usingChebyshev waveletsrdquoCommunications in Nonlinear Science andNumerical Simulation vol 15 no 9 pp 2284ndash2292 2010

[32] H Saeedi M Mohseni Moghadam N Mollahasani and G NChuev ldquoA CAS wavelet method for solving nonlinear Fredholmintegro-differential equations of fractional orderrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no3 pp 1154ndash1163 2011

[33] H Saeedi and M M Moghadam ldquoNumerical solution ofnonlinear Volterra integro-differential equations of arbitraryorder by CAS waveletsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 3 pp 1216ndash1226 2011

[34] C F Cheng Y T Tsay and T T Wu ldquoWalsh operational matri-ces for fractional calculus and their application to distributedsystemsrdquo Journal of the Franklin Institute vol 303 no 3 pp 267ndash284 1977

[35] R Y Chang and M L Wang ldquoShifted Legendre direct methodfor variational problemsrdquo Journal of Optimization Theory andApplications vol 39 no 2 pp 299ndash307 1983

[36] I R Horng and J H Chou ldquoShifted Chebyshev direct methodfor solving variational problemsrdquo International Journal of Sys-tems Science vol 16 no 7 pp 855ndash861 1985

[37] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996

[38] F Khellat and S A Yousefi ldquoThe linear Legendre motherwavelets operational matrix of integration and its applicationrdquoJournal of the Franklin Institute vol 343 no 2 pp 181ndash190 2006

[39] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp HallCRC Boca Raton Fla USA 2003

[40] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[41] M R Eslahchi M Dehghan and S Ahmadi ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

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

Abstract and Applied Analysis 5

Expanding this equation in terms of second kind Chebyshevwavelets basis we get

119889Ψ119903(119905)

119889119905= 2119896+2

119898minus1

sum119904=0

(119904+119898) odd

119904Ψ119899(119872+1)+119904

(119905) (33)

so if 119865119903119904

are chosen as

119865119903119904=

2119896+2

119904 119903 = 2 119872 + 1 119904 = 1 119903 minus 1

(119903 + 119904) odd0 otherwise

(34)

then (23) is obtained and the proof of the theorem iscompleted

Corollary 5 The operational matrix for the 119899th derivative canbe obtained from

119889119899

Ψ (119905)

119889119905119899= D119899Ψ (119905) 119899 = 1 2 (35)

whereD119899 is the 119899th power of matrix D

4 Second-Order Two-Point BoundaryValue Problems

In this section we are interested in solving linear and nonlin-ear two-point boundary value problems subject to homoge-nous or nonhomogenous initial or boundary conditionsbased on the wavelets operational matrices constructed inSection 3

41 Linear Second-Order Two-Point Boundary Value Prob-lems Consider the linear second-order differential equation

11991010158401015840

(119909) + 1198911(119909) 1199101015840

(119909) + 1198912(119909) 119910 (119909) = 119892 (119909) 119909 isin [0 1]

(36)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (37)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (38)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (39)

If we approximate 119910(119909)1198911(119909)1198912(119909) and 119892(119909) in terms of the

second kind Chebyshev wavelets basis then one can write

119910 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119888119899119898120595119899119898(119909) = 119862

119879

Ψ (119909)

1198911(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

1Ψ (119909)

1198912(119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119891119899119898120595119899119898(119909) = 119865

119879

2Ψ (119905)

119892 (119909) ≃

2119896minus1

sum119899=0

119872

sum119898=0

119892119899119898120595119899119898(119909) = 119866

119879

Ψ (119909)

(40)

where 119862119879 1198651198791 1198651198792 and 119866119879 are defined similarly as in (14)

Relations (23) and (35) enable one to approximate 1199101015840(119909) and11991010158401015840

(119909) as

1199101015840

(119909) ≃ 119862119879DΨ (119909) 119910

10158401015840

(119909) ≃ 119862119879D2Ψ (119909) (41)

Now substitution of relations (40) and (41) into (36) enablesus to define the residual 119877(119909) of this equation as

119877 (119909) = 119862119879D2Ψ (119909) + 119865119879

1Ψ (119909) (Ψ (119909))

119879D119879119862

+ 119865119879

2Ψ (119909) (Ψ (119909))

119879

119862 minus 119866119879

Ψ (119909) (42)

and application of the Petrov-Galerkin method (see [40])yields the following (2119896(119872 + 1) minus 2) linear equations in theunknown expansion coefficients 119888

119899119898 namely

int1

0

radic119909 minus 1199092Ψ119895(119909) 119877 (119909) 119889119909 = 0

119895 = 1 2 2119896

(119872 + 1) minus 2

(43)

Moreover the initial conditions (37) the boundary condi-tions (38) and the mixed boundary conditions (39) leadrespectively to the following equations

119862119879

Ψ (0) = 120572 119862119879DΨ (0) = 120573 (44)

119862119879

Ψ (0) = 120572 119862119879

Ψ (1) = 120573 (45)

1198861119862119879

Ψ (0) + 1198862119862119879DΨ (0) = 120572

1198871119862119879

Ψ (1) + 1198872119862119879DΨ (1) = 120573

(46)

Thus (43) with the two equations of (44) or (45) or (46)generates 2119896(119872 + 1) a set of linear equations which can besolved for the unknown components of the vector 119862 andhence an approximate spectral wavelets solution to119910(119909) givenin (40) can be obtained

42 Nonlinear Second-Order Two-Point Boundary Value Prob-lems Consider the nonlinear differential equation

11991010158401015840

(119909) = 119865 (119909 119910 (119909) 1199101015840

(119909)) (47)

subject to the initial conditions

119910 (0) = 120572 1199101015840

(0) = 120573 (48)

or the boundary conditions

119910 (0) = 120572 119910 (1) = 120573 (49)

or the most general mixed boundary conditions

1198861119910 (0) + 119886

21199101015840

(0) = 120572 1198871119910 (1) + 119887

21199101015840

(1) = 120573 (50)

6 Abstract and Applied Analysis

If we follow the same procedure of Section 41 approximate119910(119909) as in (40) and make use of (23) and (35) then we obtain

119862119879D2Ψ (119909) = 119865 (119909 119910 (119909) 119862119879DΨ (119909)) (51)

To find an approximate solution to 119910(119909) we compute (51)at the first 2119896(119872+1)minus2 roots of119880lowast

2119896(119872+1)

(119909)These equationswith (44) or (45) or (46) generate 2119896(119872 + 1) nonlinearequations in the expansion coefficients 119888

119899119898 which can be

solved with the aid of Newtonrsquos iterative method

5 Numerical Results and Discussions

In this section the presented algorithms in Section 4 areapplied to solve linear and nonlinear second-order initialand boundary value problems as well as Bratursquos equationsSome examples subject to different initial boundary andmixed boundary conditions are considered to illustrate theefficiency and the applicability of the proposed algorithmsAll computations are performed by using Mathematica 8

Example 1 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) + (1 minus 119909) 1199101015840

(119909) + 2119910 (119909) = (1 + 2119909 minus 1199092

) sin119909

0 lt 119909 lt 1 119910 (0) = 1 119910 (1) = 0

(52)

with the exact solution 119910(119909) = (1 minus 119909) cos119909In Table 1 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 2 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) minus sin1199091199101015840 (119909) + 119910 (119909) = sin2119909 0 lt 119909 lt 1 (53)

subject to the mixed boundary conditions

119910 (0) + 1199101015840

(0) = 1 sin (1) 119910 (1) + cos (1) 1199101015840 (1) = 0(54)

with the exact solution 119910(119909) = cos119909In Table 2 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 3 Consider the following nonlinear initial valueproblem (see [41])

411991010158401015840

(119909) minus 2(1199101015840

(119909))2

+ 119910 (119909) = 0

0 lt 119909 lt 1 119910 (0) = minus1 1199101015840

(0) = 0

(55)

with the exact solution 119910(119909) = (11990928) minus 1We solve (55) using the algorithmdescribed in Section 42

for the case corresponds to 119872 = 2 119896 = 0 to obtain anapproximate solution of119910(119909) First if wemake use of (24) and

Table 1 Maximum absolute error 119864 for Example 1

119896 119872 119864 119896 119872 119864

3 228 sdot 10minus3

2 923 sdot 10minus6

0 4 632 sdot 10minus5

1 3 862 sdot 10minus8

5 623 sdot 10minus6

4 107 sdot 10minus9

Table 2 Maximum absolute error 119864 for Example 2

119896 119872 119864 119896 119872 119864

3 252 sdot 10minus5

2 490 sdot 10minus8

0 4 211 sdot 10minus6

1 3 255 sdot 10minus9

5 901 sdot 10minus8

4 350 sdot 10minus10

(35) then the two operational matrices D and D2 are givenrespectively by

D = (0 0 0

4 0 0

0 8 0

) D2 = (0 0 0

0 0 0

32 0 0

) (56)

Moreover Ψ(119909) can be evaluated to give

Ψ (119909) = radic2

120587(

2

8119909 minus 4

321199092

minus 32119909 + 6

) (57)

If we set

119862 = (11988800 11988801 11988802)119879

= radic120587

2(1198880 1198881 1198882)119879

(58)

then (55) takes the form

4119862119879D2Ψ (119909) minus 2(119862119879DΨ (119909))

2

+ 119862119879

Ψ (119909) = 0 (59)

which is equivalent to

1198880minus 21198881+ 131119888

2minus 641198882

1+ 512119888

11198882minus 1024119888

2

2

+ 4 (1198881minus 41198882minus 256119888

11198882+ 1024119888

2

2) 119909

+ 16 (1198882minus 256119888

2

2) 1199092

= 0

(60)

We only need to satisfy this equation at the first root of119880lowast3(119909)

that is at 1199091= (2 minus radic2)4 to get

1198880minus 641198882

1+ radic2119888

1(2561198882minus 1) + (129 minus 512119888

2) 1198882= 0 (61)

Furthermore the use of initial conditions in (55) leads to thetwo equations

21198880minus 41198881+ 61198882= minus1

1198881minus 41198882= 0

(62)

The solution of the nonlinear system of (61)-(62) gives

1198880=minus123

256 119888

1=1

64 119888

2=1

256 (63)

Abstract and Applied Analysis 7

and consequently

119910 (119909) = (minus123

2561

641

256)(

2

8119909 minus 4

321199092

minus 32119909 + 6

)

=1199092

8minus 1

(64)

which is the exact solution

Example 4 Consider the following nonlinear second-orderboundary value problem

11991010158401015840

(119909) + (1199101015840

(119909))2

minus 119910 (119909) = sinh2119909

119910 (0) = 1 119910 (1) = cosh 1(65)

with the exact solution 119910(119909) = cosh 119909In Table 3 we list the maximum absolute error 119864 for

various values of 119896 and119872

Example 5 Consider the following linear singular initialvalue problem (see [23])

11991010158401015840

(119909) + 119891 (119909) 1199101015840

(119909) + 119910 (119909) = 119892 (119909)

119910 (0) = 0 1199101015840

(0) = 1

(66)

where

119891 (119909) =

minus4119909 +4

3 0 ⩽ 119909 lt

1

4

119909 minus1

3

1

4⩽ 119909 ⩽ 1

(67)

and 119892(119909) is chosen such that the exact solution of (66) is119910(119909) = 119890

119909 sin119909(1 + 1199092) We apply the algorithm describedin Section 41 with 119896 = 2 In Table 4 we list the maximumabsolute error for various values of119872 while in Figure 1 theexact solution is comparedwith the three numerical solutionscorresponding to 119896 = 2119872 = 2 119896 = 2119872 = 4 119896 = 2 and119872 = 6 and also in Table 5 we give a comparison between theerror obtained in [23] and the present method 119896 = 2119872 = 8

Example 6 Consider the following Bratu equation (see [17ndash20])

11991010158401015840

(119909) + 120582119890119910(119909)

= 0 119910 (0) = 119910 (1) = 0 0 ⩽ 119909 ⩽ 1 (68)

with the analytical solution

119910 (119909) = minus2 ln [cosh ((1205794) (2119909 minus 1))cosh (1205794)

] (69)

where 120579 is the solution of the nonlinear equation 120579 =

radic2120582 cosh 120579The presented algorithm in Section 42 is appliedto numerically solve (68) for the three cases correspondingto 120582 = 1 2 and 351 which yield 120579 = 151716 235755 and466781 respectively In Table 6 the maximum absolute error119864 is listed for 119896 = 0 and various values of119872 while in Table 7we give a comparison between the best errors obtained by

Table 3 Maximum absolute error 119864 for Example 4

119896 119872 119864 119896 119872 119864

03 569 sdot 10

minus4

1 2 112 sdot 10minus6

4 135 sdot 10minus5

3 197 sdot 10minus8

Table 4 Maximum absolute error 119864 for Example 5

119872 119864 119872 119864

2 298 sdot 10minus4 6 185 sdot 10

minus10

4 170 sdot 10minus5 8 912 sdot 10

minus11

Table 5 Comparison between different errors for Example 5

119909 02 04 06 08

LegendreWavelets [23] 430 sdot 10

minus12

124 sdot 10minus11

297 sdot 10minus12

435 sdot 10minus12

SCWOMD 647 sdot 10minus13

121 sdot 10minus12

174 sdot 10minus12

232 sdot 10minus12

Table 6 Maximum absolute error 119864 for Example 6

119872 120582 119864 120582 119864 120582 119864

10

1

2162 sdot 10minus7

2

2758 sdot 10minus5

351

7658 sdot 10minus4

12 4453 sdot 10minus9

8082 sdot 10minus6

8665 sdot 10minus5

14 8461 sdot 10minus11

1461 sdot 10minus9

2384 sdot 10minus8

16 1546 sdot 10minus12

5342 sdot 10minus10

6587 sdot 10minus9

18 2782 sdot 10minus14

1644 sdot 10minus11

2387 sdot 10minus10

20 7216 sdot 10minus16

4024 sdot 10minus12

1058 sdot 10minus10

Table 7 Comparison between the best errors for Example 6 for 119896 =0

SCWOMDError in [17] Error in [18] Error in [19] Error in [20]722 sdot 10

minus16

102 sdot 10minus6

889 sdot 10minus6

135 sdot 10minus5

301 sdot 10minus3

our method and some other numerical methods used tosolve Example 6 This table shows that our method is moreaccurate if compared with the methods developed in [17ndash20] In addition Figure 2 illustrates a comparison betweendifferent solutions obtained by our algorithm in case of 120582 = 1119896 = 0 and119872 = 1 2 3

Remark 6 It is worth noting here that the obtained numericalresults in the previous six solved examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecompared favorably with the known analytical solutions

6 Concluding Remarks

In this paper an algorithm for obtaining a numerical spectralsolution for second-order linear and nonlinear boundaryvalue problems is discussed The derivation of this algorithmis essentially based on constructing the shifted second kindChebyshev wavelets operational matrix of differentiationOne of the main advantages of the presented algorithm isits availability for application on both linear and nonlin-ear second-order boundary value problems including some

8 Abstract and Applied Analysis

Exactk = 2M = 2

k = 2M = 4

k = 2M = 6

x

00 02 04 06 08 10002

005

010

020

050

100

y(x)

Figure 1 Numerical and exact solutions of Example 5

00 02 04 06 08 100010

0100

0050

0020

0030

0015

0150

0070

Exactk = 0M = 1

k = 0M = 2

k = 0M = 3

x

y(x)

Figure 2 Different solutions of Example 6

singular equations and also Bratu type equations Anotheradvantage of the developed algorithm is that high accurateapproximate solutions are achieved using a small number ofthe second kindChebyshev waveletsThe obtained numericalresults are compared favorably with the analytical ones

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support Also they would like to thank the referee forhis valuable comments and suggestions which improved thepaper into its present form

References

[1] EHDoha andWMAbd-Elhameed ldquoEfficient spectral-Galer-kin algorithms for direct solution of second-order equationsusing ultraspherical polynomialsrdquo SIAM Journal on ScientificComputing vol 24 no 2 pp 548ndash571 2002

[2] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[4] E H Doha W M Abd-Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[5] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourth kindsrdquo Abstract and Applied Analysis vol2013 Article ID 542839 9 pages 2013

[6] A Constantmldes Applied Numerical Methods With PersonalComputers McGraw-Hill New York NY USA 1987

[7] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[8] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013

[9] N Daoudi-Merzagui F Derrab and A Boucherif ldquoSubhar-monic solutions of nonautonomous second order differentialequations with singular nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 903281 20 pages 2012

[10] M Turkyilmazoglu ldquoEffective computation of exact and ana-lytic approximate solutions to singular nonlinear equationsof Lane-Emden-Fowler typerdquo Applied Mathematical Modellingvol 37 pp 7539ndash7548 2013

[11] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[12] G Bratu ldquoSur les equations integrales non lineairesrdquo Bulletin dela Societe Mathematique de France vol 42 pp 113ndash142 1914

[13] I M Gelrsquofand ldquoSome problems in the theory of quasilinearequationsrdquoAmericanMathematical Society Translations vol 29pp 295ndash381 1963

[14] A Serghini Mounim and B M de Dormale ldquoFrom the fittingtechniques to accurate schemes for the Liouville-Bratu-Gelfandproblemrdquo Numerical Methods for Partial Differential Equationsvol 22 no 4 pp 761ndash775 2006

[15] M I Syam and A Hamdan ldquoAn efficient method for solvingBratu equationsrdquo Applied Mathematics and Computation vol176 no 2 pp 704ndash713 2006

[16] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[17] S Abbasbandy M S Hashemi and C-S Liu ldquoThe Lie-groupshooting method for solving the Bratu equationrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no11 pp 4238ndash4249 2011

[18] H Caglar N Caglar M Ozer A Valarıstos and A N Anag-nostopoulos ldquoB-spline method for solving Bratursquos problemrdquoInternational Journal of Computer Mathematics vol 87 no 8pp 1885ndash1891 2010

Abstract and Applied Analysis 9

[19] S A Khuri ldquoA new approach to Bratursquos problemrdquo AppliedMathematics and Computation vol 147 no 1 pp 131ndash136 2004

[20] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvingboundary value problemsrdquo Journal of Computational Physicsvol 159 no 2 pp 125ndash138 2000

[21] J P Boyd ldquoOne-point pseudospectral collocation for the one-dimensional Bratu equationrdquoAppliedMathematics and Compu-tation vol 217 no 12 pp 5553ndash5565 2011

[22] H Jafari S A Yousefi M A Firoozjaee S Momani and CM Khalique ldquoApplication of Legendre wavelets for solvingfractional differential equationsrdquo Computers amp Mathematicswith Applications vol 62 no 3 pp 1038ndash1045 2011

[23] F Mohammadi and M M Hosseini ldquoA new Legendre waveletoperational matrix of derivative and its applications in solvingthe singular ordinary differential equationsrdquo Journal of theFranklin Institute vol 348 no 8 pp 1787ndash1796 2011

[24] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[25] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[26] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[27] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[28] M Dehghan and A Saadatmandi ldquoChebyshev finite differencemethod for Fredholm integro-differential equationrdquo Interna-tional Journal of Computer Mathematics vol 85 no 1 pp 123ndash130 2008

[29] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[30] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[31] Y Li ldquoSolving a nonlinear fractional differential equation usingChebyshev waveletsrdquoCommunications in Nonlinear Science andNumerical Simulation vol 15 no 9 pp 2284ndash2292 2010

[32] H Saeedi M Mohseni Moghadam N Mollahasani and G NChuev ldquoA CAS wavelet method for solving nonlinear Fredholmintegro-differential equations of fractional orderrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no3 pp 1154ndash1163 2011

[33] H Saeedi and M M Moghadam ldquoNumerical solution ofnonlinear Volterra integro-differential equations of arbitraryorder by CAS waveletsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 3 pp 1216ndash1226 2011

[34] C F Cheng Y T Tsay and T T Wu ldquoWalsh operational matri-ces for fractional calculus and their application to distributedsystemsrdquo Journal of the Franklin Institute vol 303 no 3 pp 267ndash284 1977

[35] R Y Chang and M L Wang ldquoShifted Legendre direct methodfor variational problemsrdquo Journal of Optimization Theory andApplications vol 39 no 2 pp 299ndash307 1983

[36] I R Horng and J H Chou ldquoShifted Chebyshev direct methodfor solving variational problemsrdquo International Journal of Sys-tems Science vol 16 no 7 pp 855ndash861 1985

[37] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996

[38] F Khellat and S A Yousefi ldquoThe linear Legendre motherwavelets operational matrix of integration and its applicationrdquoJournal of the Franklin Institute vol 343 no 2 pp 181ndash190 2006

[39] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp HallCRC Boca Raton Fla USA 2003

[40] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[41] M R Eslahchi M Dehghan and S Ahmadi ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

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

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

If we follow the same procedure of Section 41 approximate119910(119909) as in (40) and make use of (23) and (35) then we obtain

119862119879D2Ψ (119909) = 119865 (119909 119910 (119909) 119862119879DΨ (119909)) (51)

To find an approximate solution to 119910(119909) we compute (51)at the first 2119896(119872+1)minus2 roots of119880lowast

2119896(119872+1)

(119909)These equationswith (44) or (45) or (46) generate 2119896(119872 + 1) nonlinearequations in the expansion coefficients 119888

119899119898 which can be

solved with the aid of Newtonrsquos iterative method

5 Numerical Results and Discussions

In this section the presented algorithms in Section 4 areapplied to solve linear and nonlinear second-order initialand boundary value problems as well as Bratursquos equationsSome examples subject to different initial boundary andmixed boundary conditions are considered to illustrate theefficiency and the applicability of the proposed algorithmsAll computations are performed by using Mathematica 8

Example 1 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) + (1 minus 119909) 1199101015840

(119909) + 2119910 (119909) = (1 + 2119909 minus 1199092

) sin119909

0 lt 119909 lt 1 119910 (0) = 1 119910 (1) = 0

(52)

with the exact solution 119910(119909) = (1 minus 119909) cos119909In Table 1 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 2 Consider the following linear second-orderboundary value problem

11991010158401015840

(119909) minus sin1199091199101015840 (119909) + 119910 (119909) = sin2119909 0 lt 119909 lt 1 (53)

subject to the mixed boundary conditions

119910 (0) + 1199101015840

(0) = 1 sin (1) 119910 (1) + cos (1) 1199101015840 (1) = 0(54)

with the exact solution 119910(119909) = cos119909In Table 2 the maximum absolute error 119864 is listed for

various values of 119896 and119872

Example 3 Consider the following nonlinear initial valueproblem (see [41])

411991010158401015840

(119909) minus 2(1199101015840

(119909))2

+ 119910 (119909) = 0

0 lt 119909 lt 1 119910 (0) = minus1 1199101015840

(0) = 0

(55)

with the exact solution 119910(119909) = (11990928) minus 1We solve (55) using the algorithmdescribed in Section 42

for the case corresponds to 119872 = 2 119896 = 0 to obtain anapproximate solution of119910(119909) First if wemake use of (24) and

Table 1 Maximum absolute error 119864 for Example 1

119896 119872 119864 119896 119872 119864

3 228 sdot 10minus3

2 923 sdot 10minus6

0 4 632 sdot 10minus5

1 3 862 sdot 10minus8

5 623 sdot 10minus6

4 107 sdot 10minus9

Table 2 Maximum absolute error 119864 for Example 2

119896 119872 119864 119896 119872 119864

3 252 sdot 10minus5

2 490 sdot 10minus8

0 4 211 sdot 10minus6

1 3 255 sdot 10minus9

5 901 sdot 10minus8

4 350 sdot 10minus10

(35) then the two operational matrices D and D2 are givenrespectively by

D = (0 0 0

4 0 0

0 8 0

) D2 = (0 0 0

0 0 0

32 0 0

) (56)

Moreover Ψ(119909) can be evaluated to give

Ψ (119909) = radic2

120587(

2

8119909 minus 4

321199092

minus 32119909 + 6

) (57)

If we set

119862 = (11988800 11988801 11988802)119879

= radic120587

2(1198880 1198881 1198882)119879

(58)

then (55) takes the form

4119862119879D2Ψ (119909) minus 2(119862119879DΨ (119909))

2

+ 119862119879

Ψ (119909) = 0 (59)

which is equivalent to

1198880minus 21198881+ 131119888

2minus 641198882

1+ 512119888

11198882minus 1024119888

2

2

+ 4 (1198881minus 41198882minus 256119888

11198882+ 1024119888

2

2) 119909

+ 16 (1198882minus 256119888

2

2) 1199092

= 0

(60)

We only need to satisfy this equation at the first root of119880lowast3(119909)

that is at 1199091= (2 minus radic2)4 to get

1198880minus 641198882

1+ radic2119888

1(2561198882minus 1) + (129 minus 512119888

2) 1198882= 0 (61)

Furthermore the use of initial conditions in (55) leads to thetwo equations

21198880minus 41198881+ 61198882= minus1

1198881minus 41198882= 0

(62)

The solution of the nonlinear system of (61)-(62) gives

1198880=minus123

256 119888

1=1

64 119888

2=1

256 (63)

Abstract and Applied Analysis 7

and consequently

119910 (119909) = (minus123

2561

641

256)(

2

8119909 minus 4

321199092

minus 32119909 + 6

)

=1199092

8minus 1

(64)

which is the exact solution

Example 4 Consider the following nonlinear second-orderboundary value problem

11991010158401015840

(119909) + (1199101015840

(119909))2

minus 119910 (119909) = sinh2119909

119910 (0) = 1 119910 (1) = cosh 1(65)

with the exact solution 119910(119909) = cosh 119909In Table 3 we list the maximum absolute error 119864 for

various values of 119896 and119872

Example 5 Consider the following linear singular initialvalue problem (see [23])

11991010158401015840

(119909) + 119891 (119909) 1199101015840

(119909) + 119910 (119909) = 119892 (119909)

119910 (0) = 0 1199101015840

(0) = 1

(66)

where

119891 (119909) =

minus4119909 +4

3 0 ⩽ 119909 lt

1

4

119909 minus1

3

1

4⩽ 119909 ⩽ 1

(67)

and 119892(119909) is chosen such that the exact solution of (66) is119910(119909) = 119890

119909 sin119909(1 + 1199092) We apply the algorithm describedin Section 41 with 119896 = 2 In Table 4 we list the maximumabsolute error for various values of119872 while in Figure 1 theexact solution is comparedwith the three numerical solutionscorresponding to 119896 = 2119872 = 2 119896 = 2119872 = 4 119896 = 2 and119872 = 6 and also in Table 5 we give a comparison between theerror obtained in [23] and the present method 119896 = 2119872 = 8

Example 6 Consider the following Bratu equation (see [17ndash20])

11991010158401015840

(119909) + 120582119890119910(119909)

= 0 119910 (0) = 119910 (1) = 0 0 ⩽ 119909 ⩽ 1 (68)

with the analytical solution

119910 (119909) = minus2 ln [cosh ((1205794) (2119909 minus 1))cosh (1205794)

] (69)

where 120579 is the solution of the nonlinear equation 120579 =

radic2120582 cosh 120579The presented algorithm in Section 42 is appliedto numerically solve (68) for the three cases correspondingto 120582 = 1 2 and 351 which yield 120579 = 151716 235755 and466781 respectively In Table 6 the maximum absolute error119864 is listed for 119896 = 0 and various values of119872 while in Table 7we give a comparison between the best errors obtained by

Table 3 Maximum absolute error 119864 for Example 4

119896 119872 119864 119896 119872 119864

03 569 sdot 10

minus4

1 2 112 sdot 10minus6

4 135 sdot 10minus5

3 197 sdot 10minus8

Table 4 Maximum absolute error 119864 for Example 5

119872 119864 119872 119864

2 298 sdot 10minus4 6 185 sdot 10

minus10

4 170 sdot 10minus5 8 912 sdot 10

minus11

Table 5 Comparison between different errors for Example 5

119909 02 04 06 08

LegendreWavelets [23] 430 sdot 10

minus12

124 sdot 10minus11

297 sdot 10minus12

435 sdot 10minus12

SCWOMD 647 sdot 10minus13

121 sdot 10minus12

174 sdot 10minus12

232 sdot 10minus12

Table 6 Maximum absolute error 119864 for Example 6

119872 120582 119864 120582 119864 120582 119864

10

1

2162 sdot 10minus7

2

2758 sdot 10minus5

351

7658 sdot 10minus4

12 4453 sdot 10minus9

8082 sdot 10minus6

8665 sdot 10minus5

14 8461 sdot 10minus11

1461 sdot 10minus9

2384 sdot 10minus8

16 1546 sdot 10minus12

5342 sdot 10minus10

6587 sdot 10minus9

18 2782 sdot 10minus14

1644 sdot 10minus11

2387 sdot 10minus10

20 7216 sdot 10minus16

4024 sdot 10minus12

1058 sdot 10minus10

Table 7 Comparison between the best errors for Example 6 for 119896 =0

SCWOMDError in [17] Error in [18] Error in [19] Error in [20]722 sdot 10

minus16

102 sdot 10minus6

889 sdot 10minus6

135 sdot 10minus5

301 sdot 10minus3

our method and some other numerical methods used tosolve Example 6 This table shows that our method is moreaccurate if compared with the methods developed in [17ndash20] In addition Figure 2 illustrates a comparison betweendifferent solutions obtained by our algorithm in case of 120582 = 1119896 = 0 and119872 = 1 2 3

Remark 6 It is worth noting here that the obtained numericalresults in the previous six solved examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecompared favorably with the known analytical solutions

6 Concluding Remarks

In this paper an algorithm for obtaining a numerical spectralsolution for second-order linear and nonlinear boundaryvalue problems is discussed The derivation of this algorithmis essentially based on constructing the shifted second kindChebyshev wavelets operational matrix of differentiationOne of the main advantages of the presented algorithm isits availability for application on both linear and nonlin-ear second-order boundary value problems including some

8 Abstract and Applied Analysis

Exactk = 2M = 2

k = 2M = 4

k = 2M = 6

x

00 02 04 06 08 10002

005

010

020

050

100

y(x)

Figure 1 Numerical and exact solutions of Example 5

00 02 04 06 08 100010

0100

0050

0020

0030

0015

0150

0070

Exactk = 0M = 1

k = 0M = 2

k = 0M = 3

x

y(x)

Figure 2 Different solutions of Example 6

singular equations and also Bratu type equations Anotheradvantage of the developed algorithm is that high accurateapproximate solutions are achieved using a small number ofthe second kindChebyshev waveletsThe obtained numericalresults are compared favorably with the analytical ones

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support Also they would like to thank the referee forhis valuable comments and suggestions which improved thepaper into its present form

References

[1] EHDoha andWMAbd-Elhameed ldquoEfficient spectral-Galer-kin algorithms for direct solution of second-order equationsusing ultraspherical polynomialsrdquo SIAM Journal on ScientificComputing vol 24 no 2 pp 548ndash571 2002

[2] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[4] E H Doha W M Abd-Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[5] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourth kindsrdquo Abstract and Applied Analysis vol2013 Article ID 542839 9 pages 2013

[6] A Constantmldes Applied Numerical Methods With PersonalComputers McGraw-Hill New York NY USA 1987

[7] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[8] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013

[9] N Daoudi-Merzagui F Derrab and A Boucherif ldquoSubhar-monic solutions of nonautonomous second order differentialequations with singular nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 903281 20 pages 2012

[10] M Turkyilmazoglu ldquoEffective computation of exact and ana-lytic approximate solutions to singular nonlinear equationsof Lane-Emden-Fowler typerdquo Applied Mathematical Modellingvol 37 pp 7539ndash7548 2013

[11] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[12] G Bratu ldquoSur les equations integrales non lineairesrdquo Bulletin dela Societe Mathematique de France vol 42 pp 113ndash142 1914

[13] I M Gelrsquofand ldquoSome problems in the theory of quasilinearequationsrdquoAmericanMathematical Society Translations vol 29pp 295ndash381 1963

[14] A Serghini Mounim and B M de Dormale ldquoFrom the fittingtechniques to accurate schemes for the Liouville-Bratu-Gelfandproblemrdquo Numerical Methods for Partial Differential Equationsvol 22 no 4 pp 761ndash775 2006

[15] M I Syam and A Hamdan ldquoAn efficient method for solvingBratu equationsrdquo Applied Mathematics and Computation vol176 no 2 pp 704ndash713 2006

[16] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[17] S Abbasbandy M S Hashemi and C-S Liu ldquoThe Lie-groupshooting method for solving the Bratu equationrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no11 pp 4238ndash4249 2011

[18] H Caglar N Caglar M Ozer A Valarıstos and A N Anag-nostopoulos ldquoB-spline method for solving Bratursquos problemrdquoInternational Journal of Computer Mathematics vol 87 no 8pp 1885ndash1891 2010

Abstract and Applied Analysis 9

[19] S A Khuri ldquoA new approach to Bratursquos problemrdquo AppliedMathematics and Computation vol 147 no 1 pp 131ndash136 2004

[20] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvingboundary value problemsrdquo Journal of Computational Physicsvol 159 no 2 pp 125ndash138 2000

[21] J P Boyd ldquoOne-point pseudospectral collocation for the one-dimensional Bratu equationrdquoAppliedMathematics and Compu-tation vol 217 no 12 pp 5553ndash5565 2011

[22] H Jafari S A Yousefi M A Firoozjaee S Momani and CM Khalique ldquoApplication of Legendre wavelets for solvingfractional differential equationsrdquo Computers amp Mathematicswith Applications vol 62 no 3 pp 1038ndash1045 2011

[23] F Mohammadi and M M Hosseini ldquoA new Legendre waveletoperational matrix of derivative and its applications in solvingthe singular ordinary differential equationsrdquo Journal of theFranklin Institute vol 348 no 8 pp 1787ndash1796 2011

[24] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[25] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[26] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[27] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[28] M Dehghan and A Saadatmandi ldquoChebyshev finite differencemethod for Fredholm integro-differential equationrdquo Interna-tional Journal of Computer Mathematics vol 85 no 1 pp 123ndash130 2008

[29] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[30] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[31] Y Li ldquoSolving a nonlinear fractional differential equation usingChebyshev waveletsrdquoCommunications in Nonlinear Science andNumerical Simulation vol 15 no 9 pp 2284ndash2292 2010

[32] H Saeedi M Mohseni Moghadam N Mollahasani and G NChuev ldquoA CAS wavelet method for solving nonlinear Fredholmintegro-differential equations of fractional orderrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no3 pp 1154ndash1163 2011

[33] H Saeedi and M M Moghadam ldquoNumerical solution ofnonlinear Volterra integro-differential equations of arbitraryorder by CAS waveletsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 3 pp 1216ndash1226 2011

[34] C F Cheng Y T Tsay and T T Wu ldquoWalsh operational matri-ces for fractional calculus and their application to distributedsystemsrdquo Journal of the Franklin Institute vol 303 no 3 pp 267ndash284 1977

[35] R Y Chang and M L Wang ldquoShifted Legendre direct methodfor variational problemsrdquo Journal of Optimization Theory andApplications vol 39 no 2 pp 299ndash307 1983

[36] I R Horng and J H Chou ldquoShifted Chebyshev direct methodfor solving variational problemsrdquo International Journal of Sys-tems Science vol 16 no 7 pp 855ndash861 1985

[37] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996

[38] F Khellat and S A Yousefi ldquoThe linear Legendre motherwavelets operational matrix of integration and its applicationrdquoJournal of the Franklin Institute vol 343 no 2 pp 181ndash190 2006

[39] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp HallCRC Boca Raton Fla USA 2003

[40] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[41] M R Eslahchi M Dehghan and S Ahmadi ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

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

Abstract and Applied Analysis 7

and consequently

119910 (119909) = (minus123

2561

641

256)(

2

8119909 minus 4

321199092

minus 32119909 + 6

)

=1199092

8minus 1

(64)

which is the exact solution

Example 4 Consider the following nonlinear second-orderboundary value problem

11991010158401015840

(119909) + (1199101015840

(119909))2

minus 119910 (119909) = sinh2119909

119910 (0) = 1 119910 (1) = cosh 1(65)

with the exact solution 119910(119909) = cosh 119909In Table 3 we list the maximum absolute error 119864 for

various values of 119896 and119872

Example 5 Consider the following linear singular initialvalue problem (see [23])

11991010158401015840

(119909) + 119891 (119909) 1199101015840

(119909) + 119910 (119909) = 119892 (119909)

119910 (0) = 0 1199101015840

(0) = 1

(66)

where

119891 (119909) =

minus4119909 +4

3 0 ⩽ 119909 lt

1

4

119909 minus1

3

1

4⩽ 119909 ⩽ 1

(67)

and 119892(119909) is chosen such that the exact solution of (66) is119910(119909) = 119890

119909 sin119909(1 + 1199092) We apply the algorithm describedin Section 41 with 119896 = 2 In Table 4 we list the maximumabsolute error for various values of119872 while in Figure 1 theexact solution is comparedwith the three numerical solutionscorresponding to 119896 = 2119872 = 2 119896 = 2119872 = 4 119896 = 2 and119872 = 6 and also in Table 5 we give a comparison between theerror obtained in [23] and the present method 119896 = 2119872 = 8

Example 6 Consider the following Bratu equation (see [17ndash20])

11991010158401015840

(119909) + 120582119890119910(119909)

= 0 119910 (0) = 119910 (1) = 0 0 ⩽ 119909 ⩽ 1 (68)

with the analytical solution

119910 (119909) = minus2 ln [cosh ((1205794) (2119909 minus 1))cosh (1205794)

] (69)

where 120579 is the solution of the nonlinear equation 120579 =

radic2120582 cosh 120579The presented algorithm in Section 42 is appliedto numerically solve (68) for the three cases correspondingto 120582 = 1 2 and 351 which yield 120579 = 151716 235755 and466781 respectively In Table 6 the maximum absolute error119864 is listed for 119896 = 0 and various values of119872 while in Table 7we give a comparison between the best errors obtained by

Table 3 Maximum absolute error 119864 for Example 4

119896 119872 119864 119896 119872 119864

03 569 sdot 10

minus4

1 2 112 sdot 10minus6

4 135 sdot 10minus5

3 197 sdot 10minus8

Table 4 Maximum absolute error 119864 for Example 5

119872 119864 119872 119864

2 298 sdot 10minus4 6 185 sdot 10

minus10

4 170 sdot 10minus5 8 912 sdot 10

minus11

Table 5 Comparison between different errors for Example 5

119909 02 04 06 08

LegendreWavelets [23] 430 sdot 10

minus12

124 sdot 10minus11

297 sdot 10minus12

435 sdot 10minus12

SCWOMD 647 sdot 10minus13

121 sdot 10minus12

174 sdot 10minus12

232 sdot 10minus12

Table 6 Maximum absolute error 119864 for Example 6

119872 120582 119864 120582 119864 120582 119864

10

1

2162 sdot 10minus7

2

2758 sdot 10minus5

351

7658 sdot 10minus4

12 4453 sdot 10minus9

8082 sdot 10minus6

8665 sdot 10minus5

14 8461 sdot 10minus11

1461 sdot 10minus9

2384 sdot 10minus8

16 1546 sdot 10minus12

5342 sdot 10minus10

6587 sdot 10minus9

18 2782 sdot 10minus14

1644 sdot 10minus11

2387 sdot 10minus10

20 7216 sdot 10minus16

4024 sdot 10minus12

1058 sdot 10minus10

Table 7 Comparison between the best errors for Example 6 for 119896 =0

SCWOMDError in [17] Error in [18] Error in [19] Error in [20]722 sdot 10

minus16

102 sdot 10minus6

889 sdot 10minus6

135 sdot 10minus5

301 sdot 10minus3

our method and some other numerical methods used tosolve Example 6 This table shows that our method is moreaccurate if compared with the methods developed in [17ndash20] In addition Figure 2 illustrates a comparison betweendifferent solutions obtained by our algorithm in case of 120582 = 1119896 = 0 and119872 = 1 2 3

Remark 6 It is worth noting here that the obtained numericalresults in the previous six solved examples are very accuratealthough the number of retained modes in the spectralexpansion is very few and again the numerical results arecompared favorably with the known analytical solutions

6 Concluding Remarks

In this paper an algorithm for obtaining a numerical spectralsolution for second-order linear and nonlinear boundaryvalue problems is discussed The derivation of this algorithmis essentially based on constructing the shifted second kindChebyshev wavelets operational matrix of differentiationOne of the main advantages of the presented algorithm isits availability for application on both linear and nonlin-ear second-order boundary value problems including some

8 Abstract and Applied Analysis

Exactk = 2M = 2

k = 2M = 4

k = 2M = 6

x

00 02 04 06 08 10002

005

010

020

050

100

y(x)

Figure 1 Numerical and exact solutions of Example 5

00 02 04 06 08 100010

0100

0050

0020

0030

0015

0150

0070

Exactk = 0M = 1

k = 0M = 2

k = 0M = 3

x

y(x)

Figure 2 Different solutions of Example 6

singular equations and also Bratu type equations Anotheradvantage of the developed algorithm is that high accurateapproximate solutions are achieved using a small number ofthe second kindChebyshev waveletsThe obtained numericalresults are compared favorably with the analytical ones

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support Also they would like to thank the referee forhis valuable comments and suggestions which improved thepaper into its present form

References

[1] EHDoha andWMAbd-Elhameed ldquoEfficient spectral-Galer-kin algorithms for direct solution of second-order equationsusing ultraspherical polynomialsrdquo SIAM Journal on ScientificComputing vol 24 no 2 pp 548ndash571 2002

[2] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[4] E H Doha W M Abd-Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[5] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourth kindsrdquo Abstract and Applied Analysis vol2013 Article ID 542839 9 pages 2013

[6] A Constantmldes Applied Numerical Methods With PersonalComputers McGraw-Hill New York NY USA 1987

[7] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[8] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013

[9] N Daoudi-Merzagui F Derrab and A Boucherif ldquoSubhar-monic solutions of nonautonomous second order differentialequations with singular nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 903281 20 pages 2012

[10] M Turkyilmazoglu ldquoEffective computation of exact and ana-lytic approximate solutions to singular nonlinear equationsof Lane-Emden-Fowler typerdquo Applied Mathematical Modellingvol 37 pp 7539ndash7548 2013

[11] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[12] G Bratu ldquoSur les equations integrales non lineairesrdquo Bulletin dela Societe Mathematique de France vol 42 pp 113ndash142 1914

[13] I M Gelrsquofand ldquoSome problems in the theory of quasilinearequationsrdquoAmericanMathematical Society Translations vol 29pp 295ndash381 1963

[14] A Serghini Mounim and B M de Dormale ldquoFrom the fittingtechniques to accurate schemes for the Liouville-Bratu-Gelfandproblemrdquo Numerical Methods for Partial Differential Equationsvol 22 no 4 pp 761ndash775 2006

[15] M I Syam and A Hamdan ldquoAn efficient method for solvingBratu equationsrdquo Applied Mathematics and Computation vol176 no 2 pp 704ndash713 2006

[16] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[17] S Abbasbandy M S Hashemi and C-S Liu ldquoThe Lie-groupshooting method for solving the Bratu equationrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no11 pp 4238ndash4249 2011

[18] H Caglar N Caglar M Ozer A Valarıstos and A N Anag-nostopoulos ldquoB-spline method for solving Bratursquos problemrdquoInternational Journal of Computer Mathematics vol 87 no 8pp 1885ndash1891 2010

Abstract and Applied Analysis 9

[19] S A Khuri ldquoA new approach to Bratursquos problemrdquo AppliedMathematics and Computation vol 147 no 1 pp 131ndash136 2004

[20] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvingboundary value problemsrdquo Journal of Computational Physicsvol 159 no 2 pp 125ndash138 2000

[21] J P Boyd ldquoOne-point pseudospectral collocation for the one-dimensional Bratu equationrdquoAppliedMathematics and Compu-tation vol 217 no 12 pp 5553ndash5565 2011

[22] H Jafari S A Yousefi M A Firoozjaee S Momani and CM Khalique ldquoApplication of Legendre wavelets for solvingfractional differential equationsrdquo Computers amp Mathematicswith Applications vol 62 no 3 pp 1038ndash1045 2011

[23] F Mohammadi and M M Hosseini ldquoA new Legendre waveletoperational matrix of derivative and its applications in solvingthe singular ordinary differential equationsrdquo Journal of theFranklin Institute vol 348 no 8 pp 1787ndash1796 2011

[24] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[25] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[26] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[27] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[28] M Dehghan and A Saadatmandi ldquoChebyshev finite differencemethod for Fredholm integro-differential equationrdquo Interna-tional Journal of Computer Mathematics vol 85 no 1 pp 123ndash130 2008

[29] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[30] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[31] Y Li ldquoSolving a nonlinear fractional differential equation usingChebyshev waveletsrdquoCommunications in Nonlinear Science andNumerical Simulation vol 15 no 9 pp 2284ndash2292 2010

[32] H Saeedi M Mohseni Moghadam N Mollahasani and G NChuev ldquoA CAS wavelet method for solving nonlinear Fredholmintegro-differential equations of fractional orderrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no3 pp 1154ndash1163 2011

[33] H Saeedi and M M Moghadam ldquoNumerical solution ofnonlinear Volterra integro-differential equations of arbitraryorder by CAS waveletsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 3 pp 1216ndash1226 2011

[34] C F Cheng Y T Tsay and T T Wu ldquoWalsh operational matri-ces for fractional calculus and their application to distributedsystemsrdquo Journal of the Franklin Institute vol 303 no 3 pp 267ndash284 1977

[35] R Y Chang and M L Wang ldquoShifted Legendre direct methodfor variational problemsrdquo Journal of Optimization Theory andApplications vol 39 no 2 pp 299ndash307 1983

[36] I R Horng and J H Chou ldquoShifted Chebyshev direct methodfor solving variational problemsrdquo International Journal of Sys-tems Science vol 16 no 7 pp 855ndash861 1985

[37] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996

[38] F Khellat and S A Yousefi ldquoThe linear Legendre motherwavelets operational matrix of integration and its applicationrdquoJournal of the Franklin Institute vol 343 no 2 pp 181ndash190 2006

[39] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp HallCRC Boca Raton Fla USA 2003

[40] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[41] M R Eslahchi M Dehghan and S Ahmadi ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

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

8 Abstract and Applied Analysis

Exactk = 2M = 2

k = 2M = 4

k = 2M = 6

x

00 02 04 06 08 10002

005

010

020

050

100

y(x)

Figure 1 Numerical and exact solutions of Example 5

00 02 04 06 08 100010

0100

0050

0020

0030

0015

0150

0070

Exactk = 0M = 1

k = 0M = 2

k = 0M = 3

x

y(x)

Figure 2 Different solutions of Example 6

singular equations and also Bratu type equations Anotheradvantage of the developed algorithm is that high accurateapproximate solutions are achieved using a small number ofthe second kindChebyshev waveletsThe obtained numericalresults are compared favorably with the analytical ones

Acknowledgments

This paper was funded by the Deanship of Scientific Research(DSR) King Abdulaziz University Jeddah The authorstherefore acknowledge with thanks DSR technical and finan-cial support Also they would like to thank the referee forhis valuable comments and suggestions which improved thepaper into its present form

References

[1] EHDoha andWMAbd-Elhameed ldquoEfficient spectral-Galer-kin algorithms for direct solution of second-order equationsusing ultraspherical polynomialsrdquo SIAM Journal on ScientificComputing vol 24 no 2 pp 548ndash571 2002

[2] E H Doha W M Abd-Elhameed and M A Bassuony ldquoNewalgorithms for solving high even-order differential equationsusing third and fourth Chebyshev-Galerkin methodsrdquo Journalof Computational Physics vol 236 pp 563ndash579 2013

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalizedJacobi polynomialsrdquo Quaestiones Mathematicae vol 36 no 1pp 15ndash38 2013

[4] E H Doha W M Abd-Elhameed and Y H Youssri ldquoSecondkind Chebyshev operational matrix algorithm for solving dif-ferential equations of Lane-Emden typerdquo New Astronomy vol23-24 pp 113ndash117 2013

[5] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourth kindsrdquo Abstract and Applied Analysis vol2013 Article ID 542839 9 pages 2013

[6] A Constantmldes Applied Numerical Methods With PersonalComputers McGraw-Hill New York NY USA 1987

[7] D E Newland An Introduction to Random Vibrations Spectraland Wavelet Analysis Longman Scientific and Technical NewYork NY USA 1993

[8] M Pasic ldquoFite-Wintner-Leighton-type oscillation criteria forsecond-order differential equations with nonlinear dampingrdquoAbstract and Applied Analysis vol 2013 Article ID 852180 10pages 2013

[9] N Daoudi-Merzagui F Derrab and A Boucherif ldquoSubhar-monic solutions of nonautonomous second order differentialequations with singular nonlinearitiesrdquo Abstract and AppliedAnalysis vol 2012 Article ID 903281 20 pages 2012

[10] M Turkyilmazoglu ldquoEffective computation of exact and ana-lytic approximate solutions to singular nonlinear equationsof Lane-Emden-Fowler typerdquo Applied Mathematical Modellingvol 37 pp 7539ndash7548 2013

[11] E H Doha and W M Abd-Elhameed ldquoEfficient solutions ofmultidimensional sixth-order boundary value problems usingsymmetric generalized Jacobi-Galerkin methodrdquo Abstract andApplied Analysis vol 2012 Article ID 749370 19 pages 2012

[12] G Bratu ldquoSur les equations integrales non lineairesrdquo Bulletin dela Societe Mathematique de France vol 42 pp 113ndash142 1914

[13] I M Gelrsquofand ldquoSome problems in the theory of quasilinearequationsrdquoAmericanMathematical Society Translations vol 29pp 295ndash381 1963

[14] A Serghini Mounim and B M de Dormale ldquoFrom the fittingtechniques to accurate schemes for the Liouville-Bratu-Gelfandproblemrdquo Numerical Methods for Partial Differential Equationsvol 22 no 4 pp 761ndash775 2006

[15] M I Syam and A Hamdan ldquoAn efficient method for solvingBratu equationsrdquo Applied Mathematics and Computation vol176 no 2 pp 704ndash713 2006

[16] A-M Wazwaz ldquoAdomian decomposition method for a reliabletreatment of the Bratu-type equationsrdquo Applied Mathematicsand Computation vol 166 no 3 pp 652ndash663 2005

[17] S Abbasbandy M S Hashemi and C-S Liu ldquoThe Lie-groupshooting method for solving the Bratu equationrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no11 pp 4238ndash4249 2011

[18] H Caglar N Caglar M Ozer A Valarıstos and A N Anag-nostopoulos ldquoB-spline method for solving Bratursquos problemrdquoInternational Journal of Computer Mathematics vol 87 no 8pp 1885ndash1891 2010

Abstract and Applied Analysis 9

[19] S A Khuri ldquoA new approach to Bratursquos problemrdquo AppliedMathematics and Computation vol 147 no 1 pp 131ndash136 2004

[20] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvingboundary value problemsrdquo Journal of Computational Physicsvol 159 no 2 pp 125ndash138 2000

[21] J P Boyd ldquoOne-point pseudospectral collocation for the one-dimensional Bratu equationrdquoAppliedMathematics and Compu-tation vol 217 no 12 pp 5553ndash5565 2011

[22] H Jafari S A Yousefi M A Firoozjaee S Momani and CM Khalique ldquoApplication of Legendre wavelets for solvingfractional differential equationsrdquo Computers amp Mathematicswith Applications vol 62 no 3 pp 1038ndash1045 2011

[23] F Mohammadi and M M Hosseini ldquoA new Legendre waveletoperational matrix of derivative and its applications in solvingthe singular ordinary differential equationsrdquo Journal of theFranklin Institute vol 348 no 8 pp 1787ndash1796 2011

[24] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[25] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[26] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[27] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[28] M Dehghan and A Saadatmandi ldquoChebyshev finite differencemethod for Fredholm integro-differential equationrdquo Interna-tional Journal of Computer Mathematics vol 85 no 1 pp 123ndash130 2008

[29] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[30] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[31] Y Li ldquoSolving a nonlinear fractional differential equation usingChebyshev waveletsrdquoCommunications in Nonlinear Science andNumerical Simulation vol 15 no 9 pp 2284ndash2292 2010

[32] H Saeedi M Mohseni Moghadam N Mollahasani and G NChuev ldquoA CAS wavelet method for solving nonlinear Fredholmintegro-differential equations of fractional orderrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no3 pp 1154ndash1163 2011

[33] H Saeedi and M M Moghadam ldquoNumerical solution ofnonlinear Volterra integro-differential equations of arbitraryorder by CAS waveletsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 3 pp 1216ndash1226 2011

[34] C F Cheng Y T Tsay and T T Wu ldquoWalsh operational matri-ces for fractional calculus and their application to distributedsystemsrdquo Journal of the Franklin Institute vol 303 no 3 pp 267ndash284 1977

[35] R Y Chang and M L Wang ldquoShifted Legendre direct methodfor variational problemsrdquo Journal of Optimization Theory andApplications vol 39 no 2 pp 299ndash307 1983

[36] I R Horng and J H Chou ldquoShifted Chebyshev direct methodfor solving variational problemsrdquo International Journal of Sys-tems Science vol 16 no 7 pp 855ndash861 1985

[37] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996

[38] F Khellat and S A Yousefi ldquoThe linear Legendre motherwavelets operational matrix of integration and its applicationrdquoJournal of the Franklin Institute vol 343 no 2 pp 181ndash190 2006

[39] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp HallCRC Boca Raton Fla USA 2003

[40] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[41] M R Eslahchi M Dehghan and S Ahmadi ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

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

Abstract and Applied Analysis 9

[19] S A Khuri ldquoA new approach to Bratursquos problemrdquo AppliedMathematics and Computation vol 147 no 1 pp 131ndash136 2004

[20] E Deeba S A Khuri and S Xie ldquoAn algorithm for solvingboundary value problemsrdquo Journal of Computational Physicsvol 159 no 2 pp 125ndash138 2000

[21] J P Boyd ldquoOne-point pseudospectral collocation for the one-dimensional Bratu equationrdquoAppliedMathematics and Compu-tation vol 217 no 12 pp 5553ndash5565 2011

[22] H Jafari S A Yousefi M A Firoozjaee S Momani and CM Khalique ldquoApplication of Legendre wavelets for solvingfractional differential equationsrdquo Computers amp Mathematicswith Applications vol 62 no 3 pp 1038ndash1045 2011

[23] F Mohammadi and M M Hosseini ldquoA new Legendre waveletoperational matrix of derivative and its applications in solvingthe singular ordinary differential equationsrdquo Journal of theFranklin Institute vol 348 no 8 pp 1787ndash1796 2011

[24] M Razzaghi and S Yousefi ldquoThe Legendre wavelets operationalmatrix of integrationrdquo International Journal of Systems Sciencevol 32 no 4 pp 495ndash502 2001

[25] M Razzaghi and S Yousefi ldquoLegendre wavelets method for thesolution of nonlinear problems in the calculus of variationsrdquoMathematical and Computer Modelling vol 34 no 1-2 pp 45ndash54 2001

[26] M Razzaghi and S Yousefi ldquoLegendre wavelets method forconstrained optimal control problemsrdquo Mathematical Methodsin the Applied Sciences vol 25 no 7 pp 529ndash539 2002

[27] E Babolian and F Fattahzadeh ldquoNumerical solution of differ-ential equations by using Chebyshev wavelet operational matrixof integrationrdquo Applied Mathematics and Computation vol 188no 1 pp 417ndash426 2007

[28] M Dehghan and A Saadatmandi ldquoChebyshev finite differencemethod for Fredholm integro-differential equationrdquo Interna-tional Journal of Computer Mathematics vol 85 no 1 pp 123ndash130 2008

[29] A Saadatmandi and M Dehghan ldquoA new operational matrixfor solving fractional-order differential equationsrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1326ndash13362010

[30] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[31] Y Li ldquoSolving a nonlinear fractional differential equation usingChebyshev waveletsrdquoCommunications in Nonlinear Science andNumerical Simulation vol 15 no 9 pp 2284ndash2292 2010

[32] H Saeedi M Mohseni Moghadam N Mollahasani and G NChuev ldquoA CAS wavelet method for solving nonlinear Fredholmintegro-differential equations of fractional orderrdquo Communica-tions in Nonlinear Science and Numerical Simulation vol 16 no3 pp 1154ndash1163 2011

[33] H Saeedi and M M Moghadam ldquoNumerical solution ofnonlinear Volterra integro-differential equations of arbitraryorder by CAS waveletsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 16 no 3 pp 1216ndash1226 2011

[34] C F Cheng Y T Tsay and T T Wu ldquoWalsh operational matri-ces for fractional calculus and their application to distributedsystemsrdquo Journal of the Franklin Institute vol 303 no 3 pp 267ndash284 1977

[35] R Y Chang and M L Wang ldquoShifted Legendre direct methodfor variational problemsrdquo Journal of Optimization Theory andApplications vol 39 no 2 pp 299ndash307 1983

[36] I R Horng and J H Chou ldquoShifted Chebyshev direct methodfor solving variational problemsrdquo International Journal of Sys-tems Science vol 16 no 7 pp 855ndash861 1985

[37] J S Gu and W S Jiang ldquoThe Haar wavelets operational matrixof integrationrdquo International Journal of Systems Science vol 27no 7 pp 623ndash628 1996

[38] F Khellat and S A Yousefi ldquoThe linear Legendre motherwavelets operational matrix of integration and its applicationrdquoJournal of the Franklin Institute vol 343 no 2 pp 181ndash190 2006

[39] J C Mason and D C Handscomb Chebyshev PolynomialsChapman amp HallCRC Boca Raton Fla USA 2003

[40] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods in Fluid Dynamics Springer Series in Com-putational Physics Springer New York NY USA 1988

[41] M R Eslahchi M Dehghan and S Ahmadi ldquoThe generalJacobi matrix method for solving some nonlinear ordinarydifferential equationsrdquoAppliedMathematical Modelling vol 36no 8 pp 3387ndash3398 2012

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