Research ArticleAnalytical Solutions for Systems of Singular PartialDifferential-Algebraic Equations

U Filobello-Nino1 H Vazquez-Leal1 B Benhammouda2 A Perez-Sesma1

V M Jimenez-Fernandez1 J Cervantes-Perez1 A Sarmiento-Reyes3

J Huerta-Chua4 L J Morales-Mendoza5 M Gonzalez-Lee5 A Diaz-Sanchez3

D Pereyra-Diacuteaz1 and R Loacutepez-Martiacutenez6

1Facultad de Instrumentacion Electronica Universidad Veracruzana Circuito Gonzalo Aguirre Beltran SN91000 Xalapa VER Mexico2Higher Colleges of Technology Abu Dhabi Menrsquos College PO Box 25035 Abu Dhabi UAE3Instituto Nacional de Astrofısica Optica y Electronica Luis Enrique Erro 1 72840 Santa Marıa Tonantzintla PUE Mexico4Facultad de Ingenierıa Civil Universidad Veracruzana Venustiano Carranza SN Colonia Revolucion93390 Poza Rica VER Mexico5Departamento de Ingenierıa Electronica Universidad Veracruzana Venustiano Carranza SNColonia Revolucion 93390 Poza Rica VER Mexico6Facultad de Matematicas Universidad Veracruzana Circuito Gonzalo Aguirre Beltran SN 91000 Xalapa VER Mexico

Correspondence should be addressed to H Vazquez-Leal hvazquezuvmx

Received 13 October 2014 Accepted 17 December 2014

Academic Editor Baodong Zheng

Copyright copy 2015 U Filobello-Nino et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper proposes power series method (PSM) in order to find solutions for singular partial differential-algebraic equations(SPDAEs) We will solve three examples to show that PSMmethod can be used to search for analytical solutions of SPDAEs Whatis more we will see that in some cases Pade posttreatment besides enlarging the domain of convergence may be employed inorder to get the exact solution from the truncated series solutions of PSM

1 Introduction

The importance of research on partial differential-algebraicequations (PDAEs) is that they are used in the mathematicalmodeling of many phenomena both practical and theoret-ical These systems arise for example in nanoelectronicselectrical networks and mechanical systems among manyothers Despite the importance of this topic it may beconsidered relatively new and little known

Although the case of constant-coefficient linear PDAEshas been investigated by means of numerical methods forinstance in [1 2] perhaps themore relevant aspect of PDAEsboth linear and nonlinear is the concept of index Thedifferentiation index is defined as the minimum number oftimes that all or part of the PDAEs must be differentiatedwith respect to time in order to obtain the time derivative of

the solution as a continuous function of the solution and itsspace derivatives [3] A fact that justifies the search for othermethods of solution to these equations is that the solutionsof higher index PDAEs (index greater than one) becomevery complicated even for numerical methods and manyapplication problems lead to PDAEs with different indicesA further difficulty to be considered that arises and affectsalso other kinds of systems of differential equations as well asdifferential equations is the presence of singularities whichare related to points at which some terms of the differentialequations become infinite or undefined

In recent years several methods focused on approxi-mating nonlinear and linear problems as an alternative toclassical methods have been reported such as those based onvariational approaches [4ndash7] tanh method [8] exp-function[9 10] Adomianrsquos decomposition method [11ndash16] parameter

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 752523 9 pageshttpdxdoiorg1011552015752523

2 Discrete Dynamics in Nature and Society

expansion [17] homotopy perturbation method [7 16 18ndash46] homotopy analysis method [47] homotopy asymptoticmethod [48] series method [49 50] and perturbationmethod [51ndash54] among many others Also a few exact solu-tions to nonlinear differential equations have been reportedoccasionally [55]

This study shows that power seriesmethod (PSM) [56 57]is able to address the above difficulties to obtain power seriessolutions for singular partial differential-algebraic equations(SPDAEs) that is PDAEswith singular pointsThese systemsturn out to be difficult even for numerical methods Moregenerally we will see that the combination of PSM andPade posttreatment could be effective to improve the PSMrsquostruncated series solutions in convergence rate what is moresometimes it ends up giving the exact solution of the systemsuch as what will happen in our third case study

This paper is organized as follows In Section 2 weintroduce the basic idea of power series method Section 3provides a brief explanation of application of PSM to solveSPDAEs Section 4 presents three case studies one singu-lar nonlinear index-one system one singular linear index-two system and one singular nonlinear index-two systemBesides a discussion on the results is presented in Section 5Finally a brief conclusion is given in Section 6

2 Basic Concept of Power Series Method

It can be considered that a nonlinear differential equation canbe expressed as

119860 (119906) minus 119891 (119905) = 0 119905 isin Ω (1)

with the following boundary condition

119861(119906120597119906

120597119899) = 0 119905 isin Γ (2)

where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119905) is a known analytical function and Γ is thedomain boundary forΩ

PSM [49 50] assumes that the solution of a differentialequation can be written in the following form

119906 (119905) =

infin

sum

119899=0

119906119899119905119899

(3)

where 1199060 1199061 are unknown functions to be determined by

series methodThe method of solution for differential equations can be

summarized as follows

(1) Equation (3) is substituted into (1) and then weregroup the resulting polynomial equation in termsof powers of 119905

(2) We equate each coefficient of the above-mentionedpolynomial to zero

(3) As a consequence a linear algebraic system for theunknowns of (3) is obtained

(4) To conclude the solution of the above system allowsobtaining the coefficients 119906

0 1199061

3 Application of PSM to Solve PDAE Systems

Sincemany applications problems in science and engineeringare often modeled by semiexplicit PDAEs we considertherefore the following class of PDAEs

1199061119905= 120601 (119906 119906

119909 119906119909119909) (4)

0 = 120595 (119906 119906119909 119906119909119909) (119905 119909) isin (0 119879) times (119886 119887) (5)

where 119906119896 [0 119879]times [119886 119887] rarr 119877

119898119896 119896 = 1 2 and 119887 gt 119886 in otherwords 119906 = (119906

1 1199062)

For clarification the method is described for the generalsystem (4)-(5) where the number of unknowns is givenby 1198981+ 1198982 In this notation 119906

1(differential unknown)

has 1198981components and 119906

2(algebraic unknown) has 119898

2

components In fact 1198981and 119898

2can take any values greater

than or equal to one so that the number of unknowns in (4)-(5) is greater than or equal to 2

System (4)-(5) is subject to the initial condition

1199061(0 119909) = 119892 (119909) 119886 le 119909 le 119887 (6)

and some suitable boundary conditions

119861 (119906 (119905 119886) 119906 (119905 119887) 119906119909(119905 119886) 119906

119909(119905 119887)) = 0 0 le 119905 le 119879 (7)

where 119892(119909) is a given functionWe assume that the solution to initial boundary-value

problem (4)ndash(7) exists and is unique and sufficiently smoothTo simplify the exposition of the PSM we integrate first

(4) with respect to 119905 and use the initial condition (6) to obtain

1199061(119905 119909) minus 119892 (119909) minus int

119905

0

120601 (119906 119906119909 119906119909119909) 119889119905 = 0 (8)

It is important to note that the time integration of (4) is notrelevant to the solution procedure presented here so one canapply the PSM directly to (4)

A fact that justifies the use of PSM is that in general termsgetting solutions for PDAEs becomes very complicated evenfor numerical methods Moreover there are not systematicanalytical or numerical methods to solve these problems

In view of PSM we assume the solution components119906119896(119905 119909) 119896 = 1 2 have the form

119906119896(119905 119909) = 119906

1198960(119909) + 119906

1198961(119909) 119905 + 119906

1198962(119909) 1199052

+ sdot sdot sdot (9)

where 119906119896119899(119909) 119896 = 1 2 119899 = 0 1 2 are unknown functions

to be determined later on by the PSMThen substitute (9) into system (4)-(5) and equate the

coefficients of powers of 119905 in the resulting equation to zero toobtain an algebraic linear system for the coefficients whosesolution is employed in (9) with the end of obtaining asolution for (4)ndash(7) in series form These series may havelimited regions of convergence even if we take a large numberof terms Therefore in some cases it will be convenient toapply the Pade resummationmethod to PSM truncated seriesto enlarge the convergence region as depicted in the nextsectionA relevant fact is that the steps outlined in this sectionwill be also sufficient to obtain satisfactory solutions for themost difficult case of SPDAEs

Discrete Dynamics in Nature and Society 3

4 Case Studies

The objective of this section is employing PSM in order tosolve three SPDAE systems

Our results will show the efficiency of the presentedmethod

41 Nonlinear Index-One SPDAE (following Section 31198981= 1

and1198982= 1) Consider the following

1199061119905minus 1199061119909119909

+ 11990611199061119909+1199062

119909= 1199093

minus 6119909119905 + 31199052

1199095

+1199054

1199092 (10)

1199061+ 1199062= 1199051199093

+1199054

119909 119905 gt 0 (11)

subject to the initial conditions

1199061(0 119909) = 0 (12)

In order to apply PSM we integrate (10) with respect to 119905 anduse the initial condition (12) to obtain1199061(119905 119909)

= int

119905

0

[1199061119909119909

minus 11990611199061119909minus1199062

119909+ 1199093

minus 6119909119905 + 31199052

1199095

+1199054

1199092]119889119905

(13)

PSM assumes that 119906(119905 119909) and V(119905 119909) can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (14)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (15)

where 11990610(119909) 11990611(119909) 11990612(119909) 11990620(119909) 11990621(119909) 11990622(119909) are un-

known functionsThis case study is simplified substituting (14) and (15) into

(11) to getinfin

sum

119899=0

1199062119899119905119899

=1199054

119909+ 1199051199093

minus

infin

sum

119899=0

1199061119899119905119899

(16)

On the other hand substituting (14) through (16) into (13)leads to

infin

sum

119899=0

1199061119899119905119899

= int

119905

0

[

infin

sum

119899=0

11990610158401015840

1119899119905119899

minus

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898119905119899+119898

minus1

119909(1199054

119909+ 1199051199093

minus

infin

sum

119899=0

1199061119899119905119899

)

+1199093

minus 6119909119905 + 31199052

1199095

+1199054

1199092]119889119905

(17)

From here on the dash notation in 1199061015840 denotes the ordinaryderivative with respect to 119909

Integrating the above result it is obtained thatinfin

sum

119899=0

1199061119899119905119899

=

infin

sum

119899=0

11990610158401015840

1119899

119905119899+1

119899 + 1minus

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898

119905119899+119898+1

119899 + 119898 + 1

minus1199092

1199052

2+ 1199093

119905 minus 31199091199052

+ 1199095

1199053

+1

119909

infin

sum

119899=0

1199061119899119905119899+1

119899 + 1

(18)

Standardizing the summation index and grouping we get therecursive formula

119906101199050

minus 1199093

119905 + (3119909 +1199092

2) 1199052

minus 1199095

1199053

+

infin

sum

119896=1

[1199061119896minus11990610158401015840

1119896minus1

119896+

infin

sum

119898=0

1199061119896minus119898minus1

1199061015840

1119898

119896minus1199061119896minus1

119896119909] 119905119896

= 0

(19)

Equating the coefficients of powers of 119905 to zero in (19) weobtain

119896 = 0

1199060= 0

119896 = 1

11990611= 11990610158401015840

10minus 1199061015840

1011990610+1

11990911990610+ 1199093

(20)

after employing (20) it is obtained that

11990611= 1199093

119896 = 2

11990612=11990610158401015840

11

2minus1199061015840

1011990611

2minus1199061015840

1111990610

2+11990611

2119909minus1199092

2minus 3119909

(21)

substituting (20) and (21) in the above equation it is obtainedthat

11990612= 0

119896 = 3

11990613=11990610158401015840

12

3minus1199061015840

1011990612

3minus1199061015840

1111990611

3minus1199061015840

1211990610

3+11990612

3119909+ 1199095

(22)

after substituting (20) (21) and (22) in the last equation weget

11990613= 0

119896 = 4

11990614=11990610158401015840

13

4minus1199061015840

1011990613

4minus1199061015840

1111990612

4minus1199061015840

1211990611

4minus1199061015840

1311990610

4+11990613

4119909

(23)

after employing (20) (21) (22) and (23) we get

11990614= 0

119896 = 5

11990615=11990610158401015840

14

5minus1199061015840

1011990614

5minus1199061015840

1111990613

5minus1199061015840

1211990612

5

minus1199061015840

1311990611

5minus1199061015840

1411990610

5+11990614

5119909

(24)

4 Discrete Dynamics in Nature and Society

the substitution of (20) (21) (22) (23) and (24) leads to

11990615= 0 (25)

in the same way we obtain

11990616= 11990617= 11990618= sdot sdot sdot = 0 (26)

Substituting (20) through (26) into (14) leads us to

1199061(119905 119909) = 119909

3

119905 (27)

Finally substituting (27) into (11) leads to

1199062(119905 119909) =

1199054

119909 (28)

Thus (27) and (28) are the exact solution for SPDAE system(10)ndash(12)

42 Linear Index-Two SPDAEwithVariable Coefficients (1198981=

2 1198982= 1) Consider the following

1199061119905= 1199092

1199061119909119909

minus 31199061+ 1199063+

1199092

1 + 119905 (29)

1199062119905= 1199092

1199062119909119909

minus 31199062+ 1199063+

1199092

1 + 119905 (30)

0 = 1199061+ 1199062minus 21199092 ln (1 + 119905) (31)

subject to the initial conditions

1199061(0 119909) = 0 119906

2(0 119909) = 0

minus1 lt 119905 le 1 minusinfin lt 119909 lt infin

(32)

The integration of (29) and (30) with respect to 119905 and usingthe initial conditions (32) lead to

1199061(119905 119909) = int

119905

0

[1199092

1199061119909119909

minus 31199061+ 1199063] 119889119905 + 119909

2 ln (1 + 119905) (33)

1199062(119905 119909) = int

119905

0

[1199092

1199062119909119909

minus 31199062+ 1199063] 119889119905 + 119909

2 ln (1 + 119905) (34)

assuming that 1199061(119905 119909) 119906

2(119905 119909) and 119906

3(119905 119909) can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (35)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (36)

1199063(119905 119909) = 119906

30(119909) + 119906

31(119909) 119905 + 119906

32(119909) 1199052

+ sdot sdot sdot (37)

where 11990610(119909) 11990611(119909) 119906

20(119909) 11990621(119909) 119906

30(119909) 11990631(119909)

are unknown functions to be determined later on by thePSM method

After substituting (35) and (37) into (33) we get

119906101199050

+

infin

sum

119896=1

1

119896[1198961199061119896minus 1199092

11990610158401015840

1119896minus1+ 31199061119896minus1

minus1199063119896minus1

minus 1199092

(minus1)119896minus1

] 119905119896

= 0

(38)

where we have standardized the summation index and em-ployed the following Taylor series expansion

ln (1 + 119905) =infin

sum

119899=1

(minus1)119899minus1

119899119905119899

minus1 lt 119905 le 1 (39)

In the same way the substitution of (36) and (37) into (34)leads to

119906201199050

+

infin

sum

119896=1

1

119896[1198961199062119896minus 1199092

11990610158401015840

2119896minus1+ 31199062119896minus1

minus1199063119896minus1

minus 1199092

(minus1)119896minus1

] 119905119896

= 0

(40)

On the other hand after substituting (35) (36) and (39) into(31) we have

infin

sum

119896=1

[1199061119896+ 1199062119896minus21199092

119896(minus1)119896minus1

] 119905119896

= 0 (41)

where we have employed the following results deduced from(38) and (40)

11990610= 11990620= 0 (42)

Equations (38) (40) and (41) give rise to the followingformulas

1199061119899=1199092

11990610158401015840

1119899minus1minus 31199061119899minus1

+ 1199063119899minus1

+ (minus1)119899minus1

1199092

119899 119899 ge 1 (43)

1199062119899=1199092

11990610158401015840

2119899minus1minus 31199062119899minus1

+ 1199063119899minus1

+ (minus1)119899minus1

1199092

119899 119899 ge 1 (44)

1199061119899+ 1199062119899=21199092

(minus1)119899minus1

119899 119899 ge 1 (45)

Combining the result of adding (43) and (44) with (45) weobtain

1199063119899minus1

= minus1

2(11990610158401015840

1119899minus1+ 11990610158401015840

2119899minus1) 1199092

+3

2(1199061119899minus1

+ 1199062119899minus1

) 119899 ge 1

(46)

The substitution of (46) into (43) and (44) respectively leadsus to

1199061119899=

1

2119899(1199092

11990610158401015840

1119899minus1minus 31199061119899minus1

+ 31199062119899minus1

minus1199092

11990610158401015840

2119899minus1+ 2 (minus1)

119899minus1

1199092

) 119899 ge 1

1199062119899=

1

2119899(1199092

11990610158401015840

2119899minus1minus 31199062119899minus1

+ 31199061119899minus1

minus1199092

11990610158401015840

1119899minus1+ 2 (minus1)

119899minus1

1199092

) 119899 ge 1

(47)

Discrete Dynamics in Nature and Society 5

From recursion formulas (46) and (47) we get the functions

11990610(119909) = 0 119906

11(119909) = 119909

2

11990612(119909) =

minus1199092

2

11990613=1199092

3 119906

14=minus1199092

4sdot sdot sdot

(48)

11990620(119909) = 0 119906

21(119909) = 119909

2

11990622(119909) =

minus1199092

2

11990623=1199092

3 119906

24=minus1199092

4sdot sdot sdot

(49)

11990630(119909) = 0 119906

31(119909) = 119909

2

11990632(119909) =

minus1199092

2

11990633=1199092

3 119906

34=minus1199092

4sdot sdot sdot

(50)

After substituting (48) through (50) into series (35) (36) and(37) respectively we get

1199061(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (51)

1199062(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (52)

1199063(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (53)

After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that

1199061(119905 119909) = 119909

2 ln (1 + 119905)

1199062(119905 119909) = 119909

2 ln (1 + 119905)

1199063(119905 119909) = 119909

2 ln (1 + 119905)

(54)

which is the exact solution of (29)ndash(32) (see (39))

43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898

2= 1) Finally consider the following

1199061119905= 119891 (119909) 119906

1119909119909+ 11990611199061119909minus1 minus 119905

1 + 1199051199063 (55)

1199062119905= 119892 (119909) 119906

2119909119909minus 11990621199062119909+1 + 119905

1 minus 1199051199063 (56)

0 = 1199061(1 + 119905) minus 119906

2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1

(57)

subject to the initial conditions

1199061(0 119909) = 119909 119906

2(0 119909) = 119909 119906

3(0 119909) = 2119909 (58)

where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt

infin

The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to

1199061(119905 119909) = 119909 + int

119905

0

[119891 (119909) 1199061119909119909

+ 11990611199061119909minus1 minus 119905

1 + 1199051199063] 119889119905 (59)

1199062(119905 119909) = 119909 + int

119905

0

[119892 (119909) 1199062119909119909

minus 11990621199062119909+1 + 119905

1 minus 1199051199063] 119889119905 (60)

PSM assumes once again that 1199061(119905 119909) 119906

2(119905 119909) and 119906

3(119905 119909)

can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (61)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (62)

1199063(119905 119909) = 119906

30(119909) + 119906

31(119909) 119905 + 119906

32(119909) 1199052

+ sdot sdot sdot (63)

where 11990610(119909) 11990611(119909) 119906

20(119909) 11990621(119909) 119906

30(119909) 11990631(119909)

are unknown functions to be determined later on by thePSM method

Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin

sum

119899=0

1199061119899119905119899

= 119909 + int

119905

0

119891 (119909)

infin

sum

119899=0

11990610158401015840

1119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898119905119899+119898

119889119905

minus int

119905

0

(1 minus 119905)

infin

sum

119899=0

infin

sum

119898=0

(minus1)119899

1199063119898119905119899+119898

119889119905

(64)infin

sum

119899=0

1199062119899119905119899

= 119909 + int

119905

0

119892 (119909)

infin

sum

119899=0

11990610158401015840

2119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990621198991199061015840

2119898119905119899+119898

119889119905

minus int

119905

0

(1 + 119905)

infin

sum

119899=0

infin

sum

119898=0

1199063119898119905119899+119898

119889119905

(65)

where we have employed the Taylor series expansions

1

1 minus 119905=

infin

sum

119899=0

119905119899

1

1 + 119905=

infin

sum

119899=0

(minus1)119899

119905119899

(66)

After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively

minus 11990610+ 119909 minus 119906

30119905 minus

1

2(11990631minus 211990630) 1199052

minus1

3(11990632minus 211990631+ 211990630) 1199053

minus1

4(11990633minus 211990632+ 211990631minus 211990630) 1199054

+

infin

sum

119896=1

[119891 (119909) 119906

10158401015840

1119896minus1

119896+

infin

sum

119898=0

1199061015840

11198981199061119896minus119898minus1

119896minus 1199061119896] 119905119896

= 0

minus 11990620+ 119909 + 119906

30119905 +

1

2(11990631+ 211990630) 1199052

+1

3(11990632+ 211990631+ 211990630) 1199053

6 Discrete Dynamics in Nature and Society

+1

4(11990633+ 211990632+ 211990631+ 211990630) 1199054

+

infin

sum

119896=1

[119892 (119909) 119906

10158401015840

2119896minus1

119896minus

infin

sum

119898=0

1199061015840

21198981199062119896minus119898minus1

119896minus 1199062119896] 119905119896

= 0

(67)

From (57) we obtain

infin

sum

119898=0

1199062119898119905119898

= (1 + 119905)

infin

sum

119899=0

infin

sum

119895=0

119905119899+119895

1199061119895 (68)

after using again the first series of (66)After standardizing the summation index we get a third

recurrence formula from (68)

1199062119896=

infin

sum

119899=0

[1199061119896minus119899

+ 1199061119896minus119899minus1

] where 119896 = 0 1 2 3

(69)

From recursion formulas (67) and (69) we get the followingcoupled equations

11990610= 11990610(0 119909) (70)

11990611= 119891 (119909) 119906

10158401015840

10+ 1199061015840

1011990610minus 11990630 (71)

11990612= 119891 (119909)

11990610158401015840

11

2+1199061015840

1011990611+ 1199061015840

1111990610

2+11990631

2 (72)

11990613= 119891 (119909)

11990610158401015840

12

3+1199061015840

1011990612+ 1199061015840

1111990611+ 1199061015840

1211990610

3

minus11990632+ 211990630minus 211990631

3

(73)

11990614=11990610158401015840

13

4+1199061015840

1011990613+ 1199061015840

1111990612+ 1199061015840

1211990611+ 1199061015840

1311990610

4

minus11990633minus 211990632+ 211990631minus 211990630

4

(74)

11990620= 11990620(0 119909) (75)

11990621= 119892 (119909) 119906

10158401015840

20minus 1199061015840

2011990620+ 11990630 (76)

11990622= 119892 (119909)

11990610158401015840

21

2minus1199061015840

2011990621+ 1199061015840

2111990620

2+11990631+ 211990630

2 (77)

11990623= 119892 (119909)

11990610158401015840

22

3minus1199061015840

2011990622+ 1199061015840

2111990621+ 1199061015840

2211990620

3

+11990632+ 211990630+ 211990631

3

(78)

11990624= 119892 (119909)

11990610158401015840

23

4minus1199061015840

2011990623+ 1199061015840

2111990622+ 1199061015840

2211990621+ 1199061015840

2311990620

4

+11990633+ 211990632+ 211990631+ 211990630

4

(79)

11990620= 11990610 (80)

11990621= 211990610+ 11990611 (81)

11990622= 11990612+ 211990611+ 211990610 (82)

11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)

11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610

(84)

From (70) through (84) we get the functions

11990610= 119909 119906

11= minus119909 119906

12= 119909

11990613= minus119909 119906

14= 119909 sdot sdot sdot

(85)

11990620= 119909 119906

21= 119909 119906

22= 119909

11990623= 119909 119906

24= 119909 sdot sdot sdot

(86)

11990630= 2119909 119906

31= 0 119906

32= 2119909

11990633= 0 119906

34= 2119909

(87)

Substituting (85) through (87) into series (61) (62) and (63)respectively we get

1199061(119905 119909) = 119909 (1 minus 119905 + 119905

2

minus 1199053

+ 1199054

+ sdot sdot sdot ) (88)

1199062(119905 119909) = 119909 (1 + 119905 + 119905

2

+ 1199053

+ 1199054

+ sdot sdot sdot ) (89)

1199063(119905 119909) = 2119909 (1 + 119905

2

+ 1199054

+ 1199056

+ sdot sdot sdot ) (90)

After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms

1199061(119905 119909) =

119909

1 + 119905

1199062(119905 119909) =

119909

1 minus 119905

1199063(119905 119909) =

2119909

1 minus 1199052

(91)

which is the exact solution of (55)ndash(58) where we haveemployed (66) and

1

1 minus 1199052=

infin

sum

119899=0

1199052119899

(92)

This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general

Discrete Dynamics in Nature and Society 7

terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)

5 Discussion

In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution

Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs

One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs

Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems

6 Conclusion

By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above

all of index greater than one combining PSM and Laplace-Pade resummation

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024

References

[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998

[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999

[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000

[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007

[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007

[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008

[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008

[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005

[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007

[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008

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

[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002

[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional

8 Discrete Dynamics in Nature and Society

decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012

[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011

[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011

[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011

[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007

[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006

[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008

[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009

[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005

[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012

[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009

[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008

[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010

[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011

[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012

[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012

[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012

[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012

[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009

[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009

[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012

[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009

[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013

[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014

[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012

[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012

[42] 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

[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011

[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011

[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra

Discrete Dynamics in Nature and Society 9

and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012

[46] 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

[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011

[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956

[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906

[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995

[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013

[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013

[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013

[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014

[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014

[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011

[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

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

2 Discrete Dynamics in Nature and Society

expansion [17] homotopy perturbation method [7 16 18ndash46] homotopy analysis method [47] homotopy asymptoticmethod [48] series method [49 50] and perturbationmethod [51ndash54] among many others Also a few exact solu-tions to nonlinear differential equations have been reportedoccasionally [55]

This study shows that power seriesmethod (PSM) [56 57]is able to address the above difficulties to obtain power seriessolutions for singular partial differential-algebraic equations(SPDAEs) that is PDAEswith singular pointsThese systemsturn out to be difficult even for numerical methods Moregenerally we will see that the combination of PSM andPade posttreatment could be effective to improve the PSMrsquostruncated series solutions in convergence rate what is moresometimes it ends up giving the exact solution of the systemsuch as what will happen in our third case study

This paper is organized as follows In Section 2 weintroduce the basic idea of power series method Section 3provides a brief explanation of application of PSM to solveSPDAEs Section 4 presents three case studies one singu-lar nonlinear index-one system one singular linear index-two system and one singular nonlinear index-two systemBesides a discussion on the results is presented in Section 5Finally a brief conclusion is given in Section 6

2 Basic Concept of Power Series Method

It can be considered that a nonlinear differential equation canbe expressed as

119860 (119906) minus 119891 (119905) = 0 119905 isin Ω (1)

with the following boundary condition

119861(119906120597119906

120597119899) = 0 119905 isin Γ (2)

where 119860 is a general differential operator 119861 is a boundaryoperator 119891(119905) is a known analytical function and Γ is thedomain boundary forΩ

PSM [49 50] assumes that the solution of a differentialequation can be written in the following form

119906 (119905) =

infin

sum

119899=0

119906119899119905119899

(3)

where 1199060 1199061 are unknown functions to be determined by

series methodThe method of solution for differential equations can be

summarized as follows

(1) Equation (3) is substituted into (1) and then weregroup the resulting polynomial equation in termsof powers of 119905

(2) We equate each coefficient of the above-mentionedpolynomial to zero

(3) As a consequence a linear algebraic system for theunknowns of (3) is obtained

(4) To conclude the solution of the above system allowsobtaining the coefficients 119906

0 1199061

3 Application of PSM to Solve PDAE Systems

Sincemany applications problems in science and engineeringare often modeled by semiexplicit PDAEs we considertherefore the following class of PDAEs

1199061119905= 120601 (119906 119906

119909 119906119909119909) (4)

0 = 120595 (119906 119906119909 119906119909119909) (119905 119909) isin (0 119879) times (119886 119887) (5)

where 119906119896 [0 119879]times [119886 119887] rarr 119877

119898119896 119896 = 1 2 and 119887 gt 119886 in otherwords 119906 = (119906

1 1199062)

For clarification the method is described for the generalsystem (4)-(5) where the number of unknowns is givenby 1198981+ 1198982 In this notation 119906

1(differential unknown)

has 1198981components and 119906

2(algebraic unknown) has 119898

2

components In fact 1198981and 119898

2can take any values greater

than or equal to one so that the number of unknowns in (4)-(5) is greater than or equal to 2

System (4)-(5) is subject to the initial condition

1199061(0 119909) = 119892 (119909) 119886 le 119909 le 119887 (6)

and some suitable boundary conditions

119861 (119906 (119905 119886) 119906 (119905 119887) 119906119909(119905 119886) 119906

119909(119905 119887)) = 0 0 le 119905 le 119879 (7)

where 119892(119909) is a given functionWe assume that the solution to initial boundary-value

problem (4)ndash(7) exists and is unique and sufficiently smoothTo simplify the exposition of the PSM we integrate first

(4) with respect to 119905 and use the initial condition (6) to obtain

1199061(119905 119909) minus 119892 (119909) minus int

119905

0

120601 (119906 119906119909 119906119909119909) 119889119905 = 0 (8)

It is important to note that the time integration of (4) is notrelevant to the solution procedure presented here so one canapply the PSM directly to (4)

A fact that justifies the use of PSM is that in general termsgetting solutions for PDAEs becomes very complicated evenfor numerical methods Moreover there are not systematicanalytical or numerical methods to solve these problems

In view of PSM we assume the solution components119906119896(119905 119909) 119896 = 1 2 have the form

119906119896(119905 119909) = 119906

1198960(119909) + 119906

1198961(119909) 119905 + 119906

1198962(119909) 1199052

+ sdot sdot sdot (9)

where 119906119896119899(119909) 119896 = 1 2 119899 = 0 1 2 are unknown functions

to be determined later on by the PSMThen substitute (9) into system (4)-(5) and equate the

coefficients of powers of 119905 in the resulting equation to zero toobtain an algebraic linear system for the coefficients whosesolution is employed in (9) with the end of obtaining asolution for (4)ndash(7) in series form These series may havelimited regions of convergence even if we take a large numberof terms Therefore in some cases it will be convenient toapply the Pade resummationmethod to PSM truncated seriesto enlarge the convergence region as depicted in the nextsectionA relevant fact is that the steps outlined in this sectionwill be also sufficient to obtain satisfactory solutions for themost difficult case of SPDAEs

Discrete Dynamics in Nature and Society 3

4 Case Studies

The objective of this section is employing PSM in order tosolve three SPDAE systems

Our results will show the efficiency of the presentedmethod

41 Nonlinear Index-One SPDAE (following Section 31198981= 1

and1198982= 1) Consider the following

1199061119905minus 1199061119909119909

+ 11990611199061119909+1199062

119909= 1199093

minus 6119909119905 + 31199052

1199095

+1199054

1199092 (10)

1199061+ 1199062= 1199051199093

+1199054

119909 119905 gt 0 (11)

subject to the initial conditions

1199061(0 119909) = 0 (12)

In order to apply PSM we integrate (10) with respect to 119905 anduse the initial condition (12) to obtain1199061(119905 119909)

= int

119905

0

[1199061119909119909

minus 11990611199061119909minus1199062

119909+ 1199093

minus 6119909119905 + 31199052

1199095

+1199054

1199092]119889119905

(13)

PSM assumes that 119906(119905 119909) and V(119905 119909) can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (14)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (15)

where 11990610(119909) 11990611(119909) 11990612(119909) 11990620(119909) 11990621(119909) 11990622(119909) are un-

known functionsThis case study is simplified substituting (14) and (15) into

(11) to getinfin

sum

119899=0

1199062119899119905119899

=1199054

119909+ 1199051199093

minus

infin

sum

119899=0

1199061119899119905119899

(16)

On the other hand substituting (14) through (16) into (13)leads to

infin

sum

119899=0

1199061119899119905119899

= int

119905

0

[

infin

sum

119899=0

11990610158401015840

1119899119905119899

minus

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898119905119899+119898

minus1

119909(1199054

119909+ 1199051199093

minus

infin

sum

119899=0

1199061119899119905119899

)

+1199093

minus 6119909119905 + 31199052

1199095

+1199054

1199092]119889119905

(17)

From here on the dash notation in 1199061015840 denotes the ordinaryderivative with respect to 119909

Integrating the above result it is obtained thatinfin

sum

119899=0

1199061119899119905119899

=

infin

sum

119899=0

11990610158401015840

1119899

119905119899+1

119899 + 1minus

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898

119905119899+119898+1

119899 + 119898 + 1

minus1199092

1199052

2+ 1199093

119905 minus 31199091199052

+ 1199095

1199053

+1

119909

infin

sum

119899=0

1199061119899119905119899+1

119899 + 1

(18)

Standardizing the summation index and grouping we get therecursive formula

119906101199050

minus 1199093

119905 + (3119909 +1199092

2) 1199052

minus 1199095

1199053

+

infin

sum

119896=1

[1199061119896minus11990610158401015840

1119896minus1

119896+

infin

sum

119898=0

1199061119896minus119898minus1

1199061015840

1119898

119896minus1199061119896minus1

119896119909] 119905119896

= 0

(19)

Equating the coefficients of powers of 119905 to zero in (19) weobtain

119896 = 0

1199060= 0

119896 = 1

11990611= 11990610158401015840

10minus 1199061015840

1011990610+1

11990911990610+ 1199093

(20)

after employing (20) it is obtained that

11990611= 1199093

119896 = 2

11990612=11990610158401015840

11

2minus1199061015840

1011990611

2minus1199061015840

1111990610

2+11990611

2119909minus1199092

2minus 3119909

(21)

substituting (20) and (21) in the above equation it is obtainedthat

11990612= 0

119896 = 3

11990613=11990610158401015840

12

3minus1199061015840

1011990612

3minus1199061015840

1111990611

3minus1199061015840

1211990610

3+11990612

3119909+ 1199095

(22)

after substituting (20) (21) and (22) in the last equation weget

11990613= 0

119896 = 4

11990614=11990610158401015840

13

4minus1199061015840

1011990613

4minus1199061015840

1111990612

4minus1199061015840

1211990611

4minus1199061015840

1311990610

4+11990613

4119909

(23)

after employing (20) (21) (22) and (23) we get

11990614= 0

119896 = 5

11990615=11990610158401015840

14

5minus1199061015840

1011990614

5minus1199061015840

1111990613

5minus1199061015840

1211990612

5

minus1199061015840

1311990611

5minus1199061015840

1411990610

5+11990614

5119909

(24)

4 Discrete Dynamics in Nature and Society

the substitution of (20) (21) (22) (23) and (24) leads to

11990615= 0 (25)

in the same way we obtain

11990616= 11990617= 11990618= sdot sdot sdot = 0 (26)

Substituting (20) through (26) into (14) leads us to

1199061(119905 119909) = 119909

3

119905 (27)

Finally substituting (27) into (11) leads to

1199062(119905 119909) =

1199054

119909 (28)

Thus (27) and (28) are the exact solution for SPDAE system(10)ndash(12)

42 Linear Index-Two SPDAEwithVariable Coefficients (1198981=

2 1198982= 1) Consider the following

1199061119905= 1199092

1199061119909119909

minus 31199061+ 1199063+

1199092

1 + 119905 (29)

1199062119905= 1199092

1199062119909119909

minus 31199062+ 1199063+

1199092

1 + 119905 (30)

0 = 1199061+ 1199062minus 21199092 ln (1 + 119905) (31)

subject to the initial conditions

1199061(0 119909) = 0 119906

2(0 119909) = 0

minus1 lt 119905 le 1 minusinfin lt 119909 lt infin

(32)

The integration of (29) and (30) with respect to 119905 and usingthe initial conditions (32) lead to

1199061(119905 119909) = int

119905

0

[1199092

1199061119909119909

minus 31199061+ 1199063] 119889119905 + 119909

2 ln (1 + 119905) (33)

1199062(119905 119909) = int

119905

0

[1199092

1199062119909119909

minus 31199062+ 1199063] 119889119905 + 119909

2 ln (1 + 119905) (34)

assuming that 1199061(119905 119909) 119906

2(119905 119909) and 119906

3(119905 119909) can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (35)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (36)

1199063(119905 119909) = 119906

30(119909) + 119906

31(119909) 119905 + 119906

32(119909) 1199052

+ sdot sdot sdot (37)

where 11990610(119909) 11990611(119909) 119906

20(119909) 11990621(119909) 119906

30(119909) 11990631(119909)

are unknown functions to be determined later on by thePSM method

After substituting (35) and (37) into (33) we get

119906101199050

+

infin

sum

119896=1

1

119896[1198961199061119896minus 1199092

11990610158401015840

1119896minus1+ 31199061119896minus1

minus1199063119896minus1

minus 1199092

(minus1)119896minus1

] 119905119896

= 0

(38)

where we have standardized the summation index and em-ployed the following Taylor series expansion

ln (1 + 119905) =infin

sum

119899=1

(minus1)119899minus1

119899119905119899

minus1 lt 119905 le 1 (39)

In the same way the substitution of (36) and (37) into (34)leads to

119906201199050

+

infin

sum

119896=1

1

119896[1198961199062119896minus 1199092

11990610158401015840

2119896minus1+ 31199062119896minus1

minus1199063119896minus1

minus 1199092

(minus1)119896minus1

] 119905119896

= 0

(40)

On the other hand after substituting (35) (36) and (39) into(31) we have

infin

sum

119896=1

[1199061119896+ 1199062119896minus21199092

119896(minus1)119896minus1

] 119905119896

= 0 (41)

where we have employed the following results deduced from(38) and (40)

11990610= 11990620= 0 (42)

Equations (38) (40) and (41) give rise to the followingformulas

1199061119899=1199092

11990610158401015840

1119899minus1minus 31199061119899minus1

+ 1199063119899minus1

+ (minus1)119899minus1

1199092

119899 119899 ge 1 (43)

1199062119899=1199092

11990610158401015840

2119899minus1minus 31199062119899minus1

+ 1199063119899minus1

+ (minus1)119899minus1

1199092

119899 119899 ge 1 (44)

1199061119899+ 1199062119899=21199092

(minus1)119899minus1

119899 119899 ge 1 (45)

Combining the result of adding (43) and (44) with (45) weobtain

1199063119899minus1

= minus1

2(11990610158401015840

1119899minus1+ 11990610158401015840

2119899minus1) 1199092

+3

2(1199061119899minus1

+ 1199062119899minus1

) 119899 ge 1

(46)

The substitution of (46) into (43) and (44) respectively leadsus to

1199061119899=

1

2119899(1199092

11990610158401015840

1119899minus1minus 31199061119899minus1

+ 31199062119899minus1

minus1199092

11990610158401015840

2119899minus1+ 2 (minus1)

119899minus1

1199092

) 119899 ge 1

1199062119899=

1

2119899(1199092

11990610158401015840

2119899minus1minus 31199062119899minus1

+ 31199061119899minus1

minus1199092

11990610158401015840

1119899minus1+ 2 (minus1)

119899minus1

1199092

) 119899 ge 1

(47)

Discrete Dynamics in Nature and Society 5

From recursion formulas (46) and (47) we get the functions

11990610(119909) = 0 119906

11(119909) = 119909

2

11990612(119909) =

minus1199092

2

11990613=1199092

3 119906

14=minus1199092

4sdot sdot sdot

(48)

11990620(119909) = 0 119906

21(119909) = 119909

2

11990622(119909) =

minus1199092

2

11990623=1199092

3 119906

24=minus1199092

4sdot sdot sdot

(49)

11990630(119909) = 0 119906

31(119909) = 119909

2

11990632(119909) =

minus1199092

2

11990633=1199092

3 119906

34=minus1199092

4sdot sdot sdot

(50)

After substituting (48) through (50) into series (35) (36) and(37) respectively we get

1199061(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (51)

1199062(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (52)

1199063(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (53)

After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that

1199061(119905 119909) = 119909

2 ln (1 + 119905)

1199062(119905 119909) = 119909

2 ln (1 + 119905)

1199063(119905 119909) = 119909

2 ln (1 + 119905)

(54)

which is the exact solution of (29)ndash(32) (see (39))

43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898

2= 1) Finally consider the following

1199061119905= 119891 (119909) 119906

1119909119909+ 11990611199061119909minus1 minus 119905

1 + 1199051199063 (55)

1199062119905= 119892 (119909) 119906

2119909119909minus 11990621199062119909+1 + 119905

1 minus 1199051199063 (56)

0 = 1199061(1 + 119905) minus 119906

2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1

(57)

subject to the initial conditions

1199061(0 119909) = 119909 119906

2(0 119909) = 119909 119906

3(0 119909) = 2119909 (58)

where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt

infin

The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to

1199061(119905 119909) = 119909 + int

119905

0

[119891 (119909) 1199061119909119909

+ 11990611199061119909minus1 minus 119905

1 + 1199051199063] 119889119905 (59)

1199062(119905 119909) = 119909 + int

119905

0

[119892 (119909) 1199062119909119909

minus 11990621199062119909+1 + 119905

1 minus 1199051199063] 119889119905 (60)

PSM assumes once again that 1199061(119905 119909) 119906

2(119905 119909) and 119906

3(119905 119909)

can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (61)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (62)

1199063(119905 119909) = 119906

30(119909) + 119906

31(119909) 119905 + 119906

32(119909) 1199052

+ sdot sdot sdot (63)

where 11990610(119909) 11990611(119909) 119906

20(119909) 11990621(119909) 119906

30(119909) 11990631(119909)

are unknown functions to be determined later on by thePSM method

Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin

sum

119899=0

1199061119899119905119899

= 119909 + int

119905

0

119891 (119909)

infin

sum

119899=0

11990610158401015840

1119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898119905119899+119898

119889119905

minus int

119905

0

(1 minus 119905)

infin

sum

119899=0

infin

sum

119898=0

(minus1)119899

1199063119898119905119899+119898

119889119905

(64)infin

sum

119899=0

1199062119899119905119899

= 119909 + int

119905

0

119892 (119909)

infin

sum

119899=0

11990610158401015840

2119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990621198991199061015840

2119898119905119899+119898

119889119905

minus int

119905

0

(1 + 119905)

infin

sum

119899=0

infin

sum

119898=0

1199063119898119905119899+119898

119889119905

(65)

where we have employed the Taylor series expansions

1

1 minus 119905=

infin

sum

119899=0

119905119899

1

1 + 119905=

infin

sum

119899=0

(minus1)119899

119905119899

(66)

After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively

minus 11990610+ 119909 minus 119906

30119905 minus

1

2(11990631minus 211990630) 1199052

minus1

3(11990632minus 211990631+ 211990630) 1199053

minus1

4(11990633minus 211990632+ 211990631minus 211990630) 1199054

+

infin

sum

119896=1

[119891 (119909) 119906

10158401015840

1119896minus1

119896+

infin

sum

119898=0

1199061015840

11198981199061119896minus119898minus1

119896minus 1199061119896] 119905119896

= 0

minus 11990620+ 119909 + 119906

30119905 +

1

2(11990631+ 211990630) 1199052

+1

3(11990632+ 211990631+ 211990630) 1199053

6 Discrete Dynamics in Nature and Society

+1

4(11990633+ 211990632+ 211990631+ 211990630) 1199054

+

infin

sum

119896=1

[119892 (119909) 119906

10158401015840

2119896minus1

119896minus

infin

sum

119898=0

1199061015840

21198981199062119896minus119898minus1

119896minus 1199062119896] 119905119896

= 0

(67)

From (57) we obtain

infin

sum

119898=0

1199062119898119905119898

= (1 + 119905)

infin

sum

119899=0

infin

sum

119895=0

119905119899+119895

1199061119895 (68)

after using again the first series of (66)After standardizing the summation index we get a third

recurrence formula from (68)

1199062119896=

infin

sum

119899=0

[1199061119896minus119899

+ 1199061119896minus119899minus1

] where 119896 = 0 1 2 3

(69)

From recursion formulas (67) and (69) we get the followingcoupled equations

11990610= 11990610(0 119909) (70)

11990611= 119891 (119909) 119906

10158401015840

10+ 1199061015840

1011990610minus 11990630 (71)

11990612= 119891 (119909)

11990610158401015840

11

2+1199061015840

1011990611+ 1199061015840

1111990610

2+11990631

2 (72)

11990613= 119891 (119909)

11990610158401015840

12

3+1199061015840

1011990612+ 1199061015840

1111990611+ 1199061015840

1211990610

3

minus11990632+ 211990630minus 211990631

3

(73)

11990614=11990610158401015840

13

4+1199061015840

1011990613+ 1199061015840

1111990612+ 1199061015840

1211990611+ 1199061015840

1311990610

4

minus11990633minus 211990632+ 211990631minus 211990630

4

(74)

11990620= 11990620(0 119909) (75)

11990621= 119892 (119909) 119906

10158401015840

20minus 1199061015840

2011990620+ 11990630 (76)

11990622= 119892 (119909)

11990610158401015840

21

2minus1199061015840

2011990621+ 1199061015840

2111990620

2+11990631+ 211990630

2 (77)

11990623= 119892 (119909)

11990610158401015840

22

3minus1199061015840

2011990622+ 1199061015840

2111990621+ 1199061015840

2211990620

3

+11990632+ 211990630+ 211990631

3

(78)

11990624= 119892 (119909)

11990610158401015840

23

4minus1199061015840

2011990623+ 1199061015840

2111990622+ 1199061015840

2211990621+ 1199061015840

2311990620

4

+11990633+ 211990632+ 211990631+ 211990630

4

(79)

11990620= 11990610 (80)

11990621= 211990610+ 11990611 (81)

11990622= 11990612+ 211990611+ 211990610 (82)

11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)

11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610

(84)

From (70) through (84) we get the functions

11990610= 119909 119906

11= minus119909 119906

12= 119909

11990613= minus119909 119906

14= 119909 sdot sdot sdot

(85)

11990620= 119909 119906

21= 119909 119906

22= 119909

11990623= 119909 119906

24= 119909 sdot sdot sdot

(86)

11990630= 2119909 119906

31= 0 119906

32= 2119909

11990633= 0 119906

34= 2119909

(87)

Substituting (85) through (87) into series (61) (62) and (63)respectively we get

1199061(119905 119909) = 119909 (1 minus 119905 + 119905

2

minus 1199053

+ 1199054

+ sdot sdot sdot ) (88)

1199062(119905 119909) = 119909 (1 + 119905 + 119905

2

+ 1199053

+ 1199054

+ sdot sdot sdot ) (89)

1199063(119905 119909) = 2119909 (1 + 119905

2

+ 1199054

+ 1199056

+ sdot sdot sdot ) (90)

After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms

1199061(119905 119909) =

119909

1 + 119905

1199062(119905 119909) =

119909

1 minus 119905

1199063(119905 119909) =

2119909

1 minus 1199052

(91)

which is the exact solution of (55)ndash(58) where we haveemployed (66) and

1

1 minus 1199052=

infin

sum

119899=0

1199052119899

(92)

This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general

Discrete Dynamics in Nature and Society 7

terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)

5 Discussion

In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution

Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs

One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs

Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems

6 Conclusion

By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above

all of index greater than one combining PSM and Laplace-Pade resummation

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024

References

[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998

[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999

[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000

[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007

[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007

[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008

[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008

[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005

[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007

[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008

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

[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002

[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional

8 Discrete Dynamics in Nature and Society

decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012

[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011

[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011

[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011

[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007

[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006

[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008

[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009

[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005

[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012

[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009

[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008

[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010

[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011

[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012

[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012

[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012

[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012

[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009

[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009

[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012

[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009

[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013

[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014

[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012

[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012

[42] 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

[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011

[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011

[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra

Discrete Dynamics in Nature and Society 9

and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012

[46] 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

[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011

[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956

[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906

[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995

[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013

[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013

[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013

[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014

[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014

[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011

[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

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

Discrete Dynamics in Nature and Society 3

4 Case Studies

The objective of this section is employing PSM in order tosolve three SPDAE systems

Our results will show the efficiency of the presentedmethod

41 Nonlinear Index-One SPDAE (following Section 31198981= 1

and1198982= 1) Consider the following

1199061119905minus 1199061119909119909

+ 11990611199061119909+1199062

119909= 1199093

minus 6119909119905 + 31199052

1199095

+1199054

1199092 (10)

1199061+ 1199062= 1199051199093

+1199054

119909 119905 gt 0 (11)

subject to the initial conditions

1199061(0 119909) = 0 (12)

In order to apply PSM we integrate (10) with respect to 119905 anduse the initial condition (12) to obtain1199061(119905 119909)

= int

119905

0

[1199061119909119909

minus 11990611199061119909minus1199062

119909+ 1199093

minus 6119909119905 + 31199052

1199095

+1199054

1199092]119889119905

(13)

PSM assumes that 119906(119905 119909) and V(119905 119909) can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (14)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (15)

where 11990610(119909) 11990611(119909) 11990612(119909) 11990620(119909) 11990621(119909) 11990622(119909) are un-

known functionsThis case study is simplified substituting (14) and (15) into

(11) to getinfin

sum

119899=0

1199062119899119905119899

=1199054

119909+ 1199051199093

minus

infin

sum

119899=0

1199061119899119905119899

(16)

On the other hand substituting (14) through (16) into (13)leads to

infin

sum

119899=0

1199061119899119905119899

= int

119905

0

[

infin

sum

119899=0

11990610158401015840

1119899119905119899

minus

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898119905119899+119898

minus1

119909(1199054

119909+ 1199051199093

minus

infin

sum

119899=0

1199061119899119905119899

)

+1199093

minus 6119909119905 + 31199052

1199095

+1199054

1199092]119889119905

(17)

From here on the dash notation in 1199061015840 denotes the ordinaryderivative with respect to 119909

Integrating the above result it is obtained thatinfin

sum

119899=0

1199061119899119905119899

=

infin

sum

119899=0

11990610158401015840

1119899

119905119899+1

119899 + 1minus

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898

119905119899+119898+1

119899 + 119898 + 1

minus1199092

1199052

2+ 1199093

119905 minus 31199091199052

+ 1199095

1199053

+1

119909

infin

sum

119899=0

1199061119899119905119899+1

119899 + 1

(18)

Standardizing the summation index and grouping we get therecursive formula

119906101199050

minus 1199093

119905 + (3119909 +1199092

2) 1199052

minus 1199095

1199053

+

infin

sum

119896=1

[1199061119896minus11990610158401015840

1119896minus1

119896+

infin

sum

119898=0

1199061119896minus119898minus1

1199061015840

1119898

119896minus1199061119896minus1

119896119909] 119905119896

= 0

(19)

Equating the coefficients of powers of 119905 to zero in (19) weobtain

119896 = 0

1199060= 0

119896 = 1

11990611= 11990610158401015840

10minus 1199061015840

1011990610+1

11990911990610+ 1199093

(20)

after employing (20) it is obtained that

11990611= 1199093

119896 = 2

11990612=11990610158401015840

11

2minus1199061015840

1011990611

2minus1199061015840

1111990610

2+11990611

2119909minus1199092

2minus 3119909

(21)

substituting (20) and (21) in the above equation it is obtainedthat

11990612= 0

119896 = 3

11990613=11990610158401015840

12

3minus1199061015840

1011990612

3minus1199061015840

1111990611

3minus1199061015840

1211990610

3+11990612

3119909+ 1199095

(22)

after substituting (20) (21) and (22) in the last equation weget

11990613= 0

119896 = 4

11990614=11990610158401015840

13

4minus1199061015840

1011990613

4minus1199061015840

1111990612

4minus1199061015840

1211990611

4minus1199061015840

1311990610

4+11990613

4119909

(23)

after employing (20) (21) (22) and (23) we get

11990614= 0

119896 = 5

11990615=11990610158401015840

14

5minus1199061015840

1011990614

5minus1199061015840

1111990613

5minus1199061015840

1211990612

5

minus1199061015840

1311990611

5minus1199061015840

1411990610

5+11990614

5119909

(24)

4 Discrete Dynamics in Nature and Society

the substitution of (20) (21) (22) (23) and (24) leads to

11990615= 0 (25)

in the same way we obtain

11990616= 11990617= 11990618= sdot sdot sdot = 0 (26)

Substituting (20) through (26) into (14) leads us to

1199061(119905 119909) = 119909

3

119905 (27)

Finally substituting (27) into (11) leads to

1199062(119905 119909) =

1199054

119909 (28)

Thus (27) and (28) are the exact solution for SPDAE system(10)ndash(12)

42 Linear Index-Two SPDAEwithVariable Coefficients (1198981=

2 1198982= 1) Consider the following

1199061119905= 1199092

1199061119909119909

minus 31199061+ 1199063+

1199092

1 + 119905 (29)

1199062119905= 1199092

1199062119909119909

minus 31199062+ 1199063+

1199092

1 + 119905 (30)

0 = 1199061+ 1199062minus 21199092 ln (1 + 119905) (31)

subject to the initial conditions

1199061(0 119909) = 0 119906

2(0 119909) = 0

minus1 lt 119905 le 1 minusinfin lt 119909 lt infin

(32)

The integration of (29) and (30) with respect to 119905 and usingthe initial conditions (32) lead to

1199061(119905 119909) = int

119905

0

[1199092

1199061119909119909

minus 31199061+ 1199063] 119889119905 + 119909

2 ln (1 + 119905) (33)

1199062(119905 119909) = int

119905

0

[1199092

1199062119909119909

minus 31199062+ 1199063] 119889119905 + 119909

2 ln (1 + 119905) (34)

assuming that 1199061(119905 119909) 119906

2(119905 119909) and 119906

3(119905 119909) can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (35)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (36)

1199063(119905 119909) = 119906

30(119909) + 119906

31(119909) 119905 + 119906

32(119909) 1199052

+ sdot sdot sdot (37)

where 11990610(119909) 11990611(119909) 119906

20(119909) 11990621(119909) 119906

30(119909) 11990631(119909)

are unknown functions to be determined later on by thePSM method

After substituting (35) and (37) into (33) we get

119906101199050

+

infin

sum

119896=1

1

119896[1198961199061119896minus 1199092

11990610158401015840

1119896minus1+ 31199061119896minus1

minus1199063119896minus1

minus 1199092

(minus1)119896minus1

] 119905119896

= 0

(38)

where we have standardized the summation index and em-ployed the following Taylor series expansion

ln (1 + 119905) =infin

sum

119899=1

(minus1)119899minus1

119899119905119899

minus1 lt 119905 le 1 (39)

In the same way the substitution of (36) and (37) into (34)leads to

119906201199050

+

infin

sum

119896=1

1

119896[1198961199062119896minus 1199092

11990610158401015840

2119896minus1+ 31199062119896minus1

minus1199063119896minus1

minus 1199092

(minus1)119896minus1

] 119905119896

= 0

(40)

On the other hand after substituting (35) (36) and (39) into(31) we have

infin

sum

119896=1

[1199061119896+ 1199062119896minus21199092

119896(minus1)119896minus1

] 119905119896

= 0 (41)

where we have employed the following results deduced from(38) and (40)

11990610= 11990620= 0 (42)

Equations (38) (40) and (41) give rise to the followingformulas

1199061119899=1199092

11990610158401015840

1119899minus1minus 31199061119899minus1

+ 1199063119899minus1

+ (minus1)119899minus1

1199092

119899 119899 ge 1 (43)

1199062119899=1199092

11990610158401015840

2119899minus1minus 31199062119899minus1

+ 1199063119899minus1

+ (minus1)119899minus1

1199092

119899 119899 ge 1 (44)

1199061119899+ 1199062119899=21199092

(minus1)119899minus1

119899 119899 ge 1 (45)

Combining the result of adding (43) and (44) with (45) weobtain

1199063119899minus1

= minus1

2(11990610158401015840

1119899minus1+ 11990610158401015840

2119899minus1) 1199092

+3

2(1199061119899minus1

+ 1199062119899minus1

) 119899 ge 1

(46)

The substitution of (46) into (43) and (44) respectively leadsus to

1199061119899=

1

2119899(1199092

11990610158401015840

1119899minus1minus 31199061119899minus1

+ 31199062119899minus1

minus1199092

11990610158401015840

2119899minus1+ 2 (minus1)

119899minus1

1199092

) 119899 ge 1

1199062119899=

1

2119899(1199092

11990610158401015840

2119899minus1minus 31199062119899minus1

+ 31199061119899minus1

minus1199092

11990610158401015840

1119899minus1+ 2 (minus1)

119899minus1

1199092

) 119899 ge 1

(47)

Discrete Dynamics in Nature and Society 5

From recursion formulas (46) and (47) we get the functions

11990610(119909) = 0 119906

11(119909) = 119909

2

11990612(119909) =

minus1199092

2

11990613=1199092

3 119906

14=minus1199092

4sdot sdot sdot

(48)

11990620(119909) = 0 119906

21(119909) = 119909

2

11990622(119909) =

minus1199092

2

11990623=1199092

3 119906

24=minus1199092

4sdot sdot sdot

(49)

11990630(119909) = 0 119906

31(119909) = 119909

2

11990632(119909) =

minus1199092

2

11990633=1199092

3 119906

34=minus1199092

4sdot sdot sdot

(50)

After substituting (48) through (50) into series (35) (36) and(37) respectively we get

1199061(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (51)

1199062(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (52)

1199063(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (53)

After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that

1199061(119905 119909) = 119909

2 ln (1 + 119905)

1199062(119905 119909) = 119909

2 ln (1 + 119905)

1199063(119905 119909) = 119909

2 ln (1 + 119905)

(54)

which is the exact solution of (29)ndash(32) (see (39))

43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898

2= 1) Finally consider the following

1199061119905= 119891 (119909) 119906

1119909119909+ 11990611199061119909minus1 minus 119905

1 + 1199051199063 (55)

1199062119905= 119892 (119909) 119906

2119909119909minus 11990621199062119909+1 + 119905

1 minus 1199051199063 (56)

0 = 1199061(1 + 119905) minus 119906

2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1

(57)

subject to the initial conditions

1199061(0 119909) = 119909 119906

2(0 119909) = 119909 119906

3(0 119909) = 2119909 (58)

where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt

infin

The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to

1199061(119905 119909) = 119909 + int

119905

0

[119891 (119909) 1199061119909119909

+ 11990611199061119909minus1 minus 119905

1 + 1199051199063] 119889119905 (59)

1199062(119905 119909) = 119909 + int

119905

0

[119892 (119909) 1199062119909119909

minus 11990621199062119909+1 + 119905

1 minus 1199051199063] 119889119905 (60)

PSM assumes once again that 1199061(119905 119909) 119906

2(119905 119909) and 119906

3(119905 119909)

can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (61)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (62)

1199063(119905 119909) = 119906

30(119909) + 119906

31(119909) 119905 + 119906

32(119909) 1199052

+ sdot sdot sdot (63)

where 11990610(119909) 11990611(119909) 119906

20(119909) 11990621(119909) 119906

30(119909) 11990631(119909)

are unknown functions to be determined later on by thePSM method

Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin

sum

119899=0

1199061119899119905119899

= 119909 + int

119905

0

119891 (119909)

infin

sum

119899=0

11990610158401015840

1119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898119905119899+119898

119889119905

minus int

119905

0

(1 minus 119905)

infin

sum

119899=0

infin

sum

119898=0

(minus1)119899

1199063119898119905119899+119898

119889119905

(64)infin

sum

119899=0

1199062119899119905119899

= 119909 + int

119905

0

119892 (119909)

infin

sum

119899=0

11990610158401015840

2119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990621198991199061015840

2119898119905119899+119898

119889119905

minus int

119905

0

(1 + 119905)

infin

sum

119899=0

infin

sum

119898=0

1199063119898119905119899+119898

119889119905

(65)

where we have employed the Taylor series expansions

1

1 minus 119905=

infin

sum

119899=0

119905119899

1

1 + 119905=

infin

sum

119899=0

(minus1)119899

119905119899

(66)

After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively

minus 11990610+ 119909 minus 119906

30119905 minus

1

2(11990631minus 211990630) 1199052

minus1

3(11990632minus 211990631+ 211990630) 1199053

minus1

4(11990633minus 211990632+ 211990631minus 211990630) 1199054

+

infin

sum

119896=1

[119891 (119909) 119906

10158401015840

1119896minus1

119896+

infin

sum

119898=0

1199061015840

11198981199061119896minus119898minus1

119896minus 1199061119896] 119905119896

= 0

minus 11990620+ 119909 + 119906

30119905 +

1

2(11990631+ 211990630) 1199052

+1

3(11990632+ 211990631+ 211990630) 1199053

6 Discrete Dynamics in Nature and Society

+1

4(11990633+ 211990632+ 211990631+ 211990630) 1199054

+

infin

sum

119896=1

[119892 (119909) 119906

10158401015840

2119896minus1

119896minus

infin

sum

119898=0

1199061015840

21198981199062119896minus119898minus1

119896minus 1199062119896] 119905119896

= 0

(67)

From (57) we obtain

infin

sum

119898=0

1199062119898119905119898

= (1 + 119905)

infin

sum

119899=0

infin

sum

119895=0

119905119899+119895

1199061119895 (68)

after using again the first series of (66)After standardizing the summation index we get a third

recurrence formula from (68)

1199062119896=

infin

sum

119899=0

[1199061119896minus119899

+ 1199061119896minus119899minus1

] where 119896 = 0 1 2 3

(69)

From recursion formulas (67) and (69) we get the followingcoupled equations

11990610= 11990610(0 119909) (70)

11990611= 119891 (119909) 119906

10158401015840

10+ 1199061015840

1011990610minus 11990630 (71)

11990612= 119891 (119909)

11990610158401015840

11

2+1199061015840

1011990611+ 1199061015840

1111990610

2+11990631

2 (72)

11990613= 119891 (119909)

11990610158401015840

12

3+1199061015840

1011990612+ 1199061015840

1111990611+ 1199061015840

1211990610

3

minus11990632+ 211990630minus 211990631

3

(73)

11990614=11990610158401015840

13

4+1199061015840

1011990613+ 1199061015840

1111990612+ 1199061015840

1211990611+ 1199061015840

1311990610

4

minus11990633minus 211990632+ 211990631minus 211990630

4

(74)

11990620= 11990620(0 119909) (75)

11990621= 119892 (119909) 119906

10158401015840

20minus 1199061015840

2011990620+ 11990630 (76)

11990622= 119892 (119909)

11990610158401015840

21

2minus1199061015840

2011990621+ 1199061015840

2111990620

2+11990631+ 211990630

2 (77)

11990623= 119892 (119909)

11990610158401015840

22

3minus1199061015840

2011990622+ 1199061015840

2111990621+ 1199061015840

2211990620

3

+11990632+ 211990630+ 211990631

3

(78)

11990624= 119892 (119909)

11990610158401015840

23

4minus1199061015840

2011990623+ 1199061015840

2111990622+ 1199061015840

2211990621+ 1199061015840

2311990620

4

+11990633+ 211990632+ 211990631+ 211990630

4

(79)

11990620= 11990610 (80)

11990621= 211990610+ 11990611 (81)

11990622= 11990612+ 211990611+ 211990610 (82)

11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)

11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610

(84)

From (70) through (84) we get the functions

11990610= 119909 119906

11= minus119909 119906

12= 119909

11990613= minus119909 119906

14= 119909 sdot sdot sdot

(85)

11990620= 119909 119906

21= 119909 119906

22= 119909

11990623= 119909 119906

24= 119909 sdot sdot sdot

(86)

11990630= 2119909 119906

31= 0 119906

32= 2119909

11990633= 0 119906

34= 2119909

(87)

Substituting (85) through (87) into series (61) (62) and (63)respectively we get

1199061(119905 119909) = 119909 (1 minus 119905 + 119905

2

minus 1199053

+ 1199054

+ sdot sdot sdot ) (88)

1199062(119905 119909) = 119909 (1 + 119905 + 119905

2

+ 1199053

+ 1199054

+ sdot sdot sdot ) (89)

1199063(119905 119909) = 2119909 (1 + 119905

2

+ 1199054

+ 1199056

+ sdot sdot sdot ) (90)

After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms

1199061(119905 119909) =

119909

1 + 119905

1199062(119905 119909) =

119909

1 minus 119905

1199063(119905 119909) =

2119909

1 minus 1199052

(91)

which is the exact solution of (55)ndash(58) where we haveemployed (66) and

1

1 minus 1199052=

infin

sum

119899=0

1199052119899

(92)

This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general

Discrete Dynamics in Nature and Society 7

terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)

5 Discussion

In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution

Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs

One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs

Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems

6 Conclusion

By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above

all of index greater than one combining PSM and Laplace-Pade resummation

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024

References

[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998

[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999

[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000

[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007

[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007

[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008

[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008

[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005

[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007

[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008

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

[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002

[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional

8 Discrete Dynamics in Nature and Society

decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012

[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011

[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011

[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011

[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007

[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006

[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008

[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009

[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005

[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012

[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009

[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008

[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010

[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011

[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012

[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012

[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012

[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012

[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009

[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009

[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012

[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009

[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013

[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014

[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012

[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012

[42] 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

[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011

[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011

[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra

Discrete Dynamics in Nature and Society 9

and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012

[46] 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

[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011

[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956

[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906

[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995

[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013

[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013

[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013

[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014

[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014

[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011

[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

the substitution of (20) (21) (22) (23) and (24) leads to

11990615= 0 (25)

in the same way we obtain

11990616= 11990617= 11990618= sdot sdot sdot = 0 (26)

Substituting (20) through (26) into (14) leads us to

1199061(119905 119909) = 119909

3

119905 (27)

Finally substituting (27) into (11) leads to

1199062(119905 119909) =

1199054

119909 (28)

Thus (27) and (28) are the exact solution for SPDAE system(10)ndash(12)

42 Linear Index-Two SPDAEwithVariable Coefficients (1198981=

2 1198982= 1) Consider the following

1199061119905= 1199092

1199061119909119909

minus 31199061+ 1199063+

1199092

1 + 119905 (29)

1199062119905= 1199092

1199062119909119909

minus 31199062+ 1199063+

1199092

1 + 119905 (30)

0 = 1199061+ 1199062minus 21199092 ln (1 + 119905) (31)

subject to the initial conditions

1199061(0 119909) = 0 119906

2(0 119909) = 0

minus1 lt 119905 le 1 minusinfin lt 119909 lt infin

(32)

The integration of (29) and (30) with respect to 119905 and usingthe initial conditions (32) lead to

1199061(119905 119909) = int

119905

0

[1199092

1199061119909119909

minus 31199061+ 1199063] 119889119905 + 119909

2 ln (1 + 119905) (33)

1199062(119905 119909) = int

119905

0

[1199092

1199062119909119909

minus 31199062+ 1199063] 119889119905 + 119909

2 ln (1 + 119905) (34)

assuming that 1199061(119905 119909) 119906

2(119905 119909) and 119906

3(119905 119909) can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (35)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (36)

1199063(119905 119909) = 119906

30(119909) + 119906

31(119909) 119905 + 119906

32(119909) 1199052

+ sdot sdot sdot (37)

where 11990610(119909) 11990611(119909) 119906

20(119909) 11990621(119909) 119906

30(119909) 11990631(119909)

are unknown functions to be determined later on by thePSM method

After substituting (35) and (37) into (33) we get

119906101199050

+

infin

sum

119896=1

1

119896[1198961199061119896minus 1199092

11990610158401015840

1119896minus1+ 31199061119896minus1

minus1199063119896minus1

minus 1199092

(minus1)119896minus1

] 119905119896

= 0

(38)

where we have standardized the summation index and em-ployed the following Taylor series expansion

ln (1 + 119905) =infin

sum

119899=1

(minus1)119899minus1

119899119905119899

minus1 lt 119905 le 1 (39)

In the same way the substitution of (36) and (37) into (34)leads to

119906201199050

+

infin

sum

119896=1

1

119896[1198961199062119896minus 1199092

11990610158401015840

2119896minus1+ 31199062119896minus1

minus1199063119896minus1

minus 1199092

(minus1)119896minus1

] 119905119896

= 0

(40)

On the other hand after substituting (35) (36) and (39) into(31) we have

infin

sum

119896=1

[1199061119896+ 1199062119896minus21199092

119896(minus1)119896minus1

] 119905119896

= 0 (41)

where we have employed the following results deduced from(38) and (40)

11990610= 11990620= 0 (42)

Equations (38) (40) and (41) give rise to the followingformulas

1199061119899=1199092

11990610158401015840

1119899minus1minus 31199061119899minus1

+ 1199063119899minus1

+ (minus1)119899minus1

1199092

119899 119899 ge 1 (43)

1199062119899=1199092

11990610158401015840

2119899minus1minus 31199062119899minus1

+ 1199063119899minus1

+ (minus1)119899minus1

1199092

119899 119899 ge 1 (44)

1199061119899+ 1199062119899=21199092

(minus1)119899minus1

119899 119899 ge 1 (45)

Combining the result of adding (43) and (44) with (45) weobtain

1199063119899minus1

= minus1

2(11990610158401015840

1119899minus1+ 11990610158401015840

2119899minus1) 1199092

+3

2(1199061119899minus1

+ 1199062119899minus1

) 119899 ge 1

(46)

The substitution of (46) into (43) and (44) respectively leadsus to

1199061119899=

1

2119899(1199092

11990610158401015840

1119899minus1minus 31199061119899minus1

+ 31199062119899minus1

minus1199092

11990610158401015840

2119899minus1+ 2 (minus1)

119899minus1

1199092

) 119899 ge 1

1199062119899=

1

2119899(1199092

11990610158401015840

2119899minus1minus 31199062119899minus1

+ 31199061119899minus1

minus1199092

11990610158401015840

1119899minus1+ 2 (minus1)

119899minus1

1199092

) 119899 ge 1

(47)

Discrete Dynamics in Nature and Society 5

From recursion formulas (46) and (47) we get the functions

11990610(119909) = 0 119906

11(119909) = 119909

2

11990612(119909) =

minus1199092

2

11990613=1199092

3 119906

14=minus1199092

4sdot sdot sdot

(48)

11990620(119909) = 0 119906

21(119909) = 119909

2

11990622(119909) =

minus1199092

2

11990623=1199092

3 119906

24=minus1199092

4sdot sdot sdot

(49)

11990630(119909) = 0 119906

31(119909) = 119909

2

11990632(119909) =

minus1199092

2

11990633=1199092

3 119906

34=minus1199092

4sdot sdot sdot

(50)

After substituting (48) through (50) into series (35) (36) and(37) respectively we get

1199061(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (51)

1199062(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (52)

1199063(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (53)

After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that

1199061(119905 119909) = 119909

2 ln (1 + 119905)

1199062(119905 119909) = 119909

2 ln (1 + 119905)

1199063(119905 119909) = 119909

2 ln (1 + 119905)

(54)

which is the exact solution of (29)ndash(32) (see (39))

43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898

2= 1) Finally consider the following

1199061119905= 119891 (119909) 119906

1119909119909+ 11990611199061119909minus1 minus 119905

1 + 1199051199063 (55)

1199062119905= 119892 (119909) 119906

2119909119909minus 11990621199062119909+1 + 119905

1 minus 1199051199063 (56)

0 = 1199061(1 + 119905) minus 119906

2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1

(57)

subject to the initial conditions

1199061(0 119909) = 119909 119906

2(0 119909) = 119909 119906

3(0 119909) = 2119909 (58)

where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt

infin

The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to

1199061(119905 119909) = 119909 + int

119905

0

[119891 (119909) 1199061119909119909

+ 11990611199061119909minus1 minus 119905

1 + 1199051199063] 119889119905 (59)

1199062(119905 119909) = 119909 + int

119905

0

[119892 (119909) 1199062119909119909

minus 11990621199062119909+1 + 119905

1 minus 1199051199063] 119889119905 (60)

PSM assumes once again that 1199061(119905 119909) 119906

2(119905 119909) and 119906

3(119905 119909)

can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (61)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (62)

1199063(119905 119909) = 119906

30(119909) + 119906

31(119909) 119905 + 119906

32(119909) 1199052

+ sdot sdot sdot (63)

where 11990610(119909) 11990611(119909) 119906

20(119909) 11990621(119909) 119906

30(119909) 11990631(119909)

are unknown functions to be determined later on by thePSM method

Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin

sum

119899=0

1199061119899119905119899

= 119909 + int

119905

0

119891 (119909)

infin

sum

119899=0

11990610158401015840

1119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898119905119899+119898

119889119905

minus int

119905

0

(1 minus 119905)

infin

sum

119899=0

infin

sum

119898=0

(minus1)119899

1199063119898119905119899+119898

119889119905

(64)infin

sum

119899=0

1199062119899119905119899

= 119909 + int

119905

0

119892 (119909)

infin

sum

119899=0

11990610158401015840

2119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990621198991199061015840

2119898119905119899+119898

119889119905

minus int

119905

0

(1 + 119905)

infin

sum

119899=0

infin

sum

119898=0

1199063119898119905119899+119898

119889119905

(65)

where we have employed the Taylor series expansions

1

1 minus 119905=

infin

sum

119899=0

119905119899

1

1 + 119905=

infin

sum

119899=0

(minus1)119899

119905119899

(66)

After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively

minus 11990610+ 119909 minus 119906

30119905 minus

1

2(11990631minus 211990630) 1199052

minus1

3(11990632minus 211990631+ 211990630) 1199053

minus1

4(11990633minus 211990632+ 211990631minus 211990630) 1199054

+

infin

sum

119896=1

[119891 (119909) 119906

10158401015840

1119896minus1

119896+

infin

sum

119898=0

1199061015840

11198981199061119896minus119898minus1

119896minus 1199061119896] 119905119896

= 0

minus 11990620+ 119909 + 119906

30119905 +

1

2(11990631+ 211990630) 1199052

+1

3(11990632+ 211990631+ 211990630) 1199053

6 Discrete Dynamics in Nature and Society

+1

4(11990633+ 211990632+ 211990631+ 211990630) 1199054

+

infin

sum

119896=1

[119892 (119909) 119906

10158401015840

2119896minus1

119896minus

infin

sum

119898=0

1199061015840

21198981199062119896minus119898minus1

119896minus 1199062119896] 119905119896

= 0

(67)

From (57) we obtain

infin

sum

119898=0

1199062119898119905119898

= (1 + 119905)

infin

sum

119899=0

infin

sum

119895=0

119905119899+119895

1199061119895 (68)

after using again the first series of (66)After standardizing the summation index we get a third

recurrence formula from (68)

1199062119896=

infin

sum

119899=0

[1199061119896minus119899

+ 1199061119896minus119899minus1

] where 119896 = 0 1 2 3

(69)

From recursion formulas (67) and (69) we get the followingcoupled equations

11990610= 11990610(0 119909) (70)

11990611= 119891 (119909) 119906

10158401015840

10+ 1199061015840

1011990610minus 11990630 (71)

11990612= 119891 (119909)

11990610158401015840

11

2+1199061015840

1011990611+ 1199061015840

1111990610

2+11990631

2 (72)

11990613= 119891 (119909)

11990610158401015840

12

3+1199061015840

1011990612+ 1199061015840

1111990611+ 1199061015840

1211990610

3

minus11990632+ 211990630minus 211990631

3

(73)

11990614=11990610158401015840

13

4+1199061015840

1011990613+ 1199061015840

1111990612+ 1199061015840

1211990611+ 1199061015840

1311990610

4

minus11990633minus 211990632+ 211990631minus 211990630

4

(74)

11990620= 11990620(0 119909) (75)

11990621= 119892 (119909) 119906

10158401015840

20minus 1199061015840

2011990620+ 11990630 (76)

11990622= 119892 (119909)

11990610158401015840

21

2minus1199061015840

2011990621+ 1199061015840

2111990620

2+11990631+ 211990630

2 (77)

11990623= 119892 (119909)

11990610158401015840

22

3minus1199061015840

2011990622+ 1199061015840

2111990621+ 1199061015840

2211990620

3

+11990632+ 211990630+ 211990631

3

(78)

11990624= 119892 (119909)

11990610158401015840

23

4minus1199061015840

2011990623+ 1199061015840

2111990622+ 1199061015840

2211990621+ 1199061015840

2311990620

4

+11990633+ 211990632+ 211990631+ 211990630

4

(79)

11990620= 11990610 (80)

11990621= 211990610+ 11990611 (81)

11990622= 11990612+ 211990611+ 211990610 (82)

11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)

11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610

(84)

From (70) through (84) we get the functions

11990610= 119909 119906

11= minus119909 119906

12= 119909

11990613= minus119909 119906

14= 119909 sdot sdot sdot

(85)

11990620= 119909 119906

21= 119909 119906

22= 119909

11990623= 119909 119906

24= 119909 sdot sdot sdot

(86)

11990630= 2119909 119906

31= 0 119906

32= 2119909

11990633= 0 119906

34= 2119909

(87)

Substituting (85) through (87) into series (61) (62) and (63)respectively we get

1199061(119905 119909) = 119909 (1 minus 119905 + 119905

2

minus 1199053

+ 1199054

+ sdot sdot sdot ) (88)

1199062(119905 119909) = 119909 (1 + 119905 + 119905

2

+ 1199053

+ 1199054

+ sdot sdot sdot ) (89)

1199063(119905 119909) = 2119909 (1 + 119905

2

+ 1199054

+ 1199056

+ sdot sdot sdot ) (90)

After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms

1199061(119905 119909) =

119909

1 + 119905

1199062(119905 119909) =

119909

1 minus 119905

1199063(119905 119909) =

2119909

1 minus 1199052

(91)

which is the exact solution of (55)ndash(58) where we haveemployed (66) and

1

1 minus 1199052=

infin

sum

119899=0

1199052119899

(92)

This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general

Discrete Dynamics in Nature and Society 7

terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)

5 Discussion

In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution

Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs

One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs

Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems

6 Conclusion

By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above

all of index greater than one combining PSM and Laplace-Pade resummation

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024

References

[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998

[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999

[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000

[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007

[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007

[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008

[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008

[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005

[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007

[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008

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

[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002

[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional

8 Discrete Dynamics in Nature and Society

decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012

[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011

[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011

[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011

[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007

[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006

[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008

[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009

[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005

[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012

[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009

[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008

[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010

[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011

[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012

[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012

[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012

[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012

[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009

[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009

[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012

[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009

[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013

[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014

[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012

[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012

[42] 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

[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011

[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011

[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra

Discrete Dynamics in Nature and Society 9

and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012

[46] 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

[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011

[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956

[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906

[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995

[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013

[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013

[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013

[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014

[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014

[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011

[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

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

Discrete Dynamics in Nature and Society 5

From recursion formulas (46) and (47) we get the functions

11990610(119909) = 0 119906

11(119909) = 119909

2

11990612(119909) =

minus1199092

2

11990613=1199092

3 119906

14=minus1199092

4sdot sdot sdot

(48)

11990620(119909) = 0 119906

21(119909) = 119909

2

11990622(119909) =

minus1199092

2

11990623=1199092

3 119906

24=minus1199092

4sdot sdot sdot

(49)

11990630(119909) = 0 119906

31(119909) = 119909

2

11990632(119909) =

minus1199092

2

11990633=1199092

3 119906

34=minus1199092

4sdot sdot sdot

(50)

After substituting (48) through (50) into series (35) (36) and(37) respectively we get

1199061(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (51)

1199062(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (52)

1199063(119905 119909) = 119909

2

[119905 minus1199052

2+1199053

3minus1199054

4+ sdot sdot sdot ] (53)

After identifying the 119899th terms of the series (51) (52) and (53)as ((minus1)119899minus1119899)119905119899 we conclude that

1199061(119905 119909) = 119909

2 ln (1 + 119905)

1199062(119905 119909) = 119909

2 ln (1 + 119905)

1199063(119905 119909) = 119909

2 ln (1 + 119905)

(54)

which is the exact solution of (29)ndash(32) (see (39))

43 Nonlinear Index-Two SPDAE with Variable Coefficients(1198981= 2 119898

2= 1) Finally consider the following

1199061119905= 119891 (119909) 119906

1119909119909+ 11990611199061119909minus1 minus 119905

1 + 1199051199063 (55)

1199062119905= 119892 (119909) 119906

2119909119909minus 11990621199062119909+1 + 119905

1 minus 1199051199063 (56)

0 = 1199061(1 + 119905) minus 119906

2(1 minus 119905) minusinfin lt 119909 lt infin minus1 lt 119905 lt 1

(57)

subject to the initial conditions

1199061(0 119909) = 119909 119906

2(0 119909) = 119909 119906

3(0 119909) = 2119909 (58)

where 119891(119909) and 119892(119909) are analytical functions on minusinfin lt 119909 lt

infin

The integration of (55) and (56)with respect to 119905 andusingthe initial conditions (58) lead to

1199061(119905 119909) = 119909 + int

119905

0

[119891 (119909) 1199061119909119909

+ 11990611199061119909minus1 minus 119905

1 + 1199051199063] 119889119905 (59)

1199062(119905 119909) = 119909 + int

119905

0

[119892 (119909) 1199062119909119909

minus 11990621199062119909+1 + 119905

1 minus 1199051199063] 119889119905 (60)

PSM assumes once again that 1199061(119905 119909) 119906

2(119905 119909) and 119906

3(119905 119909)

can be written as

1199061(119905 119909) = 119906

10(119909) + 119906

11(119909) 119905 + 119906

12(119909) 1199052

+ sdot sdot sdot (61)

1199062(119905 119909) = 119906

20(119909) + 119906

21(119909) 119905 + 119906

22(119909) 1199052

+ sdot sdot sdot (62)

1199063(119905 119909) = 119906

30(119909) + 119906

31(119909) 119905 + 119906

32(119909) 1199052

+ sdot sdot sdot (63)

where 11990610(119909) 11990611(119909) 119906

20(119909) 11990621(119909) 119906

30(119909) 11990631(119909)

are unknown functions to be determined later on by thePSM method

Substituting (61) and (63) into (59) and also (62) and (63)into (60) respectively we getinfin

sum

119899=0

1199061119899119905119899

= 119909 + int

119905

0

119891 (119909)

infin

sum

119899=0

11990610158401015840

1119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990611198991199061015840

1119898119905119899+119898

119889119905

minus int

119905

0

(1 minus 119905)

infin

sum

119899=0

infin

sum

119898=0

(minus1)119899

1199063119898119905119899+119898

119889119905

(64)infin

sum

119899=0

1199062119899119905119899

= 119909 + int

119905

0

119892 (119909)

infin

sum

119899=0

11990610158401015840

2119899119905119899

119889119905 + int

119905

0

infin

sum

119899=0

infin

sum

119898=0

11990621198991199061015840

2119898119905119899+119898

119889119905

minus int

119905

0

(1 + 119905)

infin

sum

119899=0

infin

sum

119898=0

1199063119898119905119899+119898

119889119905

(65)

where we have employed the Taylor series expansions

1

1 minus 119905=

infin

sum

119899=0

119905119899

1

1 + 119905=

infin

sum

119899=0

(minus1)119899

119905119899

(66)

After integrating and standardizing the summation indexwe get the following recursion formulas from (64) and (65)respectively

minus 11990610+ 119909 minus 119906

30119905 minus

1

2(11990631minus 211990630) 1199052

minus1

3(11990632minus 211990631+ 211990630) 1199053

minus1

4(11990633minus 211990632+ 211990631minus 211990630) 1199054

+

infin

sum

119896=1

[119891 (119909) 119906

10158401015840

1119896minus1

119896+

infin

sum

119898=0

1199061015840

11198981199061119896minus119898minus1

119896minus 1199061119896] 119905119896

= 0

minus 11990620+ 119909 + 119906

30119905 +

1

2(11990631+ 211990630) 1199052

+1

3(11990632+ 211990631+ 211990630) 1199053

6 Discrete Dynamics in Nature and Society

+1

4(11990633+ 211990632+ 211990631+ 211990630) 1199054

+

infin

sum

119896=1

[119892 (119909) 119906

10158401015840

2119896minus1

119896minus

infin

sum

119898=0

1199061015840

21198981199062119896minus119898minus1

119896minus 1199062119896] 119905119896

= 0

(67)

From (57) we obtain

infin

sum

119898=0

1199062119898119905119898

= (1 + 119905)

infin

sum

119899=0

infin

sum

119895=0

119905119899+119895

1199061119895 (68)

after using again the first series of (66)After standardizing the summation index we get a third

recurrence formula from (68)

1199062119896=

infin

sum

119899=0

[1199061119896minus119899

+ 1199061119896minus119899minus1

] where 119896 = 0 1 2 3

(69)

From recursion formulas (67) and (69) we get the followingcoupled equations

11990610= 11990610(0 119909) (70)

11990611= 119891 (119909) 119906

10158401015840

10+ 1199061015840

1011990610minus 11990630 (71)

11990612= 119891 (119909)

11990610158401015840

11

2+1199061015840

1011990611+ 1199061015840

1111990610

2+11990631

2 (72)

11990613= 119891 (119909)

11990610158401015840

12

3+1199061015840

1011990612+ 1199061015840

1111990611+ 1199061015840

1211990610

3

minus11990632+ 211990630minus 211990631

3

(73)

11990614=11990610158401015840

13

4+1199061015840

1011990613+ 1199061015840

1111990612+ 1199061015840

1211990611+ 1199061015840

1311990610

4

minus11990633minus 211990632+ 211990631minus 211990630

4

(74)

11990620= 11990620(0 119909) (75)

11990621= 119892 (119909) 119906

10158401015840

20minus 1199061015840

2011990620+ 11990630 (76)

11990622= 119892 (119909)

11990610158401015840

21

2minus1199061015840

2011990621+ 1199061015840

2111990620

2+11990631+ 211990630

2 (77)

11990623= 119892 (119909)

11990610158401015840

22

3minus1199061015840

2011990622+ 1199061015840

2111990621+ 1199061015840

2211990620

3

+11990632+ 211990630+ 211990631

3

(78)

11990624= 119892 (119909)

11990610158401015840

23

4minus1199061015840

2011990623+ 1199061015840

2111990622+ 1199061015840

2211990621+ 1199061015840

2311990620

4

+11990633+ 211990632+ 211990631+ 211990630

4

(79)

11990620= 11990610 (80)

11990621= 211990610+ 11990611 (81)

11990622= 11990612+ 211990611+ 211990610 (82)

11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)

11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610

(84)

From (70) through (84) we get the functions

11990610= 119909 119906

11= minus119909 119906

12= 119909

11990613= minus119909 119906

14= 119909 sdot sdot sdot

(85)

11990620= 119909 119906

21= 119909 119906

22= 119909

11990623= 119909 119906

24= 119909 sdot sdot sdot

(86)

11990630= 2119909 119906

31= 0 119906

32= 2119909

11990633= 0 119906

34= 2119909

(87)

Substituting (85) through (87) into series (61) (62) and (63)respectively we get

1199061(119905 119909) = 119909 (1 minus 119905 + 119905

2

minus 1199053

+ 1199054

+ sdot sdot sdot ) (88)

1199062(119905 119909) = 119909 (1 + 119905 + 119905

2

+ 1199053

+ 1199054

+ sdot sdot sdot ) (89)

1199063(119905 119909) = 2119909 (1 + 119905

2

+ 1199054

+ 1199056

+ sdot sdot sdot ) (90)

After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms

1199061(119905 119909) =

119909

1 + 119905

1199062(119905 119909) =

119909

1 minus 119905

1199063(119905 119909) =

2119909

1 minus 1199052

(91)

which is the exact solution of (55)ndash(58) where we haveemployed (66) and

1

1 minus 1199052=

infin

sum

119899=0

1199052119899

(92)

This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general

Discrete Dynamics in Nature and Society 7

terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)

5 Discussion

In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution

Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs

One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs

Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems

6 Conclusion

By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above

all of index greater than one combining PSM and Laplace-Pade resummation

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024

References

[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998

[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999

[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000

[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007

[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007

[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008

[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008

[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005

[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007

[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008

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

[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002

[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional

8 Discrete Dynamics in Nature and Society

decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012

[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011

[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011

[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011

[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007

[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006

[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008

[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009

[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005

[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012

[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009

[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008

[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010

[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011

[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012

[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012

[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012

[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012

[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009

[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009

[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012

[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009

[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013

[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014

[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012

[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012

[42] 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

[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011

[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011

[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra

Discrete Dynamics in Nature and Society 9

and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012

[46] 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

[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011

[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956

[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906

[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995

[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013

[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013

[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013

[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014

[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014

[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011

[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

+1

4(11990633+ 211990632+ 211990631+ 211990630) 1199054

+

infin

sum

119896=1

[119892 (119909) 119906

10158401015840

2119896minus1

119896minus

infin

sum

119898=0

1199061015840

21198981199062119896minus119898minus1

119896minus 1199062119896] 119905119896

= 0

(67)

From (57) we obtain

infin

sum

119898=0

1199062119898119905119898

= (1 + 119905)

infin

sum

119899=0

infin

sum

119895=0

119905119899+119895

1199061119895 (68)

after using again the first series of (66)After standardizing the summation index we get a third

recurrence formula from (68)

1199062119896=

infin

sum

119899=0

[1199061119896minus119899

+ 1199061119896minus119899minus1

] where 119896 = 0 1 2 3

(69)

From recursion formulas (67) and (69) we get the followingcoupled equations

11990610= 11990610(0 119909) (70)

11990611= 119891 (119909) 119906

10158401015840

10+ 1199061015840

1011990610minus 11990630 (71)

11990612= 119891 (119909)

11990610158401015840

11

2+1199061015840

1011990611+ 1199061015840

1111990610

2+11990631

2 (72)

11990613= 119891 (119909)

11990610158401015840

12

3+1199061015840

1011990612+ 1199061015840

1111990611+ 1199061015840

1211990610

3

minus11990632+ 211990630minus 211990631

3

(73)

11990614=11990610158401015840

13

4+1199061015840

1011990613+ 1199061015840

1111990612+ 1199061015840

1211990611+ 1199061015840

1311990610

4

minus11990633minus 211990632+ 211990631minus 211990630

4

(74)

11990620= 11990620(0 119909) (75)

11990621= 119892 (119909) 119906

10158401015840

20minus 1199061015840

2011990620+ 11990630 (76)

11990622= 119892 (119909)

11990610158401015840

21

2minus1199061015840

2011990621+ 1199061015840

2111990620

2+11990631+ 211990630

2 (77)

11990623= 119892 (119909)

11990610158401015840

22

3minus1199061015840

2011990622+ 1199061015840

2111990621+ 1199061015840

2211990620

3

+11990632+ 211990630+ 211990631

3

(78)

11990624= 119892 (119909)

11990610158401015840

23

4minus1199061015840

2011990623+ 1199061015840

2111990622+ 1199061015840

2211990621+ 1199061015840

2311990620

4

+11990633+ 211990632+ 211990631+ 211990630

4

(79)

11990620= 11990610 (80)

11990621= 211990610+ 11990611 (81)

11990622= 11990612+ 211990611+ 211990610 (82)

11990623= 11990613+ 211990612+ 211990611+ 211990610 (83)

11990624= 11990614+ 211990613+ 211990612+ 211990611+ 211990610

(84)

From (70) through (84) we get the functions

11990610= 119909 119906

11= minus119909 119906

12= 119909

11990613= minus119909 119906

14= 119909 sdot sdot sdot

(85)

11990620= 119909 119906

21= 119909 119906

22= 119909

11990623= 119909 119906

24= 119909 sdot sdot sdot

(86)

11990630= 2119909 119906

31= 0 119906

32= 2119909

11990633= 0 119906

34= 2119909

(87)

Substituting (85) through (87) into series (61) (62) and (63)respectively we get

1199061(119905 119909) = 119909 (1 minus 119905 + 119905

2

minus 1199053

+ 1199054

+ sdot sdot sdot ) (88)

1199062(119905 119909) = 119909 (1 + 119905 + 119905

2

+ 1199053

+ 1199054

+ sdot sdot sdot ) (89)

1199063(119905 119909) = 2119909 (1 + 119905

2

+ 1199054

+ 1199056

+ sdot sdot sdot ) (90)

After identifying the 119899th terms of the above series as (minus1)119899119905119899119905119899 and 1199052119899 respectively we conclude that series (88) through(90) admit the following closed forms

1199061(119905 119909) =

119909

1 + 119905

1199062(119905 119909) =

119909

1 minus 119905

1199063(119905 119909) =

2119909

1 minus 1199052

(91)

which is the exact solution of (55)ndash(58) where we haveemployed (66) and

1

1 minus 1199052=

infin

sum

119899=0

1199052119899

(92)

This case admits an alternative way to obtain the closedsolution (91) by using Pade posttreatment [58 59] In general

Discrete Dynamics in Nature and Society 7

terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)

5 Discussion

In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution

Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs

One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs

Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems

6 Conclusion

By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above

all of index greater than one combining PSM and Laplace-Pade resummation

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024

References

[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998

[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999

[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000

[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007

[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007

[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008

[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008

[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005

[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007

[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008

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

[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002

[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional

8 Discrete Dynamics in Nature and Society

decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012

[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011

[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011

[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011

[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007

[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006

[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008

[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009

[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005

[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012

[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009

[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008

[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010

[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011

[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012

[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012

[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012

[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012

[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009

[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009

[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012

[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009

[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013

[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014

[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012

[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012

[42] 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

[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011

[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011

[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra

Discrete Dynamics in Nature and Society 9

and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012

[46] 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

[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011

[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956

[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906

[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995

[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013

[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013

[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013

[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014

[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014

[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011

[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

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

Discrete Dynamics in Nature and Society 7

terms Pade technology is employed in order to obtainsolutions for differential equations handier and computa-tionally more efficient Also it is employed to improve theconvergence of truncated series As a matter of fact theapplication of Pade [22] to series (88)ndash(90) leads to the exactsolution (91)

5 Discussion

In this study we presented the power series method (PSM)as a useful tool in the search for analytical solutions forsingular partial differential-algebraic equations (SPDAEs) Tothis end two SPDAE problems of index-two and anotherof index-one were solved by this technique leading (forthese cases) to the exact solutions For each of the casesstudied PSM essentially transformed the SPDAE into aneasily solvable algebraic system for the coefficient functionsof the proposed power series solution

Since not all the SPDAEs have exact solutions it ispossible that in some cases the series solution obtainedfrom PSM may have limited regions of convergence eventaking a large number of terms our case study three suggeststhe use of a Pade posttreatment as a possibility to improvethe domain of convergence for the PSMrsquos truncated seriesIn fact the mentioned example showed that sometimesPade approximant leads to the exact solution It should bementioned that Laplace-Pade resummation is another knownmethod employed in the literature [53] to enlarge the domainof convergence of solutions or is inclusive to find exactsolutionsThis technique which combines Laplace transformand Pade posttreatment may be used in the future researchof SPDAEs

One of the important features of our method is thatthe high complexity of SPDAE problems was effectivelyhandled by this method This is clear if one notes thatour examples were chosen to include higher-order-indexPDAEs (differentiation index greater than one) linear andnonlinear cases even with variable coefficients In additionthe last example proposed the case of a system of equationscontaining two functions entirely arbitrary The above makesthis system completely inaccessible to numerical methodsalso we add singularities which gave rise to the name ofSPDAEs

Finally the fact that there are not any standard analyticalor numerical methods to solve higher-index SPDAEs con-verts the PSM method into an attractive tool to solve suchproblems

6 Conclusion

By solving the three examples we presented PSM as a handyanduseful tool with high potential to find analytical solutionsto SPDAEs Since on one hand we proposed the way toimprove the solutions obtained by this method if necessaryand on the other hand it is based on a straightforward proce-dure our proposal will be useful for practical applications andsuitable for engineers and scientists Finally further researchshould be conducted to solve other SPDAEs systems above

all of index greater than one combining PSM and Laplace-Pade resummation

Conflict of Interests

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

Acknowledgment

The authors gratefully acknowledge the financial supportfrom the National Council for Science and Technology ofMexico (CONACyT) through Grant CB-2010-01 no 157024

References

[1] W Lucht and K Strehmel ldquoDiscretization based indices forsemilinear partial differential algebraic equationsrdquo AppliedNumerical Mathematics vol 28 no 2ndash4 pp 371ndash386 1998

[2] W Lucht K Strehmel and C Eichler-Liebenow ldquoIndexes andspecial discretization methods for linear partial differentialalgebraic equationsrdquo BIT Numerical Mathematics vol 39 no3 pp 484ndash512 1999

[3] W S Martinson and P I Barton ldquoA differentiation indexfor partial differential-algebraic equationsrdquo SIAM Journal onScientific Computing vol 21 no 6 pp 2295ndash2315 2000

[4] LM B Assas ldquoApproximate solutions for the generalized KdV-Burgersrsquo equation by Hersquos variational iteration methodrdquo PhysicaScripta vol 76 pp 161ndash164 2007

[5] J-H He ldquoVariational approach for nonlinear oscillatorsrdquoChaos Solitons amp Fractals vol 34 no 5 pp 1430ndash1439 2007

[6] MKazemnia S A ZahediMVaezi andN Tolou ldquoAssessmentof modified variational iteration method in BVPs high-orderdifferential equationsrdquo Journal of Applied Sciences vol 8 no 22pp 4192ndash4197 2008

[7] R Noorzad A T Poor and M Omidvar ldquoVariational iterationmethod and homotopy-perturbation method for solving Burg-ers equation in fluid dynamicsrdquo Journal of Applied Sciences vol8 no 2 pp 369ndash373 2008

[8] D J Evans and K R Raslan ldquoThe tanh function method forsolving some important non-linear partial differential equa-tionsrdquo International Journal of Computer Mathematics vol 82no 7 pp 897ndash905 2005

[9] F Xu ldquoA generalized soliton solution of the Konopelchenko-Dubrovsky equation using Hersquos exp-function methodrdquoZeitschrift fur Naturforschung Section A vol 62 no 12 pp685ndash688 2007

[10] J Mahmoudi N Tolou I Khatami A Barari and D D GanjildquoExplicit solution of nonlinear ZK-BBM wave equation usingExp-function methodrdquo Journal of Applied Sciences vol 8 no 2pp 358ndash363 2008

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

[12] E Babolian and J Biazar ldquoOn the order of convergence ofAdomian methodrdquo Applied Mathematics and Computation vol130 no 2-3 pp 383ndash387 2002

[13] A Kooch and M Abadyan ldquoEfficiency of modified Ado-mian decomposition for simulating the instability of nano-electromechanical switches comparison with the conventional

8 Discrete Dynamics in Nature and Society

decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012

[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011

[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011

[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011

[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007

[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006

[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008

[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009

[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005

[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012

[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009

[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008

[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010

[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011

[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012

[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012

[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012

[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012

[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009

[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009

[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012

[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009

[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013

[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014

[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012

[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012

[42] 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

[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011

[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011

[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra

Discrete Dynamics in Nature and Society 9

and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012

[46] 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

[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011

[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956

[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906

[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995

[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013

[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013

[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013

[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014

[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014

[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011

[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

decomposition methodrdquo Trends in Applied Sciences Researchvol 7 no 1 pp 57ndash67 2012

[14] A Koochi and M Abadyan ldquoEvaluating the ability of modifiedadomian decomposition method to simulate the instability offreestanding carbon nanotube comparison with conventionaldecomposition methodrdquo Journal of Applied Sciences vol 11 no19 pp 3421ndash3428 2011

[15] S Karimi Vanani S Heidari and M Avaji ldquoA low-cost numer-ical algorithm for the solution of nonlinear delay boundaryintegral equationsrdquo Journal of Applied Sciences vol 11 no 20pp 3504ndash3509 2011

[16] S H Chowdhury ldquoA comparison between the modifiedhomotopy perturbation method and adomian decompositionmethod for solving nonlinear heat transfer equationsrdquo Journalof Applied Sciences vol 11 no 7 pp 1416ndash1420 2011

[17] L-N Zhang and L Xu ldquoDetermination of the limit cycle byHersquosparameter-expansion for oscillators in a 1199063(1 + 1199062) potentialrdquoZeitschrift fur NaturforschungmdashSection A Journal of PhysicalSciences vol 62 no 7-8 pp 396ndash398 2007

[18] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[19] J-H He ldquoHomotopy perturbation techniquerdquo Computer Meth-ods in Applied Mechanics and Engineering vol 178 no 3-4 pp257ndash262 1999

[20] J-H He ldquoHomotopy perturbation method for solving bound-ary value problemsrdquo Physics Letters A vol 350 no 1-2 pp 87ndash88 2006

[21] J-H He ldquoRecent development of the homotopy perturbationmethodrdquo Topological Methods in Nonlinear Analysis vol 31 no2 pp 205ndash209 2008

[22] A Belendez C Pascual M L Alvarez D I Mendez M SYebra and A Hernandez ldquoHigher order analytical approxi-mate solutions to the nonlinear pendulum by Hersquos homotopymethodrdquo Physica Scripta vol 79 no 1 Article ID 015009 2009

[23] J-H He ldquoA coupling method of a homotopy technique and aperturbation technique for non-linear problemsrdquo InternationalJournal of Non-Linear Mechanics vol 35 no 1 pp 37ndash43 2000

[24] M El-Shahed ldquoApplication of Hersquos homotopy perturbationmethod to Volterrarsquos integro-differential equationrdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 6 no 2 pp 163ndash168 2005

[25] Y Khan H Vazquez-Leal and N Faraz ldquoAn efficient newiterative method for oscillator differential equationrdquo ScientiaIranica vol 19 no 6 pp 1473ndash1477 2012

[26] D D Ganji H Babazadeh F Noori M M Pirouz and MJanipour ldquoAn application of homotopy perturbationmethod fornon-linear Blasius equation to boundary layer flow over a flatplaterdquo International Journal of Nonlinear Science vol 7 no 4pp 399ndash404 2009

[27] D D Ganji H Mirgolbabaei M Miansari and M MiansarildquoApplication of homotopy perturbation method to solve linearand non-linear systems of ordinary differential equations anddifferential equation of order threerdquo Journal of Applied Sciencesvol 8 no 7 pp 1256ndash1261 2008

[28] A Fereidoon Y Rostamiyan M Akbarzade and D D GanjildquoApplication of Hersquos homotopy perturbation method to nonlin-ear shock damper dynamicsrdquo Archive of Applied Mechanics vol80 no 6 pp 641ndash649 2010

[29] P R Sharma and G Methi ldquoApplications of homotopy pertur-bation method to partial differential equationsrdquo Asian Journalof Mathematics amp Statistics vol 4 no 3 pp 140ndash150 2011

[30] H Aminikhah ldquoAnalytical approximation to the solution ofnonlinear Blasiusviscous flow equation by LTNHPMrdquo ISRNMathematical Analysis vol 2012 Article ID 957473 10 pages2012

[31] H Vazquez-Leal U Filobello-Nino R Castaneda-SheissaL Hernandez-Martınez and A Sarmiento-Reyes ldquoModifiedHPMs inspired by homotopy continuation methodsrdquo Mathe-matical Problems in Engineering vol 2012 Article ID 30912319 pages 2012

[32] H Vazquez-Leal R Castaneda-Sheissa U Filobello-Nino ASarmiento-Reyes and J Sanchez Orea ldquoHigh accurate simpleapproximation of normal distribution integralrdquo MathematicalProblems in Engineering vol 2012 Article ID 124029 22 pages2012

[33] U Filobello-Nino H Vazquez-Leal R Castaneda-Sheissa et alldquoAn approximate solution of Blasius equation by using HPMmethodrdquo Asian Journal of Mathematics and Statistics vol 5 no2 pp 50ndash59 2012

[34] J Biazar and H Aminikhah ldquoStudy of convergence of homo-topy perturbation method for systems of partial differentialequationsrdquoComputersampMathematics with Applications vol 58no 11-12 pp 2221ndash2230 2009

[35] J Biazar and H Ghazvini ldquoConvergence of the homotopy per-turbation method for partial differential equationsrdquo NonlinearAnalysis Real World Applications vol 10 no 5 pp 2633ndash26402009

[36] U Filobello-Nino H D Vazquez-Leal Y Khan et al ldquoHPMapplied to solve nonlinear circuits a study caserdquo AppliedMathematics Sciences vol 6 no 87 pp 4331ndash4344 2012

[37] DDGanji A R Sahouli andM Famouri ldquoAnewmodificationofHersquos homotopy perturbationmethod for rapid convergence ofnonlinear undamped oscillatorsrdquo Journal of Applied Mathemat-ics and Computing vol 30 no 1-2 pp 181ndash192 2009

[38] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoLaplacetransform-homotopy perturbationmethod as a powerful tool tosolve nonlinear problems with boundary conditions defined onfinite intervalsrdquo Computational and Applied Mathematics 2013

[39] M Bayat and I Pakar ldquoNonlinear vibration of an electrostati-cally actuatedmicrobeamrdquo Latin American Journal of Solids andStructures vol 11 no 3 pp 534ndash544 2014

[40] MM Rashidi S AM Pour T Hayat and S Obaidat ldquoAnalyticapproximate solutions for steady flow over a rotating diskin porous medium with heat transfer by homotopy analysismethodrdquo Computers and Fluids vol 54 pp 1ndash9 2012

[41] J Biazar and B Ghanbari ldquoThe homotopy perturbationmethodfor solving neutral functional-differential equations with pro-portional delaysrdquo Journal of King Saud University Science vol24 no 1 pp 33ndash37 2012

[42] 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

[43] M F Araghi and B Rezapour ldquoApplication of homotopyperturbation method to solve multidimensional schrodingerrsquosequationsrdquo Journal of Mathematical Archive vol 2 no 11 pp1ndash6 2011

[44] J Biazar andM Eslami ldquoA newhomotopy perturbationmethodfor solving systems of partial differential equationsrdquo Computersand Mathematics with Applications vol 62 no 1 pp 225ndash2342011

[45] M F Araghi and M Sotoodeh ldquoAn enhanced modifiedhomotopy perturbation method for solving nonlinear volterra

Discrete Dynamics in Nature and Society 9

and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012

[46] 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

[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011

[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956

[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906

[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995

[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013

[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013

[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013

[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014

[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014

[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011

[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

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

Discrete Dynamics in Nature and Society 9

and fredholm integro-differential equation 1rdquo World AppliedSciences Journal vol 20 no 12 pp 1646ndash1655 2012

[46] 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

[47] T Patel M N Mehta and V H Pradhan ldquoThe numericalsolution of Burgerrsquos equation arising into the irradiation oftumour tissue in biological diffusing system by HomotopyAnalysis Methodrdquo Asian Journal of Applied Sciences vol 5 no1 pp 60ndash66 2012

[48] V Marinca and N Herisanu Nonlinear Dynamical Systems inEngineering Springer Heidelberg Germany 1st edition 2011

[49] E Ince Ordinary Differential Equations Dover New York NYUSA 1956

[50] A ForsythTheory of Differential Equations CambridgeUniver-sity Press New York NY USA 1906

[51] T L Chow Classical Mechanics John Wiley amp Sons New YorkNY USA 1995

[52] M H Holmes Introduction to Perturbation Methods SpringerNew York NY USA 1995

[53] U Filobello-NinoH YVazquez-Leal A Khan et al ldquoPerturba-tionmethod and laplace-pade approximation to solve nonlinearproblemsrdquoMiskolcMathematical Notes vol 14 no 1 pp 89ndash1012013

[54] U Filobello-Nino H Vazquez-Leal K Boubaker et al ldquoPer-turbation method as a powerful tool to solve highly nonlinearproblems the case of Gelfandrsquos equationrdquo Asian Journal ofMathematics and Statistics vol 6 no 2 pp 76ndash82 2013

[55] U Filobello-Nino H Vazquez-Leal Y Khan et al ldquoA handyexact solution for flow due to a stretching boundary with partialsliprdquo Revista Mexicana de Fısica E vol 59 no 1 pp 51ndash55 2013

[56] H Vazquez-Leal ldquoExact solutions for differential-algebraicequationsrdquo Miskolc Mathematical Notes vol 15 no 1 pp 227ndash238 2014

[57] B Benhammouda and H Vazquez-Leal ldquoAnalytical solutionsfor systems of partial differential-algebraic equationsrdquo Springer-Plus vol 3 article 137 2014

[58] H Bararnia E Ghasemi S Soleimani A Barari and D DGanji ldquoHPM-Pade method on natural convection of darcianfluid about a vertical full cone embedded in porous mediardquoJournal of Porous Media vol 14 no 6 pp 545ndash553 2011

[59] G A Baker Essentials of Pade Approximants Academic PressNew York NY USA 1975

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of