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Research Article Power Series Extender Method for the Solution of Nonlinear Differential Equations Hector Vazquez-Leal 1 and Arturo Sarmiento-Reyes 2 1 Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Cto. Gonzalo Aguirre Beltr´ an, S/N, 91000 Xalapa, VER, Mexico 2 National 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; [email protected] Received 1 October 2014; Accepted 1 December 2014 Academic Editor: Salvatore Alfonzetti Copyright © 2015 H. Vazquez-Leal and A. Sarmiento-Reyes. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose a power series extender method to obtain approximate solutions of nonlinear differential equations. In order to assess the benefits of this proposal, three nonlinear problems of different kind are solved and compared against the power series solution obtained using an approximative method. e problems are homogeneous Lane-Emden equation of index, governing equation of a burning iron particle, and an explicit differential-algebraic equation related to battery model simulations. e results show that PSEM generates highly accurate handy approximations requiring only a few steps. e main advantage of PSEM is to extend the domain of convergence of the power series solutions of approximative methods as Taylor series method, homotopy perturbation method, homotopy analysis method, variational iteration method, differential transform method, and Adomian decomposition method, among many others. From the application of PSEM, it results in handy easy computable expressions that extend the domain of convergence of high order power series solutions. 1. Introduction Solving nonlinear differential equations is an important task in sciences because many physical phenomena are modelled using such equations. Recently, the generalized homotopy method (GHM) [1] was proposed as a generalization of the homotopy perturbation method (HPM). e application of such method is based on a power series matching that enables GHM to obtain complex and rich expression impossible to obtain using HPM. erefore, using as a guide the main idea of power series matching, behind GHM, we propose a power series method extender (PSEM). e key steps of this proposal are as follows. (1) First, we apply an approximative method to obtain the coefficients of a power series solution. e approx- imative method can be one from the literature like Taylor series method (TSM) [2, 3], power series method (PSM) [4], homotopy perturbation method (HPM) [5–11], perturbation method (PM), homotopy analysis method (HAM), variational iteration method (VIM), differential transform method (DTM), and Adomian decomposition method (ADM), among many others. (2) In the same fashion as GHM [1], we propose a trial function (TF) that can potentially describe the qualitative behaviour of the nonlinear problem. (3) Next, we apply the Taylor series method to a trial function (TF). (4) en, we equate the coefficients of the power series obtained in steps 1 and 3 to obtain a nonlinear algebraic equation system in terms of the coefficients of the TF, which can be solved using symbolic or numerical methods. (5) Finally, the approximate solution is obtained by sub- stituting the calculated coefficients from step 4 into the proposed TF. Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 717404, 7 pages http://dx.doi.org/10.1155/2015/717404

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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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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

Page 2: Research Article Power Series Extender Method for the ...downloads.hindawi.com › journals › mpe › 2015 › 717404.pdf · Research Article Power Series Extender Method for the

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

Page 3: Research Article Power Series Extender Method for the ...downloads.hindawi.com › journals › mpe › 2015 › 717404.pdf · Research Article Power Series Extender Method for the

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

Page 4: Research Article Power Series Extender Method for the ...downloads.hindawi.com › journals › mpe › 2015 › 717404.pdf · Research Article Power Series Extender Method for the

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

Page 5: Research Article Power Series Extender Method for the ...downloads.hindawi.com › journals › mpe › 2015 › 717404.pdf · Research Article Power Series Extender Method for the

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

Page 6: Research Article Power Series Extender Method for the ...downloads.hindawi.com › journals › mpe › 2015 › 717404.pdf · Research Article Power Series Extender Method for the

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

Page 7: Research Article Power Series Extender Method for the ...downloads.hindawi.com › journals › mpe › 2015 › 717404.pdf · Research Article Power Series Extender Method for the

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

Page 8: Research Article Power Series Extender Method for the ...downloads.hindawi.com › journals › mpe › 2015 › 717404.pdf · Research Article Power Series Extender Method for the

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


WN2000RPT WiFi Range Extender Installation Guide · Place the extender, apply power, and connect it to your WiFi network. Place the Extender and Apply Power 1. Place your extender

WN2000RPT WiFi Range Extender Installation Guide · Place the extender, apply power, and connect it to your WiFi network. Place the Extender and Apply Power 1. Place your extender

Research Article Levenberg-Marquardt Method for the ...

Research Article Levenberg-Marquardt Method for the ...

REVIEW ARTICLE Truth Method Dogen Scholarship

REVIEW ARTICLE Truth Method Dogen Scholarship

NETGEAR N300 WiFi Range Extender Model WN3000RPv3 ... · Place the extender, apply power, and connect it to your WiFi network. Place the Extender and Apply Power 1. Place your extender

NETGEAR N300 WiFi Range Extender Model WN3000RPv3 ... · Place the extender, apply power, and connect it to your WiFi network. Place the Extender and Apply Power 1. Place your extender

Universal WiFi Range Extender WN3000RP - Netgear · Extender IP Address Setup ... Universal WiFi Range Extender WN3000RP. Hardware Features. WPS Ethernet port

Universal WiFi Range Extender WN3000RP - Netgear · Extender IP Address Setup ... Universal WiFi Range Extender WN3000RP. Hardware Features. WPS Ethernet port

VGA Extender

VGA Extender

AC1200 WiFi Range Extender Model ... - support.netgear.cnsupport.netgear.cn/Upfilepath_sc/EX6150_Installation_Guide_29May2… · To use the extender in extender mode, set the Access

AC1200 WiFi Range Extender Model ... - support.netgear.cnsupport.netgear.cn/Upfilepath_sc/EX6150_Installation_Guide_29May2… · To use the extender in extender mode, set the Access

Configuring the Fabric Extender · Configuring the Fabric Extender ThischapterdescribeshowtoconfigureaCiscoNexus2000SeriesFabricExtenderusingtheCiscoNexus ...

Configuring the Fabric Extender · Configuring the Fabric Extender ThischapterdescribeshowtoconfigureaCiscoNexus2000SeriesFabricExtenderusingtheCiscoNexus ...

osing Clariion Data Replication Method Article Rossouw

osing Clariion Data Replication Method Article Rossouw

USB Extender

USB Extender

Research Article DoE Method for Operating Parameter ...

Research Article DoE Method for Operating Parameter ...

Research Article Nonuniform Overlapping Method in ...

Research Article Nonuniform Overlapping Method in ...

Research Article Research on Identification Method of ...

Research Article Research on Identification Method of ...

Wireless-N Range Extender User Manual · Extender by using the quick setup method and explain each detailed setting on the web user interface while in repeater mode. 2-1 Repeater

Wireless-N Range Extender User Manual · Extender by using the quick setup method and explain each detailed setting on the web user interface while in repeater mode. 2-1 Repeater

N300 WiFi Range Extender - Netgear · 5 Use the Extender in Extender Mode In extender mode, the extender repeats the signals from an existing WiFi router or access point. …

N300 WiFi Range Extender - Netgear · 5 Use the Extender in Extender Mode In extender mode, the extender repeats the signals from an existing WiFi router or access point. …

Research Article Fracture Mechanics Method for Word ...

Research Article Fracture Mechanics Method for Word ...

Research Article Shafting Alignment Computing Method …downloads.hindawi.com/journals/mpe/2016/1647575.pdf ·  · 2016-01-24Research Article Shafting Alignment Computing Method

Research Article Shafting Alignment Computing Method …downloads.hindawi.com/journals/mpe/2016/1647575.pdf ·  · 2016-01-24Research Article Shafting Alignment Computing Method

Business Research Method (project of 30 article)

Business Research Method (project of 30 article)

AC1900 WiFi Mesh Extender Essentials Edition · 6 Install the extender in extender mode 1. Set the Access Point/Extender switch to Extender. 2. For initial setup, place your extender

AC1900 WiFi Mesh Extender Essentials Edition · 6 Install the extender in extender mode 1. Set the Access Point/Extender switch to Extender. 2. For initial setup, place your extender

DVI / VGA KVM Extender LwLVision 7 - G&D · Leading the way in digital KVM KVM Extender Extender systems to bridge IT-distances DVI / VGA KVM Extender LwLVision 7.6

DVI / VGA KVM Extender LwLVision 7 - G&D · Leading the way in digital KVM KVM Extender Extender systems to bridge IT-distances DVI / VGA KVM Extender LwLVision 7.6