Research ArticlePower Series Extender Method for the Solution ofNonlinear Differential Equations
Hector Vazquez-Leal1 and Arturo Sarmiento-Reyes2
1Electronic Instrumentation and Atmospheric Sciences School Universidad Veracruzana Cto Gonzalo Aguirre BeltranSN 91000 Xalapa VER Mexico2National Institute for Astrophysics Optics and Electronics Luis Enrique Erro No 1 Santa Maria 72840 Tonantzintla PUE Mexico
Correspondence should be addressed to Hector Vazquez-Leal hvazquezuvmx
Received 1 October 2014 Accepted 1 December 2014
Academic Editor Salvatore Alfonzetti
Copyright copy 2015 H Vazquez-Leal and A Sarmiento-ReyesThis is an open access article distributed under theCreativeCommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
We propose a power series extender method to obtain approximate solutions of nonlinear differential equations In order to assessthe benefits of this proposal three nonlinear problems of different kind are solved and compared against the power series solutionobtained using an approximative method The problems are homogeneous Lane-Emden equation of 120572 index governing equationof a burning iron particle and an explicit differential-algebraic equation related to battery model simulationsThe results show thatPSEM generates highly accurate handy approximations requiring only a few steps The main advantage of PSEM is to extend thedomain of convergence of the power series solutions of approximative methods as Taylor series method homotopy perturbationmethod homotopy analysis method variational iteration method differential transform method and Adomian decompositionmethod amongmany others From the application of PSEM it results in handy easy computable expressions that extend the domainof convergence of high order power series solutions
1 Introduction
Solving nonlinear differential equations is an important taskin sciences because many physical phenomena are modelledusing such equations Recently the generalized homotopymethod (GHM) [1] was proposed as a generalization of thehomotopy perturbation method (HPM) The application ofsuchmethod is based on a power seriesmatching that enablesGHM to obtain complex and rich expression impossible toobtain using HPM Therefore using as a guide the mainidea of power series matching behind GHM we propose apower series method extender (PSEM) The key steps of thisproposal are as follows
(1) First we apply an approximativemethod to obtain thecoefficients of a power series solution The approx-imative method can be one from the literature likeTaylor series method (TSM) [2 3] power seriesmethod (PSM) [4] homotopy perturbation method(HPM) [5ndash11] perturbationmethod (PM) homotopy
analysismethod (HAM) variational iterationmethod(VIM) differential transform method (DTM) andAdomian decomposition method (ADM) amongmany others
(2) In the same fashion as GHM [1] we propose atrial function (TF) that can potentially describe thequalitative behaviour of the nonlinear problem
(3) Next we apply the Taylor series method to a trialfunction (TF)
(4) Then we equate the coefficients of the power seriesobtained in steps 1 and 3 to obtain a nonlinearalgebraic equation system in terms of the coefficientsof the TF which can be solved using symbolic ornumerical methods
(5) Finally the approximate solution is obtained by sub-stituting the calculated coefficients from step 4 intothe proposed TF
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 717404 7 pageshttpdxdoiorg1011552015717404
2 Mathematical Problems in Engineering
In order to study the potential of the proposed techniquethree nonlinear problems will be solved and compared versusnumerical methods homogeneous Lane-Emden equation of120572 index [12] governing equation of burning iron particle[13] and an explicit differential-algebraic equation related tobattery model simulations [14]
This paper is organized as follows In Section 2 weintroduce the basic concept of PSEMmethod In Section 3 weshow the approximation of three nonlinear differential equa-tions related to different phenomenons in physics Numericalsimulations and a discussion about the results are provided inSection 4 Finally a brief conclusion is given in Section 5
2 Basic Concept of PSEM Method
In a broad sense a nonlinear differential equation can be ex-pressed as
119871 (119906) + 119873 (119906) minus 119891 (119909) = 0 119909 isin Ω (1)
having as boundary condition
119861 (119906
120597119906
120597120578
) = 0 119909 isin Γ (2)
where 119871 and 119873 are a linear operator and a nonlinearoperator respectively 119891(119909) is a known analytic function 119861
is a boundary operator Γ is the boundary of domain Ωand 120597119906120597120578 denotes differentiation along the normal drawnoutwards from Ω [8]
Next we express the solution of (1) as a power series
119906 =
infin
sum
119896=0
V119896119909119896
(3)
where V119896(119896 = 0 1 ) are the coefficients of the power series
It is important to notice that (3) can be obtained by someapproximative method from literature as HPM HAM VIMDTM ADM TSM and PSM among others
Now we propose that the solution for (1) can be writtenas a finite sum of functions in the general form [1]
119906 = 1199060
+
119899
sum
119894=0
119891119894(119909 119906119894) (4)
or
119906 =
1199060
+ sum119899
119894=0119891119894(119909 119906119894)
1 + sum2119899
119895=119899+1119891119895(119909 119906119895)
(5)
where 119906119894are constants to be determined by PSEM 119891
119894(119909 119906119894)
are arbitrary trial functions and 119899 and 2119899 are the orders ofapproximations (4) and (5) respectively We will denominate(4) and (5) as the trial function (TF)
Next we calculate the Taylor series of (4) or (5) resultingin the power series
119906 = 1199060
+
119899
sum
119894=0
1198751198940
+
119899
sum
119894=0
infin
sum
119896=1
119875119894119896
119909119896
(6)
119906 = 1199060
+
119899
sum
119894=0
1198751198940
+
2119899
sum
119894=0
infin
sum
119896=1
119875119894119896
119909119896
(7)
respectively where Taylor coefficients 119875119896are expressed in
terms of parameters 119906119894
Finally we equatematch the coefficients of power series(6) or (7) with (3) to obtain the values of 119906
119894and substitute
them into (4) or (5) to obtain the PSEM approximationIt is important to notice that we can separately apply (4)
or (5) to obtain an approximation of (1) where the selection ofthe TF depends of the behaviour of the problem under studyIn addition it is important to remark that if we choose the 119891
119894
functions to be analytic then (6) and (7) are convergent series[15ndash17]
3 Case Studies
In the present section we will solve three case studies toshow the utility of the PSEM method to solve nonlineardifferential equations We know from literature that approx-imative methods as HPM PM HAM (using the ℎ = minus1 asconvergence control) VIM DTM ADM TSM and PSM areable to generate power series approximate solutions that inmost of the cases are equivalent to the well-known TSMPSMsolutions Therefore the main difference in such cases isthe difficulty of application of the specific approximativemethod to the particular case study For its simplicity ofapplication we will use TSM [2] to exemplify the applicationof PSEM although it is possible to use one of the availableapproximative methods from literature
31 Homogeneous Lane-Emden Equation of 120572 Index TheLane-Emden singular equation describes a wide variety ofproblems in physics as some aspects related to the stellarstructure thermal history of spherical cloud of gas isother-mal gas spheres and thermionic currents among others [12]The equation is expressed as
11991010158401015840
+
2
119909
1199101015840
+ 119910120572
= 0 0 lt 119909 le 1 0 le 120572 le 5
119910 (0) = 1 1199101015840
(0) = 0
(8)
where prime denotes differentiation with respect to 119909 and 120572
represents the index of the equationUsing the initial conditions of (8) and considering as
expansion point 1199090
= 0 it results the solution of (8) as
119910 (119909) = 119910 (0) +
1199101015840
(0)
1
1199091
+
11991010158401015840
(0)
2
1199092
+
119910101584010158401015840
(0)
3
1199093
+
119910(4)
(0)
4
1199094
+ sdot sdot sdot
(9)
where 119910(119898)
(0) (119898 = 2 3 ) are unknown constants to bedetermined by the Taylor series method
In order to apply the TSM method [3] it is necessary tomultiply (8) by 119909 to avoid the singularity resulting in
11990911991010158401015840
+ 21199101015840
+ 119909119910120572
= 0 (10)
Mathematical Problems in Engineering 3
Next we derive successively (10) and resolve the systemof equations obtained from the derivatives 119910
(119898)
(0) (119898 =
2 3 ) resulting in
11991010158401015840
= (
1
3
) ((minus119910 minus 1199091205721199101015840
) 119910120572minus1
minus 119910101584010158401015840
119909)
119910101584010158401015840
= (
1
4
) (minus120572 (119909 (120572 minus 1) 1199101015840
+ 2119910) 1199101015840
119910120572minus2
minus119910(120572minus1)
11991010158401015840
120572119909 minus 119910(4)
119909)
119910(4)
= (
1
5
) (minus3119910(120572minus2)
11991010158401015840
(119909 (120572 minus 1) 1199101015840
+ 119910) 120572
minus (120572 minus 1) 120572 (119909 (120572 minus 2) 1199101015840
+ 3119910) 11991010158402
119910120572minus3
minus119910(120572minus1)
120572119909119910101584010158401015840
minus 119910(5)
119909)
(11)
Now substituting the initial conditions of (8) into (11) itresults in
11991010158401015840
(0) = minus
1
3
119910101584010158401015840
(0)
= 0 119910(4)
(0)
= (
1
5
) 120572
(12)
From (5) we propose a specific solution for (8) with thefollowing form
119910 (119909) =
1199100
+ 11991011199091
+ 11991021199092
+ sdot sdot sdot + 119910119899119909119899
1 + 119910119899+1
1199091
+ 119910119899+2
1199092
+ sdot sdot sdot + 1199102119899
119909119899 (13)
where 119910119894(119894 = 0 1 2 ) are constants to be determined and
the order 119899 is chosen as 2 to obtain a handy approximationNext Taylor series of (13) is
119910 (119909) = 1199100
+ (1199101
minus 11991001199103) 119909
+ (1199102
minus 11991001199104
+ (minus1199101
+ 11991001199103) 1199103) 1199092
+ ( (minus1199101
+ 11991001199103) 1199104
+ (minus1199102
+ 11991001199104
+ 11991031199101
minus 11991001199102
3) 1199103) 1199093
+ ((minus1199102
+ 11991001199104
+ 11991031199101
minus 11991001199102
3) 1199104
+ (11991041199101
minus 2119910411991001199103
+ 11991031199102
minus 1199102
31199101
+11991001199103
3) 1199103) 1199094
+ sdot sdot sdot
(14)
Then we equate coefficients of 119909-powers of (9) and(14) to obtain a system of equations which can be solvedsymbolically resulting in
1199100
= 1
1199101
= 0
1199102
=
120572
20
minus
1
6
1199103
= 0
1199104
=
120572
20
(15)
Finally the PSEMapproximation is obtained by substitut-ing (15) into (13) resulting in
119910 (119909) =
1 + (12057220 minus 16) 1199092
1 + (12057220) 1199092
0 lt 119909 le 1 0 le 120572 le 5
(16)
32 Combustion of a Single Iron Particle The nonlinearenergy equation for combustion of a single iron particle [13]is
(1 minus 1205981
(120579 minus 120579infin
)) 1205791015840
+ 120579 minus 120579infin
+ 1205982
(1205794
minus 1205794
surr) minus Ψ = 0
120579 (0) = 1
(17)
where prime denotes differentiation with respect to 120591 and 120579 isthe dimensionless temperature
The values of the parameters are obtained from [13] 120579infin
=
117647 1205981
= 0051595 Ψ = 098579 120579surr = 035294 and1205982
= 0002630145546 Using the initial conditions of (17) andconsidering the expansion point 120591 = 0 yields to the followingTaylor series
120579 (120591) = 120579 (0) +
1205791015840
(0)
1
1205911
+
12057910158401015840
(0)
2
1205912
+ sdot sdot sdot (18)
where 120579(119898)
(0) (119898 = 1 2 ) are unknown constantsNow we resolve (17) for 120579
1015840 and calculate the successivederivatives to obtain 120579
(119898)
(0) (119898 = 1 2 ) resulting in
1205791015840
= minus
120579 minus 120579infin
+ 1205982
(1205794
minus 1205794
surr) minus Ψ
1 minus 1205981
(120579 minus 120579infin
)
12057910158401015840
= minus
12059811205791015840
(120579 minus 120579infin
+ 1205982
(1205794
minus 1205794
surr) minus Ψ)
(1 minus 1205981
(120579 minus 120579infin
))2
minus
1205791015840
+ 412059821205793
1205791015840
1 minus 1205981
(120579 minus 120579infin
)
(19)
4 Mathematical Problems in Engineering
Then substituting the initial conditions of (17) into (19)we get
1205791015840
(0) = minus
120579 (0) minus 120579infin
+ 1205982
(120579 (0)4
minus 1205794
surr ) minus Ψ
1 minus 1205981
(120579 (0) minus 120579infin
)
12057910158401015840
(0) = minus
12059811205791015840
(0) (120579 (0) minus 120579infin
+ 1205982
(120579 (0)4
minus 1205794
surr) minus Ψ)
(1 minus 1205981
(120579 (0) minus 120579infin
))2
minus
1205791015840
(0) + 41205982120579 (0)3
1205791015840
(0)
1 minus 1205981
(120579 (0) minus 120579infin
)
(20)
From (4) we propose a specific solution for (17) as fol-lows
120579 (120591) = 1205790
+ 1205791exp (120579
2120591) (21)
Next Taylor series of (21) is
120579 (120591) = 1205790
+ 1205791
+ 12057911205792120591 + (
1
2
) 12057911205792
21205912
+ sdot sdot sdot (22)
Then we equate coefficients of 120591-powers of (18) and(22) to obtain a system of equations which can be solvedsymbolically resulting in
1205790
=
(11989601198962
minus 1198962
1)
1198962
1205791
=
1198962
1
1198962
1205792
=
1198962
1198961
(23)
where 1198960
= 120579(0) 1198961
= 1205791015840
(0) and 1198962
= 12057910158401015840
(0)Finally substituting (23) into (21) yields to the PSEM
approximation
120579 (120591) =
(11989601198962
minus 1198962
1)
1198962
+ (
1198962
1
1198962
) exp(
1198962120591
1198961
) (24)
For comparison purposes we calculate the Taylor seriesby substituting 120579(0) = 1 and (20) into (18) yielding
120579 (120591) = 1 + 1170326453120591 minus 063241147851205912
(25)
33 Differential-Algebraic Equation Related to Battery ModelSimulation Next PSEM will be applied to a nonlinearequation [14] related to an oversimplified battery modelwhich is formulated as follows
1199111015840
= minus2119911 + 1199102
119911 (0) = 2
minus100 ln (119910) + 2119911 = 5 119910 (0) = exp (minus001)
(26)
where 119911 is the differential variable 119910 is the algebraic variableand primes denote derivative with respect to 119905
Using the initial conditions of (26) and considering theexpansion point 119905 = 0 yields to the following Taylor series
119911 (119905) = 119911 (0) +
1199111015840
(0)
1
1199051
+
11991110158401015840
(0)
2
1199052
+
119911101584010158401015840
(0)
3
1199053
+
119911(4)
(0)
4
1199054
+ sdot sdot sdot
119910 (119905) = 119910 (0) +
1199101015840
(0)
1
1199051
+
11991010158401015840
(0)
2
1199052
+
119910101584010158401015840
(0)
3
1199053
+
119910(4)
(0)
4
1199054
+ sdot sdot sdot
(27)
where derivatives 119911(119898)
(0) and 119910(119898)
(0) (119898 = 1 2 ) areunknown
Firstly we apply an implicit derivative with respect to 119905 ofthe second equation of (26) resulting in
1199111015840
= minus2119911 + 1199102
119911 (0) = 2
1199101015840
= (
1
50
) 1199111015840
119910 119910 (0) = exp (minus001)
(28)
As before we calculate the successive derivatives of (28)and evaluate the resulting equations at 119905 = 0 yielding
119911 (0) = 120574
1199111015840
(0) = 1205772
minus 2120574
11991110158401015840
(0) = (
1
25
) (1205772
minus 50) (1205772
minus 2120574)
119911101584010158401015840
(0) =
2
625
(1205772
minus 2120574) (1205774
+ 1250 + (minus120574 minus 50) 1205772
)
119911(4)
(0) =
6
15625
(minus
62500
3
+ 1205776
+ (minus2120574 minus
175
3
) 1205774
+ (
200
3
120574 + 1250 + (
2
3
) 1205742
) 1205772
)
times (1205772
minus 2120574)
119910 (0) = 120577
1199101015840
(0) =
1
50
(1205772
minus 2120574) (120577)
11991010158401015840
(0) =
3
2500
(120577) (1205772
minus 2120574) (1205772
minus
2
3
120574 minus
100
3
)
Mathematical Problems in Engineering 5
119910101584010158401015840
(0) =
3
25000
(120577) (1205772
minus 2120574) (1205774
+ (minus
140
3
minus
8
5
120574) 1205772
+40120574 +
2000
3
+
4
15
1205742
)
119910(4)
(0) =
21
1250000
(120577) (1205776
+ (minus
1240
21
minus
18
7
120574) 1205774
+ (
26000
21
+
320
3
120574 +
52
35
1205742
) 1205772
minus
4000
3
120574 minus
8
105
1205743
minus
160
7
1205742
minus
200000
21
)
times (1205772
minus 2120574)
(29)
where 120574 = 119911(0) and 120577 = 119910(0)From (4) we propose a specific solution for (26) as
follows
119911 (119905) = 1199110
+ 1199111exp (119911
2119905) + 1199113exp (119911
4119905)
119910 (119905) = 1199100
+ 1199101exp (119910
2119905) + 119910
3exp (119910
4119905)
(30)
Finally we calculate the fourth-order Taylor series of(30) equate the resulting coefficients of the same 119905-powerswith respect to (27) and solve the two systems of equationsresulting in the following PSEM approximation
119911 (119905) = 04608270703 + 1539992595 exp (minus1963069736119905)
minus 00008196648996 exp (minus4040101111119905)
119910 (119905) = 09600410114 + 002960275048 exp (minus1964621532119905)
+ 00004060718378 exp (minus4030998609119905)
(31)
For comparison purposes we calculate the Taylor seriesby substituting (29) into (27) yielding
119911 (119905) = 2 minus 3019801327119905 + 29606012221199052
minus 19326574831199053
+ 094380751921199054
119910 (119905) = 09900498337 minus 005979507603119905 + 0060428547451199052
minus 0041845484121199053
+ 0022842654121199054
(32)
4 Numerical Simulation and Discussion
For all case studies we used built-in numerical routines fromMaple 15 for comparison purposes The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant(RKF45) was used [18 19] The command was setup with atolerance of absolute error (A E) of 10
minus12
We obtained a highly accurate rational approximatesolution (16) for the singular second order Lane-Emdenequation [12] (8) as depicted in Figures 1 and 2 for 120572 =
[1 2 3 4 5] Thus the PSEM method may be useful for suchkind of problems with singularities
Additionally we solved the nonlinear energy equationfor combustion of a single iron particle [13] (17) obtaining ahighly accurate approximation (24) as depicted in Figures 3and 4 In the same figures we can observe the Taylor seriessolution (25) noticing the higher precision of the proposedsolution
Finally we approximated a simplisticmodel [14] related toa battery model simulation that is represented as a nonlineardifferential-algebraic equation of index 1 (26) The PSEMapproximation (31) is in good agreement to numerical results(RKF45) for a large period of time as depicted in Figure 5in contrast we can observe in the same figure the poorconvergence of the Taylor series solution As a matter offact if we calculate the asymptotic behaviour of (26) we willconclude that the approximation keeps its high accuracy fora very large period of time On one side setting the 119911
1015840
(119905) = 0
in (26) and solving the system it results in that the stableequilibrium point is 119864
1= [119911infin
= 04608356746 119910infin
=
09600371604] On the other side we calculate the limit when119905 rarr infin of (31) resulting in 119864
2= [119911infin
= 04608270703 119910infin
=
09600410114] Therefore the high proximity among 1198641
and 1198642exhibits the high accuracy of the proposed PSEM
approximation for the rank 119905 isin [0 infin)The PSEM method was able to calculate approximations
with a larger domain of convergence in comparison to theTSM results The reason can lie in the fact that the trialfunction of PSEM technique may potentially contain moreinformation than the Taylor power series As long as theTaylor series of the proposed TF exist we can propose a widevariety of series of functions as trigonometric hyperbolicintegrals among many others However further research isrequired to propose a methodology to choose the optimalTF At the moment users of the method should keep inmind proposing a TF that may potentially reproduce thebehaviour of the exact solution Finally it is importantto remark that as the TF is a finite sum or division ofanalytic functions as shown in (4) or (5) therefore suchsum is convergent [15ndash17] additionally we will require onlya finite number of terms of the Taylor expansions (6) or(7) to obtain the coefficients of the trial functions (4) or(5) Finally it is important to remark that PSEM can beeasily combined with HPM PM HAM VIM DTM andADM among many others Furthermore we can select theapproximativemethoddepending on its facility of applicationto the specific case study Therefore PSEM is a powerfulmalleable technique that can be applied in combinationwith many of the approximative methods reported in litera-ture
5 Conclusions
This work introduced PSEM as a useful tool with highpotential to solve nonlinear differential equations We were
6 Mathematical Problems in Engineering
1
0975
0950
0925
0900
0875
0850
0 02 04 06 08 1
x
y
PSEM120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
Figure 1 RKF45 solution for (8) (symbols) and its approximatePSEM solution (16) (solid line) for different 120572 values
120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
00006
00005
00004
00003
00002
00001
A E
0
0
02 04 06 08 1
x
Figure 2 Absolute error (A E) of (16) with respect to numerical(RKF45) solution for (8)
able to obtain accurate and handy approximations for differ-ent types of problems homogeneous singular Lane-Emdenequation nonlinear energy equation for combustion of asingle iron particle and differential-algebraic system relatedto battery model simulation It is important to remark onthe high flexibility applicability and power of this novelmethod A key point that should be remarked is that the usershould propose a finite sum of division of analytic functions(trial function) that may be chosen according to the natureof the nonlinear problem under study Further researchis required to solve other kinds of problems nonlinearfractional differential equations nonlinear partial differentialequations and nonlinear boundary valued problems amongothers
17
16
15
14
13
12
11
1
0 02 04 06 08 1
120579(120591
)
120591
PSEMRKF45Taylor series
Figure 3 RKF45 solution for (17) (solid circles) its approximatePSEM solution (24) (solid line) and its Taylor series solution (25)(diagonal crosses)
0005
0004
0003
0002
0001
0
0 02 04 06 08 1
A E
120591
PSEMTaylor series
Figure 4 Absolute error (A E) of (24) (solid square) and (25)(empty circle) with respect to numerical (RKF45) solution for (17)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technologyof Mexico (CONACyT) through Grant CB-2010-01 no
Mathematical Problems in Engineering 7
2
15
1
05
00 10 20 30 40
z(t)
t
PSEMRKF45Taylor series
(a)
0 10 20 30 40
t
PSEMRKF45Taylor series
1
099
098
097
096
095
y(t)
(b)
Figure 5 RKF45 solution for (26) (solid circles) its approximate PSEM solution (31) (solid line) and its Taylor series solution (32) (solidsquare)
157024 The authors would like to thank Roberto Castaneda-Sheissa Uriel Filobello-Nino Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[2] H Vazquez-Leal B Benhammouda U A Filobello-Nino et alldquoModified Taylor series method for solving nonlinear differen-tial equationswithmixed boundary conditions defined on finiteintervalsrdquo SpringerPlus vol 3 no 1 article 160 2014
[3] R Barrio M Rodrıguez A Abad and F Blesa ldquoBreakingthe limits the Taylor series methodrdquo Applied Mathematics andComputation vol 217 no 20 pp 7940ndash7954 2011
[4] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[5] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[6] A Barari M Omidvar A R Ghotbi and D D Ganji ldquoApplica-tion of homotopy perturbationmethod and variational iterationmethod to nonlinear oscillator differential equationsrdquo ActaApplicandae Mathematicae vol 104 no 2 pp 161ndash171 2008
[7] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[8] Y-G Wang W-H Lin and N Liu ldquoA homotopy perturbation-based method for large deflection of a cantilever beam undera terminal follower forcerdquo International Journal for Computa-tionalMethods in Engineering Science andMechanics vol 13 no3 pp 197ndash201 2012
[9] M A Fariborzi Araghi and B Rezapour ldquoApplication ofhomotopy perturbationmethod to solve multidimensionalSchrodingerrsquos equationsrdquo International Journal of MathematicalArchive vol 2 no 11 pp 1ndash6 2011
[10] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[11] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[12] A Akyuz-Dascıoglu and H Cerdiuk-Yaslan ldquoThe solutionof high-order nonlinear ordinary differential equations byChebyshev seriesrdquo Applied Mathematics and Computation vol217 no 12 pp 5658ndash5666 2011
[13] M Bidabadi and M Mafi ldquoTime variation of combustiontemperature and burning time of a single iron particlerdquo Inter-national Journal of Thermal Sciences vol 65 pp 136ndash147 2013
[14] R N Methekar V Ramadesigan J C Pirkle Jr and V R Sub-ramanian ldquoAperturbation approach for consistent initializationof index-1 explicit differential-algebraic equations arising frombatterymodel simulationsrdquoComputersampChemical Engineeringvol 35 no 11 pp 2227ndash2234 2011
[15] D G Zill A First Course in Differential Equations with Mod-eling Applications Cengage Learning Boston Mass USA 10thedition 2012
[16] W Belser Formal Power Series and Linear Systems of Meromor-phic Ordinary Differential Equations Springer 1999
[17] M Oberguggenberger and A OstermannAnalysis for Comput-er Scientists Foundations Methods and Algorithms Springer2011
[18] W H Enright K R Jackson S P Noslashrsett and P G ThomsenldquoInterpolants for Runge-Kutta formulasrdquo ACM Transactions onMathematical Software vol 12 no 3 pp 193ndash218 1986
[19] E Fehlberg ldquoKlassische runge-kutta-formeln vier ter und nied-rigerer ordnung mit schrittweitenkontrolle und ihre anwend-ung auf waermeleitungsproblemerdquo Computing vol 6 pp 61ndash711970
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Mathematical Problems in Engineering
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
In order to study the potential of the proposed techniquethree nonlinear problems will be solved and compared versusnumerical methods homogeneous Lane-Emden equation of120572 index [12] governing equation of burning iron particle[13] and an explicit differential-algebraic equation related tobattery model simulations [14]
This paper is organized as follows In Section 2 weintroduce the basic concept of PSEMmethod In Section 3 weshow the approximation of three nonlinear differential equa-tions related to different phenomenons in physics Numericalsimulations and a discussion about the results are provided inSection 4 Finally a brief conclusion is given in Section 5
2 Basic Concept of PSEM Method
In a broad sense a nonlinear differential equation can be ex-pressed as
119871 (119906) + 119873 (119906) minus 119891 (119909) = 0 119909 isin Ω (1)
having as boundary condition
119861 (119906
120597119906
120597120578
) = 0 119909 isin Γ (2)
where 119871 and 119873 are a linear operator and a nonlinearoperator respectively 119891(119909) is a known analytic function 119861
is a boundary operator Γ is the boundary of domain Ωand 120597119906120597120578 denotes differentiation along the normal drawnoutwards from Ω [8]
Next we express the solution of (1) as a power series
119906 =
infin
sum
119896=0
V119896119909119896
(3)
where V119896(119896 = 0 1 ) are the coefficients of the power series
It is important to notice that (3) can be obtained by someapproximative method from literature as HPM HAM VIMDTM ADM TSM and PSM among others
Now we propose that the solution for (1) can be writtenas a finite sum of functions in the general form [1]
119906 = 1199060
+
119899
sum
119894=0
119891119894(119909 119906119894) (4)
or
119906 =
1199060
+ sum119899
119894=0119891119894(119909 119906119894)
1 + sum2119899
119895=119899+1119891119895(119909 119906119895)
(5)
where 119906119894are constants to be determined by PSEM 119891
119894(119909 119906119894)
are arbitrary trial functions and 119899 and 2119899 are the orders ofapproximations (4) and (5) respectively We will denominate(4) and (5) as the trial function (TF)
Next we calculate the Taylor series of (4) or (5) resultingin the power series
119906 = 1199060
+
119899
sum
119894=0
1198751198940
+
119899
sum
119894=0
infin
sum
119896=1
119875119894119896
119909119896
(6)
119906 = 1199060
+
119899
sum
119894=0
1198751198940
+
2119899
sum
119894=0
infin
sum
119896=1
119875119894119896
119909119896
(7)
respectively where Taylor coefficients 119875119896are expressed in
terms of parameters 119906119894
Finally we equatematch the coefficients of power series(6) or (7) with (3) to obtain the values of 119906
119894and substitute
them into (4) or (5) to obtain the PSEM approximationIt is important to notice that we can separately apply (4)
or (5) to obtain an approximation of (1) where the selection ofthe TF depends of the behaviour of the problem under studyIn addition it is important to remark that if we choose the 119891
119894
functions to be analytic then (6) and (7) are convergent series[15ndash17]
3 Case Studies
In the present section we will solve three case studies toshow the utility of the PSEM method to solve nonlineardifferential equations We know from literature that approx-imative methods as HPM PM HAM (using the ℎ = minus1 asconvergence control) VIM DTM ADM TSM and PSM areable to generate power series approximate solutions that inmost of the cases are equivalent to the well-known TSMPSMsolutions Therefore the main difference in such cases isthe difficulty of application of the specific approximativemethod to the particular case study For its simplicity ofapplication we will use TSM [2] to exemplify the applicationof PSEM although it is possible to use one of the availableapproximative methods from literature
31 Homogeneous Lane-Emden Equation of 120572 Index TheLane-Emden singular equation describes a wide variety ofproblems in physics as some aspects related to the stellarstructure thermal history of spherical cloud of gas isother-mal gas spheres and thermionic currents among others [12]The equation is expressed as
11991010158401015840
+
2
119909
1199101015840
+ 119910120572
= 0 0 lt 119909 le 1 0 le 120572 le 5
119910 (0) = 1 1199101015840
(0) = 0
(8)
where prime denotes differentiation with respect to 119909 and 120572
represents the index of the equationUsing the initial conditions of (8) and considering as
expansion point 1199090
= 0 it results the solution of (8) as
119910 (119909) = 119910 (0) +
1199101015840
(0)
1
1199091
+
11991010158401015840
(0)
2
1199092
+
119910101584010158401015840
(0)
3
1199093
+
119910(4)
(0)
4
1199094
+ sdot sdot sdot
(9)
where 119910(119898)
(0) (119898 = 2 3 ) are unknown constants to bedetermined by the Taylor series method
In order to apply the TSM method [3] it is necessary tomultiply (8) by 119909 to avoid the singularity resulting in
11990911991010158401015840
+ 21199101015840
+ 119909119910120572
= 0 (10)
Mathematical Problems in Engineering 3
Next we derive successively (10) and resolve the systemof equations obtained from the derivatives 119910
(119898)
(0) (119898 =
2 3 ) resulting in
11991010158401015840
= (
1
3
) ((minus119910 minus 1199091205721199101015840
) 119910120572minus1
minus 119910101584010158401015840
119909)
119910101584010158401015840
= (
1
4
) (minus120572 (119909 (120572 minus 1) 1199101015840
+ 2119910) 1199101015840
119910120572minus2
minus119910(120572minus1)
11991010158401015840
120572119909 minus 119910(4)
119909)
119910(4)
= (
1
5
) (minus3119910(120572minus2)
11991010158401015840
(119909 (120572 minus 1) 1199101015840
+ 119910) 120572
minus (120572 minus 1) 120572 (119909 (120572 minus 2) 1199101015840
+ 3119910) 11991010158402
119910120572minus3
minus119910(120572minus1)
120572119909119910101584010158401015840
minus 119910(5)
119909)
(11)
Now substituting the initial conditions of (8) into (11) itresults in
11991010158401015840
(0) = minus
1
3
119910101584010158401015840
(0)
= 0 119910(4)
(0)
= (
1
5
) 120572
(12)
From (5) we propose a specific solution for (8) with thefollowing form
119910 (119909) =
1199100
+ 11991011199091
+ 11991021199092
+ sdot sdot sdot + 119910119899119909119899
1 + 119910119899+1
1199091
+ 119910119899+2
1199092
+ sdot sdot sdot + 1199102119899
119909119899 (13)
where 119910119894(119894 = 0 1 2 ) are constants to be determined and
the order 119899 is chosen as 2 to obtain a handy approximationNext Taylor series of (13) is
119910 (119909) = 1199100
+ (1199101
minus 11991001199103) 119909
+ (1199102
minus 11991001199104
+ (minus1199101
+ 11991001199103) 1199103) 1199092
+ ( (minus1199101
+ 11991001199103) 1199104
+ (minus1199102
+ 11991001199104
+ 11991031199101
minus 11991001199102
3) 1199103) 1199093
+ ((minus1199102
+ 11991001199104
+ 11991031199101
minus 11991001199102
3) 1199104
+ (11991041199101
minus 2119910411991001199103
+ 11991031199102
minus 1199102
31199101
+11991001199103
3) 1199103) 1199094
+ sdot sdot sdot
(14)
Then we equate coefficients of 119909-powers of (9) and(14) to obtain a system of equations which can be solvedsymbolically resulting in
1199100
= 1
1199101
= 0
1199102
=
120572
20
minus
1
6
1199103
= 0
1199104
=
120572
20
(15)
Finally the PSEMapproximation is obtained by substitut-ing (15) into (13) resulting in
119910 (119909) =
1 + (12057220 minus 16) 1199092
1 + (12057220) 1199092
0 lt 119909 le 1 0 le 120572 le 5
(16)
32 Combustion of a Single Iron Particle The nonlinearenergy equation for combustion of a single iron particle [13]is
(1 minus 1205981
(120579 minus 120579infin
)) 1205791015840
+ 120579 minus 120579infin
+ 1205982
(1205794
minus 1205794
surr) minus Ψ = 0
120579 (0) = 1
(17)
where prime denotes differentiation with respect to 120591 and 120579 isthe dimensionless temperature
The values of the parameters are obtained from [13] 120579infin
=
117647 1205981
= 0051595 Ψ = 098579 120579surr = 035294 and1205982
= 0002630145546 Using the initial conditions of (17) andconsidering the expansion point 120591 = 0 yields to the followingTaylor series
120579 (120591) = 120579 (0) +
1205791015840
(0)
1
1205911
+
12057910158401015840
(0)
2
1205912
+ sdot sdot sdot (18)
where 120579(119898)
(0) (119898 = 1 2 ) are unknown constantsNow we resolve (17) for 120579
1015840 and calculate the successivederivatives to obtain 120579
(119898)
(0) (119898 = 1 2 ) resulting in
1205791015840
= minus
120579 minus 120579infin
+ 1205982
(1205794
minus 1205794
surr) minus Ψ
1 minus 1205981
(120579 minus 120579infin
)
12057910158401015840
= minus
12059811205791015840
(120579 minus 120579infin
+ 1205982
(1205794
minus 1205794
surr) minus Ψ)
(1 minus 1205981
(120579 minus 120579infin
))2
minus
1205791015840
+ 412059821205793
1205791015840
1 minus 1205981
(120579 minus 120579infin
)
(19)
4 Mathematical Problems in Engineering
Then substituting the initial conditions of (17) into (19)we get
1205791015840
(0) = minus
120579 (0) minus 120579infin
+ 1205982
(120579 (0)4
minus 1205794
surr ) minus Ψ
1 minus 1205981
(120579 (0) minus 120579infin
)
12057910158401015840
(0) = minus
12059811205791015840
(0) (120579 (0) minus 120579infin
+ 1205982
(120579 (0)4
minus 1205794
surr) minus Ψ)
(1 minus 1205981
(120579 (0) minus 120579infin
))2
minus
1205791015840
(0) + 41205982120579 (0)3
1205791015840
(0)
1 minus 1205981
(120579 (0) minus 120579infin
)
(20)
From (4) we propose a specific solution for (17) as fol-lows
120579 (120591) = 1205790
+ 1205791exp (120579
2120591) (21)
Next Taylor series of (21) is
120579 (120591) = 1205790
+ 1205791
+ 12057911205792120591 + (
1
2
) 12057911205792
21205912
+ sdot sdot sdot (22)
Then we equate coefficients of 120591-powers of (18) and(22) to obtain a system of equations which can be solvedsymbolically resulting in
1205790
=
(11989601198962
minus 1198962
1)
1198962
1205791
=
1198962
1
1198962
1205792
=
1198962
1198961
(23)
where 1198960
= 120579(0) 1198961
= 1205791015840
(0) and 1198962
= 12057910158401015840
(0)Finally substituting (23) into (21) yields to the PSEM
approximation
120579 (120591) =
(11989601198962
minus 1198962
1)
1198962
+ (
1198962
1
1198962
) exp(
1198962120591
1198961
) (24)
For comparison purposes we calculate the Taylor seriesby substituting 120579(0) = 1 and (20) into (18) yielding
120579 (120591) = 1 + 1170326453120591 minus 063241147851205912
(25)
33 Differential-Algebraic Equation Related to Battery ModelSimulation Next PSEM will be applied to a nonlinearequation [14] related to an oversimplified battery modelwhich is formulated as follows
1199111015840
= minus2119911 + 1199102
119911 (0) = 2
minus100 ln (119910) + 2119911 = 5 119910 (0) = exp (minus001)
(26)
where 119911 is the differential variable 119910 is the algebraic variableand primes denote derivative with respect to 119905
Using the initial conditions of (26) and considering theexpansion point 119905 = 0 yields to the following Taylor series
119911 (119905) = 119911 (0) +
1199111015840
(0)
1
1199051
+
11991110158401015840
(0)
2
1199052
+
119911101584010158401015840
(0)
3
1199053
+
119911(4)
(0)
4
1199054
+ sdot sdot sdot
119910 (119905) = 119910 (0) +
1199101015840
(0)
1
1199051
+
11991010158401015840
(0)
2
1199052
+
119910101584010158401015840
(0)
3
1199053
+
119910(4)
(0)
4
1199054
+ sdot sdot sdot
(27)
where derivatives 119911(119898)
(0) and 119910(119898)
(0) (119898 = 1 2 ) areunknown
Firstly we apply an implicit derivative with respect to 119905 ofthe second equation of (26) resulting in
1199111015840
= minus2119911 + 1199102
119911 (0) = 2
1199101015840
= (
1
50
) 1199111015840
119910 119910 (0) = exp (minus001)
(28)
As before we calculate the successive derivatives of (28)and evaluate the resulting equations at 119905 = 0 yielding
119911 (0) = 120574
1199111015840
(0) = 1205772
minus 2120574
11991110158401015840
(0) = (
1
25
) (1205772
minus 50) (1205772
minus 2120574)
119911101584010158401015840
(0) =
2
625
(1205772
minus 2120574) (1205774
+ 1250 + (minus120574 minus 50) 1205772
)
119911(4)
(0) =
6
15625
(minus
62500
3
+ 1205776
+ (minus2120574 minus
175
3
) 1205774
+ (
200
3
120574 + 1250 + (
2
3
) 1205742
) 1205772
)
times (1205772
minus 2120574)
119910 (0) = 120577
1199101015840
(0) =
1
50
(1205772
minus 2120574) (120577)
11991010158401015840
(0) =
3
2500
(120577) (1205772
minus 2120574) (1205772
minus
2
3
120574 minus
100
3
)
Mathematical Problems in Engineering 5
119910101584010158401015840
(0) =
3
25000
(120577) (1205772
minus 2120574) (1205774
+ (minus
140
3
minus
8
5
120574) 1205772
+40120574 +
2000
3
+
4
15
1205742
)
119910(4)
(0) =
21
1250000
(120577) (1205776
+ (minus
1240
21
minus
18
7
120574) 1205774
+ (
26000
21
+
320
3
120574 +
52
35
1205742
) 1205772
minus
4000
3
120574 minus
8
105
1205743
minus
160
7
1205742
minus
200000
21
)
times (1205772
minus 2120574)
(29)
where 120574 = 119911(0) and 120577 = 119910(0)From (4) we propose a specific solution for (26) as
follows
119911 (119905) = 1199110
+ 1199111exp (119911
2119905) + 1199113exp (119911
4119905)
119910 (119905) = 1199100
+ 1199101exp (119910
2119905) + 119910
3exp (119910
4119905)
(30)
Finally we calculate the fourth-order Taylor series of(30) equate the resulting coefficients of the same 119905-powerswith respect to (27) and solve the two systems of equationsresulting in the following PSEM approximation
119911 (119905) = 04608270703 + 1539992595 exp (minus1963069736119905)
minus 00008196648996 exp (minus4040101111119905)
119910 (119905) = 09600410114 + 002960275048 exp (minus1964621532119905)
+ 00004060718378 exp (minus4030998609119905)
(31)
For comparison purposes we calculate the Taylor seriesby substituting (29) into (27) yielding
119911 (119905) = 2 minus 3019801327119905 + 29606012221199052
minus 19326574831199053
+ 094380751921199054
119910 (119905) = 09900498337 minus 005979507603119905 + 0060428547451199052
minus 0041845484121199053
+ 0022842654121199054
(32)
4 Numerical Simulation and Discussion
For all case studies we used built-in numerical routines fromMaple 15 for comparison purposes The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant(RKF45) was used [18 19] The command was setup with atolerance of absolute error (A E) of 10
minus12
We obtained a highly accurate rational approximatesolution (16) for the singular second order Lane-Emdenequation [12] (8) as depicted in Figures 1 and 2 for 120572 =
[1 2 3 4 5] Thus the PSEM method may be useful for suchkind of problems with singularities
Additionally we solved the nonlinear energy equationfor combustion of a single iron particle [13] (17) obtaining ahighly accurate approximation (24) as depicted in Figures 3and 4 In the same figures we can observe the Taylor seriessolution (25) noticing the higher precision of the proposedsolution
Finally we approximated a simplisticmodel [14] related toa battery model simulation that is represented as a nonlineardifferential-algebraic equation of index 1 (26) The PSEMapproximation (31) is in good agreement to numerical results(RKF45) for a large period of time as depicted in Figure 5in contrast we can observe in the same figure the poorconvergence of the Taylor series solution As a matter offact if we calculate the asymptotic behaviour of (26) we willconclude that the approximation keeps its high accuracy fora very large period of time On one side setting the 119911
1015840
(119905) = 0
in (26) and solving the system it results in that the stableequilibrium point is 119864
1= [119911infin
= 04608356746 119910infin
=
09600371604] On the other side we calculate the limit when119905 rarr infin of (31) resulting in 119864
2= [119911infin
= 04608270703 119910infin
=
09600410114] Therefore the high proximity among 1198641
and 1198642exhibits the high accuracy of the proposed PSEM
approximation for the rank 119905 isin [0 infin)The PSEM method was able to calculate approximations
with a larger domain of convergence in comparison to theTSM results The reason can lie in the fact that the trialfunction of PSEM technique may potentially contain moreinformation than the Taylor power series As long as theTaylor series of the proposed TF exist we can propose a widevariety of series of functions as trigonometric hyperbolicintegrals among many others However further research isrequired to propose a methodology to choose the optimalTF At the moment users of the method should keep inmind proposing a TF that may potentially reproduce thebehaviour of the exact solution Finally it is importantto remark that as the TF is a finite sum or division ofanalytic functions as shown in (4) or (5) therefore suchsum is convergent [15ndash17] additionally we will require onlya finite number of terms of the Taylor expansions (6) or(7) to obtain the coefficients of the trial functions (4) or(5) Finally it is important to remark that PSEM can beeasily combined with HPM PM HAM VIM DTM andADM among many others Furthermore we can select theapproximativemethoddepending on its facility of applicationto the specific case study Therefore PSEM is a powerfulmalleable technique that can be applied in combinationwith many of the approximative methods reported in litera-ture
5 Conclusions
This work introduced PSEM as a useful tool with highpotential to solve nonlinear differential equations We were
6 Mathematical Problems in Engineering
1
0975
0950
0925
0900
0875
0850
0 02 04 06 08 1
x
y
PSEM120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
Figure 1 RKF45 solution for (8) (symbols) and its approximatePSEM solution (16) (solid line) for different 120572 values
120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
00006
00005
00004
00003
00002
00001
A E
0
0
02 04 06 08 1
x
Figure 2 Absolute error (A E) of (16) with respect to numerical(RKF45) solution for (8)
able to obtain accurate and handy approximations for differ-ent types of problems homogeneous singular Lane-Emdenequation nonlinear energy equation for combustion of asingle iron particle and differential-algebraic system relatedto battery model simulation It is important to remark onthe high flexibility applicability and power of this novelmethod A key point that should be remarked is that the usershould propose a finite sum of division of analytic functions(trial function) that may be chosen according to the natureof the nonlinear problem under study Further researchis required to solve other kinds of problems nonlinearfractional differential equations nonlinear partial differentialequations and nonlinear boundary valued problems amongothers
17
16
15
14
13
12
11
1
0 02 04 06 08 1
120579(120591
)
120591
PSEMRKF45Taylor series
Figure 3 RKF45 solution for (17) (solid circles) its approximatePSEM solution (24) (solid line) and its Taylor series solution (25)(diagonal crosses)
0005
0004
0003
0002
0001
0
0 02 04 06 08 1
A E
120591
PSEMTaylor series
Figure 4 Absolute error (A E) of (24) (solid square) and (25)(empty circle) with respect to numerical (RKF45) solution for (17)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technologyof Mexico (CONACyT) through Grant CB-2010-01 no
Mathematical Problems in Engineering 7
2
15
1
05
00 10 20 30 40
z(t)
t
PSEMRKF45Taylor series
(a)
0 10 20 30 40
t
PSEMRKF45Taylor series
1
099
098
097
096
095
y(t)
(b)
Figure 5 RKF45 solution for (26) (solid circles) its approximate PSEM solution (31) (solid line) and its Taylor series solution (32) (solidsquare)
157024 The authors would like to thank Roberto Castaneda-Sheissa Uriel Filobello-Nino Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[2] H Vazquez-Leal B Benhammouda U A Filobello-Nino et alldquoModified Taylor series method for solving nonlinear differen-tial equationswithmixed boundary conditions defined on finiteintervalsrdquo SpringerPlus vol 3 no 1 article 160 2014
[3] R Barrio M Rodrıguez A Abad and F Blesa ldquoBreakingthe limits the Taylor series methodrdquo Applied Mathematics andComputation vol 217 no 20 pp 7940ndash7954 2011
[4] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[5] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[6] A Barari M Omidvar A R Ghotbi and D D Ganji ldquoApplica-tion of homotopy perturbationmethod and variational iterationmethod to nonlinear oscillator differential equationsrdquo ActaApplicandae Mathematicae vol 104 no 2 pp 161ndash171 2008
[7] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[8] Y-G Wang W-H Lin and N Liu ldquoA homotopy perturbation-based method for large deflection of a cantilever beam undera terminal follower forcerdquo International Journal for Computa-tionalMethods in Engineering Science andMechanics vol 13 no3 pp 197ndash201 2012
[9] M A Fariborzi Araghi and B Rezapour ldquoApplication ofhomotopy perturbationmethod to solve multidimensionalSchrodingerrsquos equationsrdquo International Journal of MathematicalArchive vol 2 no 11 pp 1ndash6 2011
[10] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[11] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[12] A Akyuz-Dascıoglu and H Cerdiuk-Yaslan ldquoThe solutionof high-order nonlinear ordinary differential equations byChebyshev seriesrdquo Applied Mathematics and Computation vol217 no 12 pp 5658ndash5666 2011
[13] M Bidabadi and M Mafi ldquoTime variation of combustiontemperature and burning time of a single iron particlerdquo Inter-national Journal of Thermal Sciences vol 65 pp 136ndash147 2013
[14] R N Methekar V Ramadesigan J C Pirkle Jr and V R Sub-ramanian ldquoAperturbation approach for consistent initializationof index-1 explicit differential-algebraic equations arising frombatterymodel simulationsrdquoComputersampChemical Engineeringvol 35 no 11 pp 2227ndash2234 2011
[15] D G Zill A First Course in Differential Equations with Mod-eling Applications Cengage Learning Boston Mass USA 10thedition 2012
[16] W Belser Formal Power Series and Linear Systems of Meromor-phic Ordinary Differential Equations Springer 1999
[17] M Oberguggenberger and A OstermannAnalysis for Comput-er Scientists Foundations Methods and Algorithms Springer2011
[18] W H Enright K R Jackson S P Noslashrsett and P G ThomsenldquoInterpolants for Runge-Kutta formulasrdquo ACM Transactions onMathematical Software vol 12 no 3 pp 193ndash218 1986
[19] E Fehlberg ldquoKlassische runge-kutta-formeln vier ter und nied-rigerer ordnung mit schrittweitenkontrolle und ihre anwend-ung auf waermeleitungsproblemerdquo Computing vol 6 pp 61ndash711970
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
Mathematical Problems in Engineering 3
Next we derive successively (10) and resolve the systemof equations obtained from the derivatives 119910
(119898)
(0) (119898 =
2 3 ) resulting in
11991010158401015840
= (
1
3
) ((minus119910 minus 1199091205721199101015840
) 119910120572minus1
minus 119910101584010158401015840
119909)
119910101584010158401015840
= (
1
4
) (minus120572 (119909 (120572 minus 1) 1199101015840
+ 2119910) 1199101015840
119910120572minus2
minus119910(120572minus1)
11991010158401015840
120572119909 minus 119910(4)
119909)
119910(4)
= (
1
5
) (minus3119910(120572minus2)
11991010158401015840
(119909 (120572 minus 1) 1199101015840
+ 119910) 120572
minus (120572 minus 1) 120572 (119909 (120572 minus 2) 1199101015840
+ 3119910) 11991010158402
119910120572minus3
minus119910(120572minus1)
120572119909119910101584010158401015840
minus 119910(5)
119909)
(11)
Now substituting the initial conditions of (8) into (11) itresults in
11991010158401015840
(0) = minus
1
3
119910101584010158401015840
(0)
= 0 119910(4)
(0)
= (
1
5
) 120572
(12)
From (5) we propose a specific solution for (8) with thefollowing form
119910 (119909) =
1199100
+ 11991011199091
+ 11991021199092
+ sdot sdot sdot + 119910119899119909119899
1 + 119910119899+1
1199091
+ 119910119899+2
1199092
+ sdot sdot sdot + 1199102119899
119909119899 (13)
where 119910119894(119894 = 0 1 2 ) are constants to be determined and
the order 119899 is chosen as 2 to obtain a handy approximationNext Taylor series of (13) is
119910 (119909) = 1199100
+ (1199101
minus 11991001199103) 119909
+ (1199102
minus 11991001199104
+ (minus1199101
+ 11991001199103) 1199103) 1199092
+ ( (minus1199101
+ 11991001199103) 1199104
+ (minus1199102
+ 11991001199104
+ 11991031199101
minus 11991001199102
3) 1199103) 1199093
+ ((minus1199102
+ 11991001199104
+ 11991031199101
minus 11991001199102
3) 1199104
+ (11991041199101
minus 2119910411991001199103
+ 11991031199102
minus 1199102
31199101
+11991001199103
3) 1199103) 1199094
+ sdot sdot sdot
(14)
Then we equate coefficients of 119909-powers of (9) and(14) to obtain a system of equations which can be solvedsymbolically resulting in
1199100
= 1
1199101
= 0
1199102
=
120572
20
minus
1
6
1199103
= 0
1199104
=
120572
20
(15)
Finally the PSEMapproximation is obtained by substitut-ing (15) into (13) resulting in
119910 (119909) =
1 + (12057220 minus 16) 1199092
1 + (12057220) 1199092
0 lt 119909 le 1 0 le 120572 le 5
(16)
32 Combustion of a Single Iron Particle The nonlinearenergy equation for combustion of a single iron particle [13]is
(1 minus 1205981
(120579 minus 120579infin
)) 1205791015840
+ 120579 minus 120579infin
+ 1205982
(1205794
minus 1205794
surr) minus Ψ = 0
120579 (0) = 1
(17)
where prime denotes differentiation with respect to 120591 and 120579 isthe dimensionless temperature
The values of the parameters are obtained from [13] 120579infin
=
117647 1205981
= 0051595 Ψ = 098579 120579surr = 035294 and1205982
= 0002630145546 Using the initial conditions of (17) andconsidering the expansion point 120591 = 0 yields to the followingTaylor series
120579 (120591) = 120579 (0) +
1205791015840
(0)
1
1205911
+
12057910158401015840
(0)
2
1205912
+ sdot sdot sdot (18)
where 120579(119898)
(0) (119898 = 1 2 ) are unknown constantsNow we resolve (17) for 120579
1015840 and calculate the successivederivatives to obtain 120579
(119898)
(0) (119898 = 1 2 ) resulting in
1205791015840
= minus
120579 minus 120579infin
+ 1205982
(1205794
minus 1205794
surr) minus Ψ
1 minus 1205981
(120579 minus 120579infin
)
12057910158401015840
= minus
12059811205791015840
(120579 minus 120579infin
+ 1205982
(1205794
minus 1205794
surr) minus Ψ)
(1 minus 1205981
(120579 minus 120579infin
))2
minus
1205791015840
+ 412059821205793
1205791015840
1 minus 1205981
(120579 minus 120579infin
)
(19)
4 Mathematical Problems in Engineering
Then substituting the initial conditions of (17) into (19)we get
1205791015840
(0) = minus
120579 (0) minus 120579infin
+ 1205982
(120579 (0)4
minus 1205794
surr ) minus Ψ
1 minus 1205981
(120579 (0) minus 120579infin
)
12057910158401015840
(0) = minus
12059811205791015840
(0) (120579 (0) minus 120579infin
+ 1205982
(120579 (0)4
minus 1205794
surr) minus Ψ)
(1 minus 1205981
(120579 (0) minus 120579infin
))2
minus
1205791015840
(0) + 41205982120579 (0)3
1205791015840
(0)
1 minus 1205981
(120579 (0) minus 120579infin
)
(20)
From (4) we propose a specific solution for (17) as fol-lows
120579 (120591) = 1205790
+ 1205791exp (120579
2120591) (21)
Next Taylor series of (21) is
120579 (120591) = 1205790
+ 1205791
+ 12057911205792120591 + (
1
2
) 12057911205792
21205912
+ sdot sdot sdot (22)
Then we equate coefficients of 120591-powers of (18) and(22) to obtain a system of equations which can be solvedsymbolically resulting in
1205790
=
(11989601198962
minus 1198962
1)
1198962
1205791
=
1198962
1
1198962
1205792
=
1198962
1198961
(23)
where 1198960
= 120579(0) 1198961
= 1205791015840
(0) and 1198962
= 12057910158401015840
(0)Finally substituting (23) into (21) yields to the PSEM
approximation
120579 (120591) =
(11989601198962
minus 1198962
1)
1198962
+ (
1198962
1
1198962
) exp(
1198962120591
1198961
) (24)
For comparison purposes we calculate the Taylor seriesby substituting 120579(0) = 1 and (20) into (18) yielding
120579 (120591) = 1 + 1170326453120591 minus 063241147851205912
(25)
33 Differential-Algebraic Equation Related to Battery ModelSimulation Next PSEM will be applied to a nonlinearequation [14] related to an oversimplified battery modelwhich is formulated as follows
1199111015840
= minus2119911 + 1199102
119911 (0) = 2
minus100 ln (119910) + 2119911 = 5 119910 (0) = exp (minus001)
(26)
where 119911 is the differential variable 119910 is the algebraic variableand primes denote derivative with respect to 119905
Using the initial conditions of (26) and considering theexpansion point 119905 = 0 yields to the following Taylor series
119911 (119905) = 119911 (0) +
1199111015840
(0)
1
1199051
+
11991110158401015840
(0)
2
1199052
+
119911101584010158401015840
(0)
3
1199053
+
119911(4)
(0)
4
1199054
+ sdot sdot sdot
119910 (119905) = 119910 (0) +
1199101015840
(0)
1
1199051
+
11991010158401015840
(0)
2
1199052
+
119910101584010158401015840
(0)
3
1199053
+
119910(4)
(0)
4
1199054
+ sdot sdot sdot
(27)
where derivatives 119911(119898)
(0) and 119910(119898)
(0) (119898 = 1 2 ) areunknown
Firstly we apply an implicit derivative with respect to 119905 ofthe second equation of (26) resulting in
1199111015840
= minus2119911 + 1199102
119911 (0) = 2
1199101015840
= (
1
50
) 1199111015840
119910 119910 (0) = exp (minus001)
(28)
As before we calculate the successive derivatives of (28)and evaluate the resulting equations at 119905 = 0 yielding
119911 (0) = 120574
1199111015840
(0) = 1205772
minus 2120574
11991110158401015840
(0) = (
1
25
) (1205772
minus 50) (1205772
minus 2120574)
119911101584010158401015840
(0) =
2
625
(1205772
minus 2120574) (1205774
+ 1250 + (minus120574 minus 50) 1205772
)
119911(4)
(0) =
6
15625
(minus
62500
3
+ 1205776
+ (minus2120574 minus
175
3
) 1205774
+ (
200
3
120574 + 1250 + (
2
3
) 1205742
) 1205772
)
times (1205772
minus 2120574)
119910 (0) = 120577
1199101015840
(0) =
1
50
(1205772
minus 2120574) (120577)
11991010158401015840
(0) =
3
2500
(120577) (1205772
minus 2120574) (1205772
minus
2
3
120574 minus
100
3
)
Mathematical Problems in Engineering 5
119910101584010158401015840
(0) =
3
25000
(120577) (1205772
minus 2120574) (1205774
+ (minus
140
3
minus
8
5
120574) 1205772
+40120574 +
2000
3
+
4
15
1205742
)
119910(4)
(0) =
21
1250000
(120577) (1205776
+ (minus
1240
21
minus
18
7
120574) 1205774
+ (
26000
21
+
320
3
120574 +
52
35
1205742
) 1205772
minus
4000
3
120574 minus
8
105
1205743
minus
160
7
1205742
minus
200000
21
)
times (1205772
minus 2120574)
(29)
where 120574 = 119911(0) and 120577 = 119910(0)From (4) we propose a specific solution for (26) as
follows
119911 (119905) = 1199110
+ 1199111exp (119911
2119905) + 1199113exp (119911
4119905)
119910 (119905) = 1199100
+ 1199101exp (119910
2119905) + 119910
3exp (119910
4119905)
(30)
Finally we calculate the fourth-order Taylor series of(30) equate the resulting coefficients of the same 119905-powerswith respect to (27) and solve the two systems of equationsresulting in the following PSEM approximation
119911 (119905) = 04608270703 + 1539992595 exp (minus1963069736119905)
minus 00008196648996 exp (minus4040101111119905)
119910 (119905) = 09600410114 + 002960275048 exp (minus1964621532119905)
+ 00004060718378 exp (minus4030998609119905)
(31)
For comparison purposes we calculate the Taylor seriesby substituting (29) into (27) yielding
119911 (119905) = 2 minus 3019801327119905 + 29606012221199052
minus 19326574831199053
+ 094380751921199054
119910 (119905) = 09900498337 minus 005979507603119905 + 0060428547451199052
minus 0041845484121199053
+ 0022842654121199054
(32)
4 Numerical Simulation and Discussion
For all case studies we used built-in numerical routines fromMaple 15 for comparison purposes The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant(RKF45) was used [18 19] The command was setup with atolerance of absolute error (A E) of 10
minus12
We obtained a highly accurate rational approximatesolution (16) for the singular second order Lane-Emdenequation [12] (8) as depicted in Figures 1 and 2 for 120572 =
[1 2 3 4 5] Thus the PSEM method may be useful for suchkind of problems with singularities
Additionally we solved the nonlinear energy equationfor combustion of a single iron particle [13] (17) obtaining ahighly accurate approximation (24) as depicted in Figures 3and 4 In the same figures we can observe the Taylor seriessolution (25) noticing the higher precision of the proposedsolution
Finally we approximated a simplisticmodel [14] related toa battery model simulation that is represented as a nonlineardifferential-algebraic equation of index 1 (26) The PSEMapproximation (31) is in good agreement to numerical results(RKF45) for a large period of time as depicted in Figure 5in contrast we can observe in the same figure the poorconvergence of the Taylor series solution As a matter offact if we calculate the asymptotic behaviour of (26) we willconclude that the approximation keeps its high accuracy fora very large period of time On one side setting the 119911
1015840
(119905) = 0
in (26) and solving the system it results in that the stableequilibrium point is 119864
1= [119911infin
= 04608356746 119910infin
=
09600371604] On the other side we calculate the limit when119905 rarr infin of (31) resulting in 119864
2= [119911infin
= 04608270703 119910infin
=
09600410114] Therefore the high proximity among 1198641
and 1198642exhibits the high accuracy of the proposed PSEM
approximation for the rank 119905 isin [0 infin)The PSEM method was able to calculate approximations
with a larger domain of convergence in comparison to theTSM results The reason can lie in the fact that the trialfunction of PSEM technique may potentially contain moreinformation than the Taylor power series As long as theTaylor series of the proposed TF exist we can propose a widevariety of series of functions as trigonometric hyperbolicintegrals among many others However further research isrequired to propose a methodology to choose the optimalTF At the moment users of the method should keep inmind proposing a TF that may potentially reproduce thebehaviour of the exact solution Finally it is importantto remark that as the TF is a finite sum or division ofanalytic functions as shown in (4) or (5) therefore suchsum is convergent [15ndash17] additionally we will require onlya finite number of terms of the Taylor expansions (6) or(7) to obtain the coefficients of the trial functions (4) or(5) Finally it is important to remark that PSEM can beeasily combined with HPM PM HAM VIM DTM andADM among many others Furthermore we can select theapproximativemethoddepending on its facility of applicationto the specific case study Therefore PSEM is a powerfulmalleable technique that can be applied in combinationwith many of the approximative methods reported in litera-ture
5 Conclusions
This work introduced PSEM as a useful tool with highpotential to solve nonlinear differential equations We were
6 Mathematical Problems in Engineering
1
0975
0950
0925
0900
0875
0850
0 02 04 06 08 1
x
y
PSEM120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
Figure 1 RKF45 solution for (8) (symbols) and its approximatePSEM solution (16) (solid line) for different 120572 values
120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
00006
00005
00004
00003
00002
00001
A E
0
0
02 04 06 08 1
x
Figure 2 Absolute error (A E) of (16) with respect to numerical(RKF45) solution for (8)
able to obtain accurate and handy approximations for differ-ent types of problems homogeneous singular Lane-Emdenequation nonlinear energy equation for combustion of asingle iron particle and differential-algebraic system relatedto battery model simulation It is important to remark onthe high flexibility applicability and power of this novelmethod A key point that should be remarked is that the usershould propose a finite sum of division of analytic functions(trial function) that may be chosen according to the natureof the nonlinear problem under study Further researchis required to solve other kinds of problems nonlinearfractional differential equations nonlinear partial differentialequations and nonlinear boundary valued problems amongothers
17
16
15
14
13
12
11
1
0 02 04 06 08 1
120579(120591
)
120591
PSEMRKF45Taylor series
Figure 3 RKF45 solution for (17) (solid circles) its approximatePSEM solution (24) (solid line) and its Taylor series solution (25)(diagonal crosses)
0005
0004
0003
0002
0001
0
0 02 04 06 08 1
A E
120591
PSEMTaylor series
Figure 4 Absolute error (A E) of (24) (solid square) and (25)(empty circle) with respect to numerical (RKF45) solution for (17)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technologyof Mexico (CONACyT) through Grant CB-2010-01 no
Mathematical Problems in Engineering 7
2
15
1
05
00 10 20 30 40
z(t)
t
PSEMRKF45Taylor series
(a)
0 10 20 30 40
t
PSEMRKF45Taylor series
1
099
098
097
096
095
y(t)
(b)
Figure 5 RKF45 solution for (26) (solid circles) its approximate PSEM solution (31) (solid line) and its Taylor series solution (32) (solidsquare)
157024 The authors would like to thank Roberto Castaneda-Sheissa Uriel Filobello-Nino Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[2] H Vazquez-Leal B Benhammouda U A Filobello-Nino et alldquoModified Taylor series method for solving nonlinear differen-tial equationswithmixed boundary conditions defined on finiteintervalsrdquo SpringerPlus vol 3 no 1 article 160 2014
[3] R Barrio M Rodrıguez A Abad and F Blesa ldquoBreakingthe limits the Taylor series methodrdquo Applied Mathematics andComputation vol 217 no 20 pp 7940ndash7954 2011
[4] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[5] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[6] A Barari M Omidvar A R Ghotbi and D D Ganji ldquoApplica-tion of homotopy perturbationmethod and variational iterationmethod to nonlinear oscillator differential equationsrdquo ActaApplicandae Mathematicae vol 104 no 2 pp 161ndash171 2008
[7] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[8] Y-G Wang W-H Lin and N Liu ldquoA homotopy perturbation-based method for large deflection of a cantilever beam undera terminal follower forcerdquo International Journal for Computa-tionalMethods in Engineering Science andMechanics vol 13 no3 pp 197ndash201 2012
[9] M A Fariborzi Araghi and B Rezapour ldquoApplication ofhomotopy perturbationmethod to solve multidimensionalSchrodingerrsquos equationsrdquo International Journal of MathematicalArchive vol 2 no 11 pp 1ndash6 2011
[10] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[11] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[12] A Akyuz-Dascıoglu and H Cerdiuk-Yaslan ldquoThe solutionof high-order nonlinear ordinary differential equations byChebyshev seriesrdquo Applied Mathematics and Computation vol217 no 12 pp 5658ndash5666 2011
[13] M Bidabadi and M Mafi ldquoTime variation of combustiontemperature and burning time of a single iron particlerdquo Inter-national Journal of Thermal Sciences vol 65 pp 136ndash147 2013
[14] R N Methekar V Ramadesigan J C Pirkle Jr and V R Sub-ramanian ldquoAperturbation approach for consistent initializationof index-1 explicit differential-algebraic equations arising frombatterymodel simulationsrdquoComputersampChemical Engineeringvol 35 no 11 pp 2227ndash2234 2011
[15] D G Zill A First Course in Differential Equations with Mod-eling Applications Cengage Learning Boston Mass USA 10thedition 2012
[16] W Belser Formal Power Series and Linear Systems of Meromor-phic Ordinary Differential Equations Springer 1999
[17] M Oberguggenberger and A OstermannAnalysis for Comput-er Scientists Foundations Methods and Algorithms Springer2011
[18] W H Enright K R Jackson S P Noslashrsett and P G ThomsenldquoInterpolants for Runge-Kutta formulasrdquo ACM Transactions onMathematical Software vol 12 no 3 pp 193ndash218 1986
[19] E Fehlberg ldquoKlassische runge-kutta-formeln vier ter und nied-rigerer ordnung mit schrittweitenkontrolle und ihre anwend-ung auf waermeleitungsproblemerdquo Computing vol 6 pp 61ndash711970
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 Mathematical Problems in Engineering
Then substituting the initial conditions of (17) into (19)we get
1205791015840
(0) = minus
120579 (0) minus 120579infin
+ 1205982
(120579 (0)4
minus 1205794
surr ) minus Ψ
1 minus 1205981
(120579 (0) minus 120579infin
)
12057910158401015840
(0) = minus
12059811205791015840
(0) (120579 (0) minus 120579infin
+ 1205982
(120579 (0)4
minus 1205794
surr) minus Ψ)
(1 minus 1205981
(120579 (0) minus 120579infin
))2
minus
1205791015840
(0) + 41205982120579 (0)3
1205791015840
(0)
1 minus 1205981
(120579 (0) minus 120579infin
)
(20)
From (4) we propose a specific solution for (17) as fol-lows
120579 (120591) = 1205790
+ 1205791exp (120579
2120591) (21)
Next Taylor series of (21) is
120579 (120591) = 1205790
+ 1205791
+ 12057911205792120591 + (
1
2
) 12057911205792
21205912
+ sdot sdot sdot (22)
Then we equate coefficients of 120591-powers of (18) and(22) to obtain a system of equations which can be solvedsymbolically resulting in
1205790
=
(11989601198962
minus 1198962
1)
1198962
1205791
=
1198962
1
1198962
1205792
=
1198962
1198961
(23)
where 1198960
= 120579(0) 1198961
= 1205791015840
(0) and 1198962
= 12057910158401015840
(0)Finally substituting (23) into (21) yields to the PSEM
approximation
120579 (120591) =
(11989601198962
minus 1198962
1)
1198962
+ (
1198962
1
1198962
) exp(
1198962120591
1198961
) (24)
For comparison purposes we calculate the Taylor seriesby substituting 120579(0) = 1 and (20) into (18) yielding
120579 (120591) = 1 + 1170326453120591 minus 063241147851205912
(25)
33 Differential-Algebraic Equation Related to Battery ModelSimulation Next PSEM will be applied to a nonlinearequation [14] related to an oversimplified battery modelwhich is formulated as follows
1199111015840
= minus2119911 + 1199102
119911 (0) = 2
minus100 ln (119910) + 2119911 = 5 119910 (0) = exp (minus001)
(26)
where 119911 is the differential variable 119910 is the algebraic variableand primes denote derivative with respect to 119905
Using the initial conditions of (26) and considering theexpansion point 119905 = 0 yields to the following Taylor series
119911 (119905) = 119911 (0) +
1199111015840
(0)
1
1199051
+
11991110158401015840
(0)
2
1199052
+
119911101584010158401015840
(0)
3
1199053
+
119911(4)
(0)
4
1199054
+ sdot sdot sdot
119910 (119905) = 119910 (0) +
1199101015840
(0)
1
1199051
+
11991010158401015840
(0)
2
1199052
+
119910101584010158401015840
(0)
3
1199053
+
119910(4)
(0)
4
1199054
+ sdot sdot sdot
(27)
where derivatives 119911(119898)
(0) and 119910(119898)
(0) (119898 = 1 2 ) areunknown
Firstly we apply an implicit derivative with respect to 119905 ofthe second equation of (26) resulting in
1199111015840
= minus2119911 + 1199102
119911 (0) = 2
1199101015840
= (
1
50
) 1199111015840
119910 119910 (0) = exp (minus001)
(28)
As before we calculate the successive derivatives of (28)and evaluate the resulting equations at 119905 = 0 yielding
119911 (0) = 120574
1199111015840
(0) = 1205772
minus 2120574
11991110158401015840
(0) = (
1
25
) (1205772
minus 50) (1205772
minus 2120574)
119911101584010158401015840
(0) =
2
625
(1205772
minus 2120574) (1205774
+ 1250 + (minus120574 minus 50) 1205772
)
119911(4)
(0) =
6
15625
(minus
62500
3
+ 1205776
+ (minus2120574 minus
175
3
) 1205774
+ (
200
3
120574 + 1250 + (
2
3
) 1205742
) 1205772
)
times (1205772
minus 2120574)
119910 (0) = 120577
1199101015840
(0) =
1
50
(1205772
minus 2120574) (120577)
11991010158401015840
(0) =
3
2500
(120577) (1205772
minus 2120574) (1205772
minus
2
3
120574 minus
100
3
)
Mathematical Problems in Engineering 5
119910101584010158401015840
(0) =
3
25000
(120577) (1205772
minus 2120574) (1205774
+ (minus
140
3
minus
8
5
120574) 1205772
+40120574 +
2000
3
+
4
15
1205742
)
119910(4)
(0) =
21
1250000
(120577) (1205776
+ (minus
1240
21
minus
18
7
120574) 1205774
+ (
26000
21
+
320
3
120574 +
52
35
1205742
) 1205772
minus
4000
3
120574 minus
8
105
1205743
minus
160
7
1205742
minus
200000
21
)
times (1205772
minus 2120574)
(29)
where 120574 = 119911(0) and 120577 = 119910(0)From (4) we propose a specific solution for (26) as
follows
119911 (119905) = 1199110
+ 1199111exp (119911
2119905) + 1199113exp (119911
4119905)
119910 (119905) = 1199100
+ 1199101exp (119910
2119905) + 119910
3exp (119910
4119905)
(30)
Finally we calculate the fourth-order Taylor series of(30) equate the resulting coefficients of the same 119905-powerswith respect to (27) and solve the two systems of equationsresulting in the following PSEM approximation
119911 (119905) = 04608270703 + 1539992595 exp (minus1963069736119905)
minus 00008196648996 exp (minus4040101111119905)
119910 (119905) = 09600410114 + 002960275048 exp (minus1964621532119905)
+ 00004060718378 exp (minus4030998609119905)
(31)
For comparison purposes we calculate the Taylor seriesby substituting (29) into (27) yielding
119911 (119905) = 2 minus 3019801327119905 + 29606012221199052
minus 19326574831199053
+ 094380751921199054
119910 (119905) = 09900498337 minus 005979507603119905 + 0060428547451199052
minus 0041845484121199053
+ 0022842654121199054
(32)
4 Numerical Simulation and Discussion
For all case studies we used built-in numerical routines fromMaple 15 for comparison purposes The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant(RKF45) was used [18 19] The command was setup with atolerance of absolute error (A E) of 10
minus12
We obtained a highly accurate rational approximatesolution (16) for the singular second order Lane-Emdenequation [12] (8) as depicted in Figures 1 and 2 for 120572 =
[1 2 3 4 5] Thus the PSEM method may be useful for suchkind of problems with singularities
Additionally we solved the nonlinear energy equationfor combustion of a single iron particle [13] (17) obtaining ahighly accurate approximation (24) as depicted in Figures 3and 4 In the same figures we can observe the Taylor seriessolution (25) noticing the higher precision of the proposedsolution
Finally we approximated a simplisticmodel [14] related toa battery model simulation that is represented as a nonlineardifferential-algebraic equation of index 1 (26) The PSEMapproximation (31) is in good agreement to numerical results(RKF45) for a large period of time as depicted in Figure 5in contrast we can observe in the same figure the poorconvergence of the Taylor series solution As a matter offact if we calculate the asymptotic behaviour of (26) we willconclude that the approximation keeps its high accuracy fora very large period of time On one side setting the 119911
1015840
(119905) = 0
in (26) and solving the system it results in that the stableequilibrium point is 119864
1= [119911infin
= 04608356746 119910infin
=
09600371604] On the other side we calculate the limit when119905 rarr infin of (31) resulting in 119864
2= [119911infin
= 04608270703 119910infin
=
09600410114] Therefore the high proximity among 1198641
and 1198642exhibits the high accuracy of the proposed PSEM
approximation for the rank 119905 isin [0 infin)The PSEM method was able to calculate approximations
with a larger domain of convergence in comparison to theTSM results The reason can lie in the fact that the trialfunction of PSEM technique may potentially contain moreinformation than the Taylor power series As long as theTaylor series of the proposed TF exist we can propose a widevariety of series of functions as trigonometric hyperbolicintegrals among many others However further research isrequired to propose a methodology to choose the optimalTF At the moment users of the method should keep inmind proposing a TF that may potentially reproduce thebehaviour of the exact solution Finally it is importantto remark that as the TF is a finite sum or division ofanalytic functions as shown in (4) or (5) therefore suchsum is convergent [15ndash17] additionally we will require onlya finite number of terms of the Taylor expansions (6) or(7) to obtain the coefficients of the trial functions (4) or(5) Finally it is important to remark that PSEM can beeasily combined with HPM PM HAM VIM DTM andADM among many others Furthermore we can select theapproximativemethoddepending on its facility of applicationto the specific case study Therefore PSEM is a powerfulmalleable technique that can be applied in combinationwith many of the approximative methods reported in litera-ture
5 Conclusions
This work introduced PSEM as a useful tool with highpotential to solve nonlinear differential equations We were
6 Mathematical Problems in Engineering
1
0975
0950
0925
0900
0875
0850
0 02 04 06 08 1
x
y
PSEM120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
Figure 1 RKF45 solution for (8) (symbols) and its approximatePSEM solution (16) (solid line) for different 120572 values
120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
00006
00005
00004
00003
00002
00001
A E
0
0
02 04 06 08 1
x
Figure 2 Absolute error (A E) of (16) with respect to numerical(RKF45) solution for (8)
able to obtain accurate and handy approximations for differ-ent types of problems homogeneous singular Lane-Emdenequation nonlinear energy equation for combustion of asingle iron particle and differential-algebraic system relatedto battery model simulation It is important to remark onthe high flexibility applicability and power of this novelmethod A key point that should be remarked is that the usershould propose a finite sum of division of analytic functions(trial function) that may be chosen according to the natureof the nonlinear problem under study Further researchis required to solve other kinds of problems nonlinearfractional differential equations nonlinear partial differentialequations and nonlinear boundary valued problems amongothers
17
16
15
14
13
12
11
1
0 02 04 06 08 1
120579(120591
)
120591
PSEMRKF45Taylor series
Figure 3 RKF45 solution for (17) (solid circles) its approximatePSEM solution (24) (solid line) and its Taylor series solution (25)(diagonal crosses)
0005
0004
0003
0002
0001
0
0 02 04 06 08 1
A E
120591
PSEMTaylor series
Figure 4 Absolute error (A E) of (24) (solid square) and (25)(empty circle) with respect to numerical (RKF45) solution for (17)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technologyof Mexico (CONACyT) through Grant CB-2010-01 no
Mathematical Problems in Engineering 7
2
15
1
05
00 10 20 30 40
z(t)
t
PSEMRKF45Taylor series
(a)
0 10 20 30 40
t
PSEMRKF45Taylor series
1
099
098
097
096
095
y(t)
(b)
Figure 5 RKF45 solution for (26) (solid circles) its approximate PSEM solution (31) (solid line) and its Taylor series solution (32) (solidsquare)
157024 The authors would like to thank Roberto Castaneda-Sheissa Uriel Filobello-Nino Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[2] H Vazquez-Leal B Benhammouda U A Filobello-Nino et alldquoModified Taylor series method for solving nonlinear differen-tial equationswithmixed boundary conditions defined on finiteintervalsrdquo SpringerPlus vol 3 no 1 article 160 2014
[3] R Barrio M Rodrıguez A Abad and F Blesa ldquoBreakingthe limits the Taylor series methodrdquo Applied Mathematics andComputation vol 217 no 20 pp 7940ndash7954 2011
[4] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[5] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[6] A Barari M Omidvar A R Ghotbi and D D Ganji ldquoApplica-tion of homotopy perturbationmethod and variational iterationmethod to nonlinear oscillator differential equationsrdquo ActaApplicandae Mathematicae vol 104 no 2 pp 161ndash171 2008
[7] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[8] Y-G Wang W-H Lin and N Liu ldquoA homotopy perturbation-based method for large deflection of a cantilever beam undera terminal follower forcerdquo International Journal for Computa-tionalMethods in Engineering Science andMechanics vol 13 no3 pp 197ndash201 2012
[9] M A Fariborzi Araghi and B Rezapour ldquoApplication ofhomotopy perturbationmethod to solve multidimensionalSchrodingerrsquos equationsrdquo International Journal of MathematicalArchive vol 2 no 11 pp 1ndash6 2011
[10] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[11] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[12] A Akyuz-Dascıoglu and H Cerdiuk-Yaslan ldquoThe solutionof high-order nonlinear ordinary differential equations byChebyshev seriesrdquo Applied Mathematics and Computation vol217 no 12 pp 5658ndash5666 2011
[13] M Bidabadi and M Mafi ldquoTime variation of combustiontemperature and burning time of a single iron particlerdquo Inter-national Journal of Thermal Sciences vol 65 pp 136ndash147 2013
[14] R N Methekar V Ramadesigan J C Pirkle Jr and V R Sub-ramanian ldquoAperturbation approach for consistent initializationof index-1 explicit differential-algebraic equations arising frombatterymodel simulationsrdquoComputersampChemical Engineeringvol 35 no 11 pp 2227ndash2234 2011
[15] D G Zill A First Course in Differential Equations with Mod-eling Applications Cengage Learning Boston Mass USA 10thedition 2012
[16] W Belser Formal Power Series and Linear Systems of Meromor-phic Ordinary Differential Equations Springer 1999
[17] M Oberguggenberger and A OstermannAnalysis for Comput-er Scientists Foundations Methods and Algorithms Springer2011
[18] W H Enright K R Jackson S P Noslashrsett and P G ThomsenldquoInterpolants for Runge-Kutta formulasrdquo ACM Transactions onMathematical Software vol 12 no 3 pp 193ndash218 1986
[19] E Fehlberg ldquoKlassische runge-kutta-formeln vier ter und nied-rigerer ordnung mit schrittweitenkontrolle und ihre anwend-ung auf waermeleitungsproblemerdquo Computing vol 6 pp 61ndash711970
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
Mathematical Problems in Engineering 5
119910101584010158401015840
(0) =
3
25000
(120577) (1205772
minus 2120574) (1205774
+ (minus
140
3
minus
8
5
120574) 1205772
+40120574 +
2000
3
+
4
15
1205742
)
119910(4)
(0) =
21
1250000
(120577) (1205776
+ (minus
1240
21
minus
18
7
120574) 1205774
+ (
26000
21
+
320
3
120574 +
52
35
1205742
) 1205772
minus
4000
3
120574 minus
8
105
1205743
minus
160
7
1205742
minus
200000
21
)
times (1205772
minus 2120574)
(29)
where 120574 = 119911(0) and 120577 = 119910(0)From (4) we propose a specific solution for (26) as
follows
119911 (119905) = 1199110
+ 1199111exp (119911
2119905) + 1199113exp (119911
4119905)
119910 (119905) = 1199100
+ 1199101exp (119910
2119905) + 119910
3exp (119910
4119905)
(30)
Finally we calculate the fourth-order Taylor series of(30) equate the resulting coefficients of the same 119905-powerswith respect to (27) and solve the two systems of equationsresulting in the following PSEM approximation
119911 (119905) = 04608270703 + 1539992595 exp (minus1963069736119905)
minus 00008196648996 exp (minus4040101111119905)
119910 (119905) = 09600410114 + 002960275048 exp (minus1964621532119905)
+ 00004060718378 exp (minus4030998609119905)
(31)
For comparison purposes we calculate the Taylor seriesby substituting (29) into (27) yielding
119911 (119905) = 2 minus 3019801327119905 + 29606012221199052
minus 19326574831199053
+ 094380751921199054
119910 (119905) = 09900498337 minus 005979507603119905 + 0060428547451199052
minus 0041845484121199053
+ 0022842654121199054
(32)
4 Numerical Simulation and Discussion
For all case studies we used built-in numerical routines fromMaple 15 for comparison purposes The Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant(RKF45) was used [18 19] The command was setup with atolerance of absolute error (A E) of 10
minus12
We obtained a highly accurate rational approximatesolution (16) for the singular second order Lane-Emdenequation [12] (8) as depicted in Figures 1 and 2 for 120572 =
[1 2 3 4 5] Thus the PSEM method may be useful for suchkind of problems with singularities
Additionally we solved the nonlinear energy equationfor combustion of a single iron particle [13] (17) obtaining ahighly accurate approximation (24) as depicted in Figures 3and 4 In the same figures we can observe the Taylor seriessolution (25) noticing the higher precision of the proposedsolution
Finally we approximated a simplisticmodel [14] related toa battery model simulation that is represented as a nonlineardifferential-algebraic equation of index 1 (26) The PSEMapproximation (31) is in good agreement to numerical results(RKF45) for a large period of time as depicted in Figure 5in contrast we can observe in the same figure the poorconvergence of the Taylor series solution As a matter offact if we calculate the asymptotic behaviour of (26) we willconclude that the approximation keeps its high accuracy fora very large period of time On one side setting the 119911
1015840
(119905) = 0
in (26) and solving the system it results in that the stableequilibrium point is 119864
1= [119911infin
= 04608356746 119910infin
=
09600371604] On the other side we calculate the limit when119905 rarr infin of (31) resulting in 119864
2= [119911infin
= 04608270703 119910infin
=
09600410114] Therefore the high proximity among 1198641
and 1198642exhibits the high accuracy of the proposed PSEM
approximation for the rank 119905 isin [0 infin)The PSEM method was able to calculate approximations
with a larger domain of convergence in comparison to theTSM results The reason can lie in the fact that the trialfunction of PSEM technique may potentially contain moreinformation than the Taylor power series As long as theTaylor series of the proposed TF exist we can propose a widevariety of series of functions as trigonometric hyperbolicintegrals among many others However further research isrequired to propose a methodology to choose the optimalTF At the moment users of the method should keep inmind proposing a TF that may potentially reproduce thebehaviour of the exact solution Finally it is importantto remark that as the TF is a finite sum or division ofanalytic functions as shown in (4) or (5) therefore suchsum is convergent [15ndash17] additionally we will require onlya finite number of terms of the Taylor expansions (6) or(7) to obtain the coefficients of the trial functions (4) or(5) Finally it is important to remark that PSEM can beeasily combined with HPM PM HAM VIM DTM andADM among many others Furthermore we can select theapproximativemethoddepending on its facility of applicationto the specific case study Therefore PSEM is a powerfulmalleable technique that can be applied in combinationwith many of the approximative methods reported in litera-ture
5 Conclusions
This work introduced PSEM as a useful tool with highpotential to solve nonlinear differential equations We were
6 Mathematical Problems in Engineering
1
0975
0950
0925
0900
0875
0850
0 02 04 06 08 1
x
y
PSEM120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
Figure 1 RKF45 solution for (8) (symbols) and its approximatePSEM solution (16) (solid line) for different 120572 values
120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
00006
00005
00004
00003
00002
00001
A E
0
0
02 04 06 08 1
x
Figure 2 Absolute error (A E) of (16) with respect to numerical(RKF45) solution for (8)
able to obtain accurate and handy approximations for differ-ent types of problems homogeneous singular Lane-Emdenequation nonlinear energy equation for combustion of asingle iron particle and differential-algebraic system relatedto battery model simulation It is important to remark onthe high flexibility applicability and power of this novelmethod A key point that should be remarked is that the usershould propose a finite sum of division of analytic functions(trial function) that may be chosen according to the natureof the nonlinear problem under study Further researchis required to solve other kinds of problems nonlinearfractional differential equations nonlinear partial differentialequations and nonlinear boundary valued problems amongothers
17
16
15
14
13
12
11
1
0 02 04 06 08 1
120579(120591
)
120591
PSEMRKF45Taylor series
Figure 3 RKF45 solution for (17) (solid circles) its approximatePSEM solution (24) (solid line) and its Taylor series solution (25)(diagonal crosses)
0005
0004
0003
0002
0001
0
0 02 04 06 08 1
A E
120591
PSEMTaylor series
Figure 4 Absolute error (A E) of (24) (solid square) and (25)(empty circle) with respect to numerical (RKF45) solution for (17)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technologyof Mexico (CONACyT) through Grant CB-2010-01 no
Mathematical Problems in Engineering 7
2
15
1
05
00 10 20 30 40
z(t)
t
PSEMRKF45Taylor series
(a)
0 10 20 30 40
t
PSEMRKF45Taylor series
1
099
098
097
096
095
y(t)
(b)
Figure 5 RKF45 solution for (26) (solid circles) its approximate PSEM solution (31) (solid line) and its Taylor series solution (32) (solidsquare)
157024 The authors would like to thank Roberto Castaneda-Sheissa Uriel Filobello-Nino Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[2] H Vazquez-Leal B Benhammouda U A Filobello-Nino et alldquoModified Taylor series method for solving nonlinear differen-tial equationswithmixed boundary conditions defined on finiteintervalsrdquo SpringerPlus vol 3 no 1 article 160 2014
[3] R Barrio M Rodrıguez A Abad and F Blesa ldquoBreakingthe limits the Taylor series methodrdquo Applied Mathematics andComputation vol 217 no 20 pp 7940ndash7954 2011
[4] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[5] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[6] A Barari M Omidvar A R Ghotbi and D D Ganji ldquoApplica-tion of homotopy perturbationmethod and variational iterationmethod to nonlinear oscillator differential equationsrdquo ActaApplicandae Mathematicae vol 104 no 2 pp 161ndash171 2008
[7] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[8] Y-G Wang W-H Lin and N Liu ldquoA homotopy perturbation-based method for large deflection of a cantilever beam undera terminal follower forcerdquo International Journal for Computa-tionalMethods in Engineering Science andMechanics vol 13 no3 pp 197ndash201 2012
[9] M A Fariborzi Araghi and B Rezapour ldquoApplication ofhomotopy perturbationmethod to solve multidimensionalSchrodingerrsquos equationsrdquo International Journal of MathematicalArchive vol 2 no 11 pp 1ndash6 2011
[10] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[11] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[12] A Akyuz-Dascıoglu and H Cerdiuk-Yaslan ldquoThe solutionof high-order nonlinear ordinary differential equations byChebyshev seriesrdquo Applied Mathematics and Computation vol217 no 12 pp 5658ndash5666 2011
[13] M Bidabadi and M Mafi ldquoTime variation of combustiontemperature and burning time of a single iron particlerdquo Inter-national Journal of Thermal Sciences vol 65 pp 136ndash147 2013
[14] R N Methekar V Ramadesigan J C Pirkle Jr and V R Sub-ramanian ldquoAperturbation approach for consistent initializationof index-1 explicit differential-algebraic equations arising frombatterymodel simulationsrdquoComputersampChemical Engineeringvol 35 no 11 pp 2227ndash2234 2011
[15] D G Zill A First Course in Differential Equations with Mod-eling Applications Cengage Learning Boston Mass USA 10thedition 2012
[16] W Belser Formal Power Series and Linear Systems of Meromor-phic Ordinary Differential Equations Springer 1999
[17] M Oberguggenberger and A OstermannAnalysis for Comput-er Scientists Foundations Methods and Algorithms Springer2011
[18] W H Enright K R Jackson S P Noslashrsett and P G ThomsenldquoInterpolants for Runge-Kutta formulasrdquo ACM Transactions onMathematical Software vol 12 no 3 pp 193ndash218 1986
[19] E Fehlberg ldquoKlassische runge-kutta-formeln vier ter und nied-rigerer ordnung mit schrittweitenkontrolle und ihre anwend-ung auf waermeleitungsproblemerdquo Computing vol 6 pp 61ndash711970
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
1
0975
0950
0925
0900
0875
0850
0 02 04 06 08 1
x
y
PSEM120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
Figure 1 RKF45 solution for (8) (symbols) and its approximatePSEM solution (16) (solid line) for different 120572 values
120572 = 5
120572 = 4
120572 = 3
120572 = 2
120572 = 1
00006
00005
00004
00003
00002
00001
A E
0
0
02 04 06 08 1
x
Figure 2 Absolute error (A E) of (16) with respect to numerical(RKF45) solution for (8)
able to obtain accurate and handy approximations for differ-ent types of problems homogeneous singular Lane-Emdenequation nonlinear energy equation for combustion of asingle iron particle and differential-algebraic system relatedto battery model simulation It is important to remark onthe high flexibility applicability and power of this novelmethod A key point that should be remarked is that the usershould propose a finite sum of division of analytic functions(trial function) that may be chosen according to the natureof the nonlinear problem under study Further researchis required to solve other kinds of problems nonlinearfractional differential equations nonlinear partial differentialequations and nonlinear boundary valued problems amongothers
17
16
15
14
13
12
11
1
0 02 04 06 08 1
120579(120591
)
120591
PSEMRKF45Taylor series
Figure 3 RKF45 solution for (17) (solid circles) its approximatePSEM solution (24) (solid line) and its Taylor series solution (25)(diagonal crosses)
0005
0004
0003
0002
0001
0
0 02 04 06 08 1
A E
120591
PSEMTaylor series
Figure 4 Absolute error (A E) of (24) (solid square) and (25)(empty circle) with respect to numerical (RKF45) solution for (17)
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technologyof Mexico (CONACyT) through Grant CB-2010-01 no
Mathematical Problems in Engineering 7
2
15
1
05
00 10 20 30 40
z(t)
t
PSEMRKF45Taylor series
(a)
0 10 20 30 40
t
PSEMRKF45Taylor series
1
099
098
097
096
095
y(t)
(b)
Figure 5 RKF45 solution for (26) (solid circles) its approximate PSEM solution (31) (solid line) and its Taylor series solution (32) (solidsquare)
157024 The authors would like to thank Roberto Castaneda-Sheissa Uriel Filobello-Nino Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[2] H Vazquez-Leal B Benhammouda U A Filobello-Nino et alldquoModified Taylor series method for solving nonlinear differen-tial equationswithmixed boundary conditions defined on finiteintervalsrdquo SpringerPlus vol 3 no 1 article 160 2014
[3] R Barrio M Rodrıguez A Abad and F Blesa ldquoBreakingthe limits the Taylor series methodrdquo Applied Mathematics andComputation vol 217 no 20 pp 7940ndash7954 2011
[4] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[5] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[6] A Barari M Omidvar A R Ghotbi and D D Ganji ldquoApplica-tion of homotopy perturbationmethod and variational iterationmethod to nonlinear oscillator differential equationsrdquo ActaApplicandae Mathematicae vol 104 no 2 pp 161ndash171 2008
[7] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[8] Y-G Wang W-H Lin and N Liu ldquoA homotopy perturbation-based method for large deflection of a cantilever beam undera terminal follower forcerdquo International Journal for Computa-tionalMethods in Engineering Science andMechanics vol 13 no3 pp 197ndash201 2012
[9] M A Fariborzi Araghi and B Rezapour ldquoApplication ofhomotopy perturbationmethod to solve multidimensionalSchrodingerrsquos equationsrdquo International Journal of MathematicalArchive vol 2 no 11 pp 1ndash6 2011
[10] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[11] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[12] A Akyuz-Dascıoglu and H Cerdiuk-Yaslan ldquoThe solutionof high-order nonlinear ordinary differential equations byChebyshev seriesrdquo Applied Mathematics and Computation vol217 no 12 pp 5658ndash5666 2011
[13] M Bidabadi and M Mafi ldquoTime variation of combustiontemperature and burning time of a single iron particlerdquo Inter-national Journal of Thermal Sciences vol 65 pp 136ndash147 2013
[14] R N Methekar V Ramadesigan J C Pirkle Jr and V R Sub-ramanian ldquoAperturbation approach for consistent initializationof index-1 explicit differential-algebraic equations arising frombatterymodel simulationsrdquoComputersampChemical Engineeringvol 35 no 11 pp 2227ndash2234 2011
[15] D G Zill A First Course in Differential Equations with Mod-eling Applications Cengage Learning Boston Mass USA 10thedition 2012
[16] W Belser Formal Power Series and Linear Systems of Meromor-phic Ordinary Differential Equations Springer 1999
[17] M Oberguggenberger and A OstermannAnalysis for Comput-er Scientists Foundations Methods and Algorithms Springer2011
[18] W H Enright K R Jackson S P Noslashrsett and P G ThomsenldquoInterpolants for Runge-Kutta formulasrdquo ACM Transactions onMathematical Software vol 12 no 3 pp 193ndash218 1986
[19] E Fehlberg ldquoKlassische runge-kutta-formeln vier ter und nied-rigerer ordnung mit schrittweitenkontrolle und ihre anwend-ung auf waermeleitungsproblemerdquo Computing vol 6 pp 61ndash711970
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
Mathematical Problems in Engineering 7
2
15
1
05
00 10 20 30 40
z(t)
t
PSEMRKF45Taylor series
(a)
0 10 20 30 40
t
PSEMRKF45Taylor series
1
099
098
097
096
095
y(t)
(b)
Figure 5 RKF45 solution for (26) (solid circles) its approximate PSEM solution (31) (solid line) and its Taylor series solution (32) (solidsquare)
157024 The authors would like to thank Roberto Castaneda-Sheissa Uriel Filobello-Nino Rogelio-Alejandro Callejas-Molina and Roberto Ruiz-Gomez for their contribution tothis project
References
[1] H Vazquez-Leal ldquoGeneralized homotopy method for solvingnonlinear differential equationsrdquo Computational and AppliedMathematics vol 33 no 1 pp 275ndash288 2014
[2] H Vazquez-Leal B Benhammouda U A Filobello-Nino et alldquoModified Taylor series method for solving nonlinear differen-tial equationswithmixed boundary conditions defined on finiteintervalsrdquo SpringerPlus vol 3 no 1 article 160 2014
[3] R Barrio M Rodrıguez A Abad and F Blesa ldquoBreakingthe limits the Taylor series methodrdquo Applied Mathematics andComputation vol 217 no 20 pp 7940ndash7954 2011
[4] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014
[5] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999
[6] A Barari M Omidvar A R Ghotbi and D D Ganji ldquoApplica-tion of homotopy perturbationmethod and variational iterationmethod to nonlinear oscillator differential equationsrdquo ActaApplicandae Mathematicae vol 104 no 2 pp 161ndash171 2008
[7] Y Khan H Vazquez-Leal and Q Wu ldquoAn efficient iteratedmethod for mathematical biology modelrdquo Neural Computingand Applications vol 23 no 3-4 pp 677ndash682 2013
[8] Y-G Wang W-H Lin and N Liu ldquoA homotopy perturbation-based method for large deflection of a cantilever beam undera terminal follower forcerdquo International Journal for Computa-tionalMethods in Engineering Science andMechanics vol 13 no3 pp 197ndash201 2012
[9] M A Fariborzi Araghi and B Rezapour ldquoApplication ofhomotopy perturbationmethod to solve multidimensionalSchrodingerrsquos equationsrdquo International Journal of MathematicalArchive vol 2 no 11 pp 1ndash6 2011
[10] M Bayat I Pakar and A Emadi ldquoVibration of electrostati-cally actuated microbeam by means of homotopy perturbationmethodrdquo Structural Engineering and Mechanics vol 48 no 6pp 823ndash831 2013
[11] M M Rashidi M T Rastegari M Asadi and O A BegldquoA study of non-Newtonian flow and heat transfer over anon-isothermal wedge using the homotopy analysis methodrdquoChemical Engineering Communications vol 199 no 2 pp 231ndash256 2012
[12] A Akyuz-Dascıoglu and H Cerdiuk-Yaslan ldquoThe solutionof high-order nonlinear ordinary differential equations byChebyshev seriesrdquo Applied Mathematics and Computation vol217 no 12 pp 5658ndash5666 2011
[13] M Bidabadi and M Mafi ldquoTime variation of combustiontemperature and burning time of a single iron particlerdquo Inter-national Journal of Thermal Sciences vol 65 pp 136ndash147 2013
[14] R N Methekar V Ramadesigan J C Pirkle Jr and V R Sub-ramanian ldquoAperturbation approach for consistent initializationof index-1 explicit differential-algebraic equations arising frombatterymodel simulationsrdquoComputersampChemical Engineeringvol 35 no 11 pp 2227ndash2234 2011
[15] D G Zill A First Course in Differential Equations with Mod-eling Applications Cengage Learning Boston Mass USA 10thedition 2012
[16] W Belser Formal Power Series and Linear Systems of Meromor-phic Ordinary Differential Equations Springer 1999
[17] M Oberguggenberger and A OstermannAnalysis for Comput-er Scientists Foundations Methods and Algorithms Springer2011
[18] W H Enright K R Jackson S P Noslashrsett and P G ThomsenldquoInterpolants for Runge-Kutta formulasrdquo ACM Transactions onMathematical Software vol 12 no 3 pp 193ndash218 1986
[19] E Fehlberg ldquoKlassische runge-kutta-formeln vier ter und nied-rigerer ordnung mit schrittweitenkontrolle und ihre anwend-ung auf waermeleitungsproblemerdquo Computing vol 6 pp 61ndash711970
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