Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 717540 9 pageshttpdxdoiorg1011552013717540

Research ArticleSolution of Fractional Partial Differential Equations in FluidMechanics by Extension of Some Iterative Method

A A Hemeda

Department of Mathematics Faculty of Science Tanta University Tanta 31527 Egypt

Correspondence should be addressed to A A Hemeda aahemedayahoocom

Received 23 September 2013 Accepted 7 November 2013

Academic Editor Tie-cheng Xia

Copyright copy 2013 A A HemedaThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

An extension of the so-called new iterativemethod (NIM) has been used to handle linear andnonlinear fractional partial differentialequationsThemain property of themethod lies in its flexibility and ability to solve nonlinear equations accurately and convenientlyTherefore a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluidmechanics The fractional derivatives are described in the Caputo sense Numerical illustrations that include the fractional waveequation fractional Burgers equation fractional KdV equation fractional Klein-Gordon equation and fractional Boussinesq-likeequation are investigated to show the pertinent features of the technique Comparison of the results obtained by the NIM withthose obtained by both Adomian decomposition method (ADM) and the variational iteration method (VIM) reveals that the NIMis very effective and convenient The basic idea described in this paper is expected to be further employed to solve other similarlinear and nonlinear problems in fractional calculus

1 Introduction

Recent advances of fractional differential equations are stim-ulated by new examples of applications in fluid mechanicsviscoelasticity mathematical biology electrochemistry andphysics For example the nonlinear oscillation of earthquakecan be modeled with fractional derivatives [1] and the fluid-dynamic traffic model with fractional derivatives [2] caneliminate the deficiency arising from the assumption ofcontinuum traffic flow Based on experimental data fractionalpartial differential equations for seepage flow in porousmediaare suggested in [3] and differential equations with fractionalorder have recently proved to be valuable tools to the mod-eling of many physical phenomena [4] Different fractionalpartial differential equations have been studied and solvedincluding the space-time-fractional diffusion-wave equation[5ndash7] the fractional advection-dispersion equation [8 9] thefractional telegraph equation [10] the fractional KdV equa-tion [11] and the linear inhomogeneous fractional partialdifferential equations [12]

The NIM [13ndash15] is a suitable approach to provide analyt-ical approximation to linear and nonlinear problems and itis particularly valuable as tool for scientists and applied

mathematicians because it provides immediate and visiblesymbolic terms of analytical solutions as well as numericalapproximate solutions to both linear and nonlinear differ-ential equations without linearization or discretization TheNIM proposed by Daftardar-Gejji and Jafari in 2006 [13]and improved by Hemeda [14] was successfully applied toa variety of linear and nonlinear equations such as algebraicequations integral equations integrodifferential equationsordinary and partial differential equations of integer and frac-tional order and systems of equations as well NIM is simpleto understand and easy to implement using computer pack-ages and yields better results [15] than the existing ADM [16]homotopy perturbation method (HPM) [17] or VIM [18]

The objective of this work is to extend the application ofthe NIM to obtain analytical solutions to some fractional par-tial differential equations in fluidmechanicsThese equationsinclude wave equation Burgers equation KdV equationKlein-Gordon equation and Boussinesq-like equation TheNIM is a computational method that yields analytical solu-tions and has certain advantages over standard numericalmethods It is free from rounding-off errors as it does notinvolve discretization and does not require large computer

2 Abstract and Applied Analysis

obtained memory or power The method introduces thesolution in the form of a convergent fractional series with ele-gantly computable termsThe corresponding solutions of theinteger order equations are found to follow as special cases ofthose of fractional order equations The obtained results oftheNIM are comparedwith those obtained by bothADM [1920] and the VIM [21ndash25] which confirm that this method isvery effective and convenient to these equations and to othersimilar equations where it has the advantage that there is noneed to calculate Adomianrsquos polynomials for the nonlinearproblems as in the ADM For more details see [26ndash31]

Throughout this work fractional partial differential equa-tions are obtained from the corresponding integer orderequations by replacing the first-order or the second-order timederivative by a fractional in the Caputo sense [32] of order 120572with 0 lt 120572 le 1 or 1 lt 120572 le 2

2 Preliminaries and Notations

In this section we give some basic definitions and propertiesof the fractional calculus theorywhich are used further in thiswork

Definition 1 A real function 119891(119905) 119905 gt 0 is said to be in thespace119862120583 120583 isin 119877 if there exists a real number 119901 (gt120583) such that119891(119905) = 119905

1199011198911(119905) where 1198911(119905) isin 119862[0infin) and it is said to be in

the space 119862119898120583if and only if 119891(119898) isin 119862120583119898 isin 119873

Definition 2 TheRiemann-Liouville fractional integral oper-ator of order 120572 ge 0 of a function 119891 isin 119862120583 120583 ge minus1 is definedas

119868120572119891 (119905) =

1

Γ (120572)

int

119905

0

(119905 minus 120591)120572minus1119891 (120591) 119889120591 120572 gt 0 119905 gt 0

1198680119891 (119905) = 119891 (119905)

(1)

Properties of the operators 119868120572 can be found in [32ndash35] wemention only the following for 119891 isin 119862120583 120583 ge minus1 120572 120573 ge 0 and120592 gt minus1

(1) 119868120572119868120573119891(119905) = 119868120572+120573119891(119905)(2) 119868120572119868120573119891(119905) = 119868120573119868120572119891(119905)(3) 119868120572119905120592 = (Γ(120592 + 1)Γ(120592 + 1 + 120572))119905120572+]

The Riemann-Liouville derivative has certain disadvan-tages when trying to model real-world phenomena withfractional differential equationsTherefore we will introducea modified fractional differential operator 119863120572 proposed byCaputo in this work on the theory of viscoelasticity [32]

Definition 3 The fractional derivative of 119891(119905) in the Caputosense is defined as

119863120572119891 (119905) = 119868

119898minus120572119863119898119891 (119905) =

1

Γ (119898 minus 120572)

int

119905

0

(119905 minus 120591)119898minus120572minus1

119891 (120591) 119889120591

for 119898 minus 1 lt 120572 le 119898 119898 isin 119873 119905 gt 0 119891 isin 119862119898minus1

(2)

Also we need here two of its basic properties

Lemma 4 If 119898 minus 1 lt 120572 le 119898 119898 isin 119873 and 119891 isin 119862119898120583 120583 ge minus1

then119863120572119868120572119891 (119905) = 119891 (119905)

119868120572119863120572119891 (119905) = 119891 (119905) minus

119898minus1

sum

119896=0

119891(119896)(0+)

119905119896

119896

119905 gt 0

(3)

The Caputo fractional derivative is considered herebecause it allows traditional initial and boundary conditionsto be included in the formulation of the problem [36] Inthis work we consider the one-dimensional linear inho-mogeneous fractional partial differential equations in fluidmechanics where the unknown function 119906(119909 119905) is assumedto be a causal function of time that is vanishing for 119905 lt 0The fractional derivative is taken in Caputo sense as follows

Definition 5 For119898 to be the smallest integer that exceeds 120572the Caputo time-fractional derivative operator of order 120572 gt 0is defined as

119863120572

119905119906 (119909 119905) =

120597120572119906

120597119905120572

=

1

Γ (119898 minus 120572)

int

119905

0

(119905 minus 120591)119898minus120572minus1 120597

119898

120597120591119898119906 (119909 120591) 119889120591

119898 minus 1 lt 120572 le 119898

120597119898

120597119905119898119906 (119909 119905) 120572 = 119898 119898 isin 119873

(4)

3 New Iterative Method (NIM)

To illustrate the basic idea of the NIM consider the followinggeneral functional equation [13 14 26 27]

119906 (119909) = 119891 (119909) + 119873 (119906 (119909)) (5)where119873 is a nonlinear operator from a Banach space 119861 rarr 119861

and 119891 is a known function (element) of the Banach space 119861We are looking for a solution 119906 of (5) having the series form

119906 (119909) =

infin

sum

119894=0

119906119894 (119909) (6)

The nonlinear operator119873 can be decomposed as

119873(

infin

sum

119894=0

119906119894) = 119873(1199060) +

infin

sum

119894=1

119873(

119894

sum

119895=0

119906119895) minus119873(

119894minus1

sum

119895=0

119906119895)

(7)From (6) and (7) (5) is equivalent toinfin

sum

119894=0

119906119894 = 119891 + 119873 (1199060) +

infin

sum

119894=1

119873(

119894

sum

119895=0

119906119895) minus119873(

119894minus1

sum

119895=0

119906119895)

(8)

The required solution for (5) can be obtained recurrentlyfrom the recurrence relation

1199060 = 119891 1199061 = 119873 (1199060)

119906119899+1 = 119873 (1199060 + 1199061 + sdot sdot sdot + 119906119899)

minus 119873 (1199060 + 1199061 + sdot sdot sdot + 119906119899minus1) 119899 = 1 2

(9)

Abstract and Applied Analysis 3

Then

(1199061 + 1199062 + sdot sdot sdot + 119906119899+1)

= 119873 (1199060 + 1199061 + sdot sdot sdot + 119906119899) 119899 = 1 2

119906 = 119891 +

infin

sum

119894=0

119906119894

(10)

The n-term approximate solution of (5) and (6) is given by

119906 (119909) = 1199060 + 1199061 + sdot sdot sdot + 119906119899minus1 (11)

Remark 6 If 119873 is a contraction that is 119873(119909) minus 119873(119910) le119896119909 minus 119910 0 lt 119896 lt 1 then

1003817100381710038171003817119906119899+1

1003817100381710038171003817le 119896119899+1 10038171003817100381710038171199060

1003817100381710038171003817 119899 = 0 1 2 (12)

Proof From (9) we have

1199060 = 11989110038171003817100381710038171199061

1003817100381710038171003817=1003817100381710038171003817119873 (1199060)

1003817100381710038171003817le 11989610038171003817100381710038171199060

1003817100381710038171003817

10038171003817100381710038171199062

1003817100381710038171003817=1003817100381710038171003817119873 (1199060 + 1199061) minus 119873 (1199060)

1003817100381710038171003817le 11989610038171003817100381710038171199061

1003817100381710038171003817le 1198962 10038171003817100381710038171199060

1003817100381710038171003817

10038171003817100381710038171199063

1003817100381710038171003817=1003817100381710038171003817119873 (1199060 + 1199061 + 1199062) minus 119873 (1199060 + 1199061)

1003817100381710038171003817

le 11989610038171003817100381710038171199062

1003817100381710038171003817le 1198963 10038171003817100381710038171199060

1003817100381710038171003817

1003817100381710038171003817119906119899+1

1003817100381710038171003817=1003817100381710038171003817119873 (1199060 + sdot sdot sdot + 119906119899) minus 119873 (1199060 + sdot sdot sdot + 119906119899minus1)

1003817100381710038171003817

le 1198961003817100381710038171003817119906119899

1003817100381710038171003817le 119896119899+1 10038171003817100381710038171199060

1003817100381710038171003817 119899 = 0 1 2

(13)

and the series suminfin119894=0119906119894 absolutely and uniformly converges to

a solution of (5) [37] which is unique in view of the Banachfixed point theorem [38] For more details about NIM see[39]

31 Reliable Algorithm To illustrate the basic idea of the reli-able algorithm we consider the general fractional partial dif-ferential equation of arbitrary fractional order

120597120572119906 (119909 119905)

120597119905120572

= 119860 (119906 120597119906) + 119861 (119909 119905) 119898 minus 1 lt 120572 le 119898 (14a)

with the initial conditions

120597119896119906 (119909 0)

120597119905119896

= ℎ119896 (119909) 119896 = 0 1 2 119898 minus 1 119898 isin 119873

(14b)

where119860 is a nonlinear function of 119906 120597119906 (partial derivatives of119906with respect to 119909 and 119905) and119861 is the source function In viewof the fractional calculus and the properties of the fractionalintegral operators the initial value problem (14a) and (14b) isequivalent to the following fractional integral equation

119906 (119909 119905) =

119898minus1

sum

119896=0

ℎ119896 (119909)

119905119896

119896

+ 119868120572

119905119861 + 119868120572

119905119860 = 119891 + 119873 (119906) (15)

where 119891 = sum119898minus1119896=0ℎ119896(119909)(119905

119896119896) + 119868

120572

119905119861 and119873(119906) = 119868120572

119905119860 We get

the solution of (15) by employing the recurrence relation (9)

Remark 7 When the general functional equation (5) is linearthe recurrence relation (9) can be simplified in the form

1199060 = 119891

119906119899+1 = 119873 (119906119899) 119899 = 0 1 2

(16)

Proof From the properties of integration in case119873 is an inte-gral operator we have

119906119899+1 = 119873 (1199060 + sdot sdot sdot + 119906119899minus1 + 119906119899) minus 119873 (1199060 + sdot sdot sdot + 119906119899minus1)

= 119868120572

119909[1199060 + sdot sdot sdot + 119906119899minus1 + 119906119899] minus 119868

120572

119909[1199060 + sdot sdot sdot + 119906119899minus1]

= 119868120572

119909[1199060] + sdot sdot sdot + 119868

120572

119909[119906119899minus1] + 119868

120572

119909[119906119899]

minus 119868120572

119909[1199060] minus sdot sdot sdot minus 119868

120572

119909[119906119899minus1]

= 119868120572

119909[119906119899] = 119873 (119906119899) 119899 = 0 1 2

(17)

In case 119873 is a differential operator we obtain the sameresult

32 Convergence ofNIM Nowwe analyze the convergence ofthe NIM for solving any general functional equation (5) Let119890 = 119906lowastminus119906 where 119906lowast is the exact solution 119906 is the approximate

solution and 119890 is the error in the solution of (5) obviously 119890satisfies (5) that is

119890 (119909) = 119891 (119909) + 119873 (119890 (119909)) (18)

and the recurrence relation (9) becomes

1198900 = 119891 1198901 = 119873 (1198900)

119890119899+1 = 119873 (1198900 + 1198901 + sdot sdot sdot + 119890119899)

minus 119873 (1198900 + 1198901 + sdot sdot sdot + 119890119899minus1) 119899 = 1 2

(19)

If 119873(119909) minus 119873(119910) le 119896119909 minus 119910 0 lt 119896 lt 1 then

1198900 = 11989110038171003817100381710038171198901

1003817100381710038171003817=1003817100381710038171003817119873 (1198900)

1003817100381710038171003817le 11989610038171003817100381710038171198900

1003817100381710038171003817

10038171003817100381710038171198902

1003817100381710038171003817=1003817100381710038171003817119873 (1198900 + 1198901) minus 119873 (1198900)

1003817100381710038171003817

le 11989610038171003817100381710038171198901

1003817100381710038171003817le 1198962 10038171003817100381710038171198900

1003817100381710038171003817

10038171003817100381710038171198903

1003817100381710038171003817=1003817100381710038171003817119873 (1198900 + 1198901 + 1198902) minus 119873 (1198900 + 1198901)

1003817100381710038171003817

le 11989610038171003817100381710038171198902

1003817100381710038171003817le 1198963 10038171003817100381710038171198900

1003817100381710038171003817

1003817100381710038171003817119890119899+1

1003817100381710038171003817=1003817100381710038171003817119873 (1198900 + sdot sdot sdot + 119890119899) minus 119873 (1198900 + sdot sdot sdot + 119890119899minus1)

1003817100381710038171003817

le 1198961003817100381710038171003817119890119899

1003817100381710038171003817le 119896119899+1 10038171003817100381710038171198900

1003817100381710038171003817 119899 = 0 1 2

(20)

Thus 119890119899+1 rarr 0 as 119899 rarr infin which proves the convergence ofthe NIM for solving the general functional equation (5)

Remark 8 For linear problems we get the solution of (5) byemploying the recurrence relation (16) in place of the recur-rence relation (9)

4 Abstract and Applied Analysis

4 Numerical Examples

41 Linear Problems To incorporate the above discussionthree linear fractional partial differential equations will bestudiedTheNIM is used to obtain the exact solution of theseproblems

Example 1 Consider the following one-dimensional linearinhomogeneous fractional wave equation

119863120572

119909119906 + 119906119909 =

1199051+120572

Γ (2 minus 120572)

sin119909 + 119905 cos119909

119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(21a)

subject to the initial condition

119906 (119909 0) = 0 (21b)

Problem (21a) and (21b) is solved in [40] by using theADM the first few components of solution are as follows

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199061 (119909 119905) = minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199062 (119909 119905) = minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909

(22)

The solution for (21a) and (21b) in series form is given by

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909 minus 1199051+120572

Γ (2 + 120572)

cos119909

+

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+3120572

Γ (2 + 3120572)

cos119909 + sdot sdot sdot

(23)

Canceling the noise terms and keeping the nonnoise termsin (23) yield the exact solution of (21a) and (21b) given by119906(119909 119905) = 119905 sin119909 which is easily verified

Also problem (21a) and (21b) is solved in [40] by usingthe VIM By beginning with 1199060 = 0 the following approxi-mations can be obtained

1199061 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199062 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199063 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909 minus 1199051+120572

Γ (2 + 120572)

cos119909

+

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+3120572

Γ (2 + 3120572)

cos119909

(24)

As in (23) canceling the noise terms and keeping the non-noise terms yield the exact solution of (21a) and (21b)

According to the NIM and by (15) and (16) we obtain

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909 (25)

Therefore the initial value problem (21a) and (21b) is equiva-lent to the integral equation

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909 minus 119868120572119905[119906119909]

(26)

Let119873(119906) = minus119868120572119905[119906119909] we can obtain the following first few

components of the new iterative solution for (21a) and (21b)

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199061 (119909 119905) = minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199062 (119909 119905) = minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909

(27)

The solution in series form is given by

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909

minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909 + sdot sdot sdot

(28)

Canceling the noise terms and keeping the non-noise termsin (28) yield the exact solution of (21a) and (21b)

From (23) (24) and (28) it is clear that the threemethodsare the same in solving (21a) and (21b)

Example 2 Consider the following one-dimensional linearinhomogeneous fractional Burgers equation

119863120572

119905119906 + 119906119909 minus 119906119909119909

=

21199052minus120572

Γ (3 minus 120572)

+ 2119909 minus 2 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(29a)

subject to the initial condition

119906 (119909 0) = 1199092 (29b)

Abstract and Applied Analysis 5

Problem (29a) and (29b) is solved in [40] by using theADM the first few components of solution are as follows

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(30)

Therefore the exact solution is given by 119906(119909 119905) = 1199092 + 1199052

Also problem (29a) and (29b) is solved in [40] by usingthe VIM with 1199060 = 119909

2 the following approximations can beobtained

1199060 (119909 119905) = sdot sdot sdot = 119906119899 (119909 119905) = 1199092+ 1199052 (31)

The exact solution 119906(119909 119905) = 1199092 + 1199052 follows immediatelyAccording to the NIM by (15) and (16) we obtain

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) (32)

Therefore the initial value problem (29a) and (29b) is equiv-alent to the integral equation

119906 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus 119868120572

119905[119906119909 minus 119906119909119909]

(33)

Let 119873(119906) = minus119868120572119905[119906119909 minus 119906119909119909] we obtain the following first few

components of the new iterative solution for (29a) and (29b)

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(34)

Therefore the exact solution 119906(119909 119905) = 1199092 + 1199052 follows imme-diately

Example 3 Consider the following one-dimensional linearinhomogeneous fractional Klein-Gordon equation

119863120572

119905119906 minus 119906119909119909 + 119906

= 61199093119905 + (119909

3minus 6119909) 119905

3 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(35a)

subject to the initial conditions

119906 (119909 0) = 0 119906119905 (119909 0) = 0 (35b)

In view of the ADM [40] the first few components ofsolution for (35a) and (35b) are derived as follows

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(36)

The solution in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(37)

In view of the VIM [40] with 1199060 = 0 the followingapproximations for (35a) and (35b) are obtained

1199061 (119909 119905) = (120572 minus 1) [61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

1199062 (119909 119905)

= 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

minus (120572 minus 1)2

times [6 (1199093minus 6119909)

1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

+ sdot sdot sdot

(38)

According to the NIM by (15) and (16) we can obtain

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

(39)

Therefore the initial value problem (35a) and (35b) is equiv-alent to the integral equation

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 119868120572

119905[119906119909119909 minus 119906]

(40)

6 Abstract and Applied Analysis

Let119873(119906) = 119868120572119905[119906119909119909 minus 119906] we can obtain the following first few

components of the new iterative solution for (35a) and (35b)

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(41)

The solution for (35a) and (35b) in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(42)

From (37) (38) and (42) the ADM the VIM and theNIM give the same solution for the classical Klein-Gordonequation (35a) and (35b) in the case 120572 = 2 which is given by

119906 (119909 119905) = 11990931199053+ (1199093minus 6119909)

61199055

Γ (6)

+ 36119909

1199055

Γ (6)

minus 36119909

61199057

Γ (8)

minus 61199093 1199055

Γ (6)

minus (1199093minus 6119909)

61199057

Γ (8)

+ sdot sdot sdot

(43)

Canceling the noise terms and keeping the non-noise terms in(43) yield the exact solution of (35a) and (35b) for the specialcase 120572 = 2 which is given by 119906(119909 119905) = 11990931199053 which is easilyverified

42 Nonlinear Problems For nonlinear equations in generalthere exists no method that yields the exact solution andtherefore only approximate solutions can be derived In thissubsection we use the NIM to provide approximate solutionsfor two kinds of nonlinear time-fractional partial differentialequations

Example 4 Consider the following one-dimensional nonlin-ear homogeneous time-fractional KdV equation

119863120572

119905119906 + 6119906119906119909 + 119906119909119909119909 = 0 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(44a)

subject to the initial condition

119906 (119909 0) =

1

2

sec ℎ2 (1199092

) (44b)

The time-fractional KdV equation (44a) and (44b) issolved in [11 40] by using the ADM The solution in seriesform is found as

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(45)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

)

1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(46)

Also problem (44a) and (44b) is solved in [40] by usingtheVIMwith1199060 = (12)sec ℎ

2(1199092) the first few approximate

solutions are1199061 (119909 119905) = 1198910 (119909)

1199062 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

1199063 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

minus 61198911 (119909) 1198911015840

1(119909)

Γ (1 + 2120572) 1199053120572

Γ(1 + 120572)2Γ (1 + 3120572)

(47)

According to the NIM by (15) and (9) we can obtain

1199060 (119909 119905) =

1

2

sec ℎ2 (1199092

) (48)

Therefore the initial value problem (44a) and (44b) is equiv-alent to the integral equation

119906 (119909 119905) =

1

2

sec ℎ2 (1199092

) minus 119868120572

119905[6119906119906119909 + 119906119909119909119909]

(49)

Let 119873(119906) = minus119868120572119905[6119906119906119909 + 119906119909119909119909] we obtain the following first

few components of the new iterative solution for (44a) and(44b)

1199060 (119909 119905) = 1198910 (119909)

1199061 (119909 119905) = 1198911 (119909)

119905120572

Γ (1 + 120572)

1199062 (119909 119905) = 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

(50)

Abstract and Applied Analysis 7

The solution in series form is given by

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(51)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

) 1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(52)

Now the approximate solution for (44a) and (44b)obtained by the NIM in (51) is the same solution obtained byboth the ADM in (45) and the VIM in (47) Therefore all thethree methods provide the same approximate solution for thetime-fractional KdV equation

Example 5 In this example we consider the one-dimensionalnonlinear homogeneous time-fractional Boussinesq-likeequation

119863120572

119905119906 + (119906

2)119909119909minus (1199062)119909119909119909119909

= 0 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(53a)

subject to the initial conditions

119906 (119909 0) =

4

3

sinh2 (1199094

) 119906119905 (119909 0) = minus

1

3

sinh(1199092

)

(53b)

The time-fractional Boussinesq-like equation (53a) and(53b) is solved by the ADM in [40] The fourth-order termapproximate solution in series form is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(54)

Also problem (53a) and (53b) is solved by the VIM in[40] The fourth-order term approximate solution with 1199060 =(43)sinh2(1199094) minus (13) sinh(1199092) sdot 119905 is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+ (120572 minus 1) [

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

]

+ (120572 minus 1)2[

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

]

+ (120572 minus 1)3[

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

]

(55)

According to the NIM by (15) and (9) we can obtain thefollowing first approximations

1199060 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

1199061 (119909 119905) =

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

1199062 (119909 119905) =

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

1199063 (119909 119905) =

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(56)

The fourth-order term approximate solution in series form isgiven by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

8 Abstract and Applied Analysis

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(57)

where

119873(119906) = minus119868120572

119905[(1199062)119909119909minus (1199062)119909119909119909119909] (58)

It is interesting to point out that for the case of 120572 = 2 theapproximate solution

119906 (119909 119905) =

2

3

[cosh (1199092

)(1 +

1

22

1199052

2

+

1

24

1199054

4

+ sdot sdot sdot ) minus 1]

minus

2

3

sinh(1199092

)[

1

2

119905 +

1

23

1199053

3

+

1

25

1199055

5

+ sdot sdot sdot ]

(59)

follows immediately upon replacing 120572 by 2 in the decompo-sition solution (54) the variational iteration solution (55) orthe new iteration solution (57) which converges to the exactsolution of the Boussinesq-like equation (53a) and (53b)when 120572 = 2 119906(119909 119905) = (43)sinh2((119909 minus 119905)4)

The advantage of the NIM in the nonlinear problemsExamples 4 and 5 is that there is no need to calculate Ado-mianrsquos polynomials as done in the ADM which means thatthe first method is simple and easy of the procedure of calcu-lations over the second Also for more details about the solu-tions of the fractional differential equations by the VIM andsome new asymptotic methods you can see [41ndash43]

5 Conclusion

NIM has been known as a powerful tool for solving manyfunctional equations such as ordinary partial differentialequations integral equations integrodifferential equationsand somany other equations In this work we have presenteda general framework of the NIM for the analytical treatmentof fractional partial differential equations in fluid mechanicsThe present work shows the validity and great potential of theNIM for solving linear and nonlinear fractional partial differ-ential equations All of the examples show that the results ofthe NIM are in excellent agreement with those obtained byboth the ADM and the VIM without calculating Adomianrsquospolynomials for the nonlinear problems as in the ADM Thebasic idea described in this work is expected to be furtheremployed to solve other similar linear and nonlinear prob-lems in fractional calculus

Conflict of Interests

The author declared that there is no conflict of interestsregarding the publication of this paper

References

[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsapplicationsrdquo in Proceedings of the International Conference onVibrating Engineering 98 Dalian China 1988

[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 no 2 pp 86ndash90 1999

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[5] K Al-Khaled and S Momani ldquoAn approximate solution fora fractional diffusion-wave equation using the decompositionmethodrdquoAppliedMathematics and Computation vol 165 no 2pp 473ndash483 2005

[6] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[7] A Hanyga ldquoMultidimensional solutions of time-fractional dif-fusion-wave equationsrdquoTheRoyal Society of London A vol 458no 2020 pp 933ndash957 2002

[8] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquo The ANZIAM Journal vol46 no 3 pp 317ndash330 2005

[9] F Huang and F Liu ldquoThe fundamental solution of the space-time fractional advection-dispersion equationrdquo Journal ofApplied Mathematics amp Computing vol 18 no 1-2 pp 339ndash3502005

[10] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005

[11] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

[12] L Debnath and D D Bhatta ldquoSolutions to few linear fractionalinhomogeneous partial differential equations in fluid mechan-icsrdquo Fractional Calculus amp Applied Analysis vol 7 no 1 pp 21ndash36 2004

[13] VDaftardar-Gejji andH Jafari ldquoAn iterativemethod for solvingnonlinear functional equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 316 no 2 pp 753ndash763 2006

[14] A A Hemeda ldquoNew iterative method application to the nth-order integro-differential equationsrdquo Information B vol 16 no6 pp 3841ndash3852 2013

[15] S Bhalekar and V Daftardar-Gejji ldquoNew iterative methodapplication to partial differential equationsrdquoAppliedMathemat-ics and Computation vol 203 no 2 pp 778ndash783 2008

[16] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer 1994

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

[18] A A Hemeda ldquoVariational iteration method for solving non-linear coupled equations in 2-dimensional space in fluid mech-anicsrdquo International Journal of Contemporary MathematicalSciences vol 7 no 37 pp 1839ndash1852 2012

[19] G Adomian ldquoA review of the decompositionmethod in appliedmathematicsrdquo Journal of Mathematical Analysis and Applica-tions vol 135 no 2 pp 501ndash544 1988

Abstract and Applied Analysis 9

[20] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008

[21] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[22] J-H He ldquoVariational principle for nano thin film lubricationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 4 no 3 pp 313ndash314 2003

[23] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[24] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[25] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[26] A A Hemeda ldquoFormulation and solution of nth-order deriva-tive fuzzy integro-differential equation using new iterativemethodwith a reliable algorithmrdquo Journal of AppliedMathemat-ics vol 2012 Article ID 325473 17 pages 2012

[27] S Bhalekar andVDaftardar-Gejji ldquoSolving evolution equationsusing a new iterative methodrdquo Numerical Methods for PartialDifferential Equations vol 26 no 4 pp 906ndash916 2010

[28] A A Hemeda ldquoVariational iteration method for solving waveequationrdquo Computers ampMathematics with Applications vol 56no 8 pp 1948ndash1953 2008

[29] A A Hemeda ldquoVariational iteration method for solving non-linear partial differential equationsrdquo Chaos Solitons and Frac-tals vol 39 no 3 pp 1297ndash1303 2009

[30] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[31] A A Hemeda ldquoHomotopy perturbation method for solvingpartial differential equations of fractional orderrdquo InternationalJournal of Mathematical Analysis vol 6 no 49ndash52 pp 2431ndash2448 2012

[32] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency indepedentmdashpart IIrdquoGeophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[33] A Y Luchko and R Groreo The Initial Value Problem forSome Fractional Differential Equations with Caputo DerivativePreprint series A80-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin 1998

[34] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[35] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[36] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] A J Jerri Introduction to Integral Equations with ApplicationsWiley-Interscience 2nd edition 1999

[39] A A Hemeda ldquoNew iterative method an application forsolving fractional physical differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 617010 9 pages 2013

[40] Z Odibat and S Momani ldquoThe variational iteration methodan efficient scheme for handling fractional partial differentialequations in fluid mechanicsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2199ndash2208 2009

[41] G-CWu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013

[42] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 pp 59ndash75 2011

[43] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

obtained memory or power The method introduces thesolution in the form of a convergent fractional series with ele-gantly computable termsThe corresponding solutions of theinteger order equations are found to follow as special cases ofthose of fractional order equations The obtained results oftheNIM are comparedwith those obtained by bothADM [1920] and the VIM [21ndash25] which confirm that this method isvery effective and convenient to these equations and to othersimilar equations where it has the advantage that there is noneed to calculate Adomianrsquos polynomials for the nonlinearproblems as in the ADM For more details see [26ndash31]

Throughout this work fractional partial differential equa-tions are obtained from the corresponding integer orderequations by replacing the first-order or the second-order timederivative by a fractional in the Caputo sense [32] of order 120572with 0 lt 120572 le 1 or 1 lt 120572 le 2

2 Preliminaries and Notations

In this section we give some basic definitions and propertiesof the fractional calculus theorywhich are used further in thiswork

Definition 1 A real function 119891(119905) 119905 gt 0 is said to be in thespace119862120583 120583 isin 119877 if there exists a real number 119901 (gt120583) such that119891(119905) = 119905

1199011198911(119905) where 1198911(119905) isin 119862[0infin) and it is said to be in

the space 119862119898120583if and only if 119891(119898) isin 119862120583119898 isin 119873

Definition 2 TheRiemann-Liouville fractional integral oper-ator of order 120572 ge 0 of a function 119891 isin 119862120583 120583 ge minus1 is definedas

119868120572119891 (119905) =

1

Γ (120572)

int

119905

0

(119905 minus 120591)120572minus1119891 (120591) 119889120591 120572 gt 0 119905 gt 0

1198680119891 (119905) = 119891 (119905)

(1)

Properties of the operators 119868120572 can be found in [32ndash35] wemention only the following for 119891 isin 119862120583 120583 ge minus1 120572 120573 ge 0 and120592 gt minus1

(1) 119868120572119868120573119891(119905) = 119868120572+120573119891(119905)(2) 119868120572119868120573119891(119905) = 119868120573119868120572119891(119905)(3) 119868120572119905120592 = (Γ(120592 + 1)Γ(120592 + 1 + 120572))119905120572+]

The Riemann-Liouville derivative has certain disadvan-tages when trying to model real-world phenomena withfractional differential equationsTherefore we will introducea modified fractional differential operator 119863120572 proposed byCaputo in this work on the theory of viscoelasticity [32]

Definition 3 The fractional derivative of 119891(119905) in the Caputosense is defined as

119863120572119891 (119905) = 119868

119898minus120572119863119898119891 (119905) =

1

Γ (119898 minus 120572)

int

119905

0

(119905 minus 120591)119898minus120572minus1

119891 (120591) 119889120591

for 119898 minus 1 lt 120572 le 119898 119898 isin 119873 119905 gt 0 119891 isin 119862119898minus1

(2)

Also we need here two of its basic properties

Lemma 4 If 119898 minus 1 lt 120572 le 119898 119898 isin 119873 and 119891 isin 119862119898120583 120583 ge minus1

then119863120572119868120572119891 (119905) = 119891 (119905)

119868120572119863120572119891 (119905) = 119891 (119905) minus

119898minus1

sum

119896=0

119891(119896)(0+)

119905119896

119896

119905 gt 0

(3)

The Caputo fractional derivative is considered herebecause it allows traditional initial and boundary conditionsto be included in the formulation of the problem [36] Inthis work we consider the one-dimensional linear inho-mogeneous fractional partial differential equations in fluidmechanics where the unknown function 119906(119909 119905) is assumedto be a causal function of time that is vanishing for 119905 lt 0The fractional derivative is taken in Caputo sense as follows

Definition 5 For119898 to be the smallest integer that exceeds 120572the Caputo time-fractional derivative operator of order 120572 gt 0is defined as

119863120572

119905119906 (119909 119905) =

120597120572119906

120597119905120572

=

1

Γ (119898 minus 120572)

int

119905

0

(119905 minus 120591)119898minus120572minus1 120597

119898

120597120591119898119906 (119909 120591) 119889120591

119898 minus 1 lt 120572 le 119898

120597119898

120597119905119898119906 (119909 119905) 120572 = 119898 119898 isin 119873

(4)

3 New Iterative Method (NIM)

To illustrate the basic idea of the NIM consider the followinggeneral functional equation [13 14 26 27]

119906 (119909) = 119891 (119909) + 119873 (119906 (119909)) (5)where119873 is a nonlinear operator from a Banach space 119861 rarr 119861

and 119891 is a known function (element) of the Banach space 119861We are looking for a solution 119906 of (5) having the series form

119906 (119909) =

infin

sum

119894=0

119906119894 (119909) (6)

The nonlinear operator119873 can be decomposed as

119873(

infin

sum

119894=0

119906119894) = 119873(1199060) +

infin

sum

119894=1

119873(

119894

sum

119895=0

119906119895) minus119873(

119894minus1

sum

119895=0

119906119895)

(7)From (6) and (7) (5) is equivalent toinfin

sum

119894=0

119906119894 = 119891 + 119873 (1199060) +

infin

sum

119894=1

119873(

119894

sum

119895=0

119906119895) minus119873(

119894minus1

sum

119895=0

119906119895)

(8)

The required solution for (5) can be obtained recurrentlyfrom the recurrence relation

1199060 = 119891 1199061 = 119873 (1199060)

119906119899+1 = 119873 (1199060 + 1199061 + sdot sdot sdot + 119906119899)

minus 119873 (1199060 + 1199061 + sdot sdot sdot + 119906119899minus1) 119899 = 1 2

(9)

Abstract and Applied Analysis 3

Then

(1199061 + 1199062 + sdot sdot sdot + 119906119899+1)

= 119873 (1199060 + 1199061 + sdot sdot sdot + 119906119899) 119899 = 1 2

119906 = 119891 +

infin

sum

119894=0

119906119894

(10)

The n-term approximate solution of (5) and (6) is given by

119906 (119909) = 1199060 + 1199061 + sdot sdot sdot + 119906119899minus1 (11)

Remark 6 If 119873 is a contraction that is 119873(119909) minus 119873(119910) le119896119909 minus 119910 0 lt 119896 lt 1 then

1003817100381710038171003817119906119899+1

1003817100381710038171003817le 119896119899+1 10038171003817100381710038171199060

1003817100381710038171003817 119899 = 0 1 2 (12)

Proof From (9) we have

1199060 = 11989110038171003817100381710038171199061

1003817100381710038171003817=1003817100381710038171003817119873 (1199060)

1003817100381710038171003817le 11989610038171003817100381710038171199060

1003817100381710038171003817

10038171003817100381710038171199062

1003817100381710038171003817=1003817100381710038171003817119873 (1199060 + 1199061) minus 119873 (1199060)

1003817100381710038171003817le 11989610038171003817100381710038171199061

1003817100381710038171003817le 1198962 10038171003817100381710038171199060

1003817100381710038171003817

10038171003817100381710038171199063

1003817100381710038171003817=1003817100381710038171003817119873 (1199060 + 1199061 + 1199062) minus 119873 (1199060 + 1199061)

1003817100381710038171003817

le 11989610038171003817100381710038171199062

1003817100381710038171003817le 1198963 10038171003817100381710038171199060

1003817100381710038171003817

1003817100381710038171003817119906119899+1

1003817100381710038171003817=1003817100381710038171003817119873 (1199060 + sdot sdot sdot + 119906119899) minus 119873 (1199060 + sdot sdot sdot + 119906119899minus1)

1003817100381710038171003817

le 1198961003817100381710038171003817119906119899

1003817100381710038171003817le 119896119899+1 10038171003817100381710038171199060

1003817100381710038171003817 119899 = 0 1 2

(13)

and the series suminfin119894=0119906119894 absolutely and uniformly converges to

a solution of (5) [37] which is unique in view of the Banachfixed point theorem [38] For more details about NIM see[39]

31 Reliable Algorithm To illustrate the basic idea of the reli-able algorithm we consider the general fractional partial dif-ferential equation of arbitrary fractional order

120597120572119906 (119909 119905)

120597119905120572

= 119860 (119906 120597119906) + 119861 (119909 119905) 119898 minus 1 lt 120572 le 119898 (14a)

with the initial conditions

120597119896119906 (119909 0)

120597119905119896

= ℎ119896 (119909) 119896 = 0 1 2 119898 minus 1 119898 isin 119873

(14b)

where119860 is a nonlinear function of 119906 120597119906 (partial derivatives of119906with respect to 119909 and 119905) and119861 is the source function In viewof the fractional calculus and the properties of the fractionalintegral operators the initial value problem (14a) and (14b) isequivalent to the following fractional integral equation

119906 (119909 119905) =

119898minus1

sum

119896=0

ℎ119896 (119909)

119905119896

119896

+ 119868120572

119905119861 + 119868120572

119905119860 = 119891 + 119873 (119906) (15)

where 119891 = sum119898minus1119896=0ℎ119896(119909)(119905

119896119896) + 119868

120572

119905119861 and119873(119906) = 119868120572

119905119860 We get

the solution of (15) by employing the recurrence relation (9)

Remark 7 When the general functional equation (5) is linearthe recurrence relation (9) can be simplified in the form

1199060 = 119891

119906119899+1 = 119873 (119906119899) 119899 = 0 1 2

(16)

Proof From the properties of integration in case119873 is an inte-gral operator we have

119906119899+1 = 119873 (1199060 + sdot sdot sdot + 119906119899minus1 + 119906119899) minus 119873 (1199060 + sdot sdot sdot + 119906119899minus1)

= 119868120572

119909[1199060 + sdot sdot sdot + 119906119899minus1 + 119906119899] minus 119868

120572

119909[1199060 + sdot sdot sdot + 119906119899minus1]

= 119868120572

119909[1199060] + sdot sdot sdot + 119868

120572

119909[119906119899minus1] + 119868

120572

119909[119906119899]

minus 119868120572

119909[1199060] minus sdot sdot sdot minus 119868

120572

119909[119906119899minus1]

= 119868120572

119909[119906119899] = 119873 (119906119899) 119899 = 0 1 2

(17)

In case 119873 is a differential operator we obtain the sameresult

32 Convergence ofNIM Nowwe analyze the convergence ofthe NIM for solving any general functional equation (5) Let119890 = 119906lowastminus119906 where 119906lowast is the exact solution 119906 is the approximate

solution and 119890 is the error in the solution of (5) obviously 119890satisfies (5) that is

119890 (119909) = 119891 (119909) + 119873 (119890 (119909)) (18)

and the recurrence relation (9) becomes

1198900 = 119891 1198901 = 119873 (1198900)

119890119899+1 = 119873 (1198900 + 1198901 + sdot sdot sdot + 119890119899)

minus 119873 (1198900 + 1198901 + sdot sdot sdot + 119890119899minus1) 119899 = 1 2

(19)

If 119873(119909) minus 119873(119910) le 119896119909 minus 119910 0 lt 119896 lt 1 then

1198900 = 11989110038171003817100381710038171198901

1003817100381710038171003817=1003817100381710038171003817119873 (1198900)

1003817100381710038171003817le 11989610038171003817100381710038171198900

1003817100381710038171003817

10038171003817100381710038171198902

1003817100381710038171003817=1003817100381710038171003817119873 (1198900 + 1198901) minus 119873 (1198900)

1003817100381710038171003817

le 11989610038171003817100381710038171198901

1003817100381710038171003817le 1198962 10038171003817100381710038171198900

1003817100381710038171003817

10038171003817100381710038171198903

1003817100381710038171003817=1003817100381710038171003817119873 (1198900 + 1198901 + 1198902) minus 119873 (1198900 + 1198901)

1003817100381710038171003817

le 11989610038171003817100381710038171198902

1003817100381710038171003817le 1198963 10038171003817100381710038171198900

1003817100381710038171003817

1003817100381710038171003817119890119899+1

1003817100381710038171003817=1003817100381710038171003817119873 (1198900 + sdot sdot sdot + 119890119899) minus 119873 (1198900 + sdot sdot sdot + 119890119899minus1)

1003817100381710038171003817

le 1198961003817100381710038171003817119890119899

1003817100381710038171003817le 119896119899+1 10038171003817100381710038171198900

1003817100381710038171003817 119899 = 0 1 2

(20)

Thus 119890119899+1 rarr 0 as 119899 rarr infin which proves the convergence ofthe NIM for solving the general functional equation (5)

Remark 8 For linear problems we get the solution of (5) byemploying the recurrence relation (16) in place of the recur-rence relation (9)

4 Abstract and Applied Analysis

4 Numerical Examples

41 Linear Problems To incorporate the above discussionthree linear fractional partial differential equations will bestudiedTheNIM is used to obtain the exact solution of theseproblems

Example 1 Consider the following one-dimensional linearinhomogeneous fractional wave equation

119863120572

119909119906 + 119906119909 =

1199051+120572

Γ (2 minus 120572)

sin119909 + 119905 cos119909

119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(21a)

subject to the initial condition

119906 (119909 0) = 0 (21b)

Problem (21a) and (21b) is solved in [40] by using theADM the first few components of solution are as follows

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199061 (119909 119905) = minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199062 (119909 119905) = minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909

(22)

The solution for (21a) and (21b) in series form is given by

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909 minus 1199051+120572

Γ (2 + 120572)

cos119909

+

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+3120572

Γ (2 + 3120572)

cos119909 + sdot sdot sdot

(23)

Canceling the noise terms and keeping the nonnoise termsin (23) yield the exact solution of (21a) and (21b) given by119906(119909 119905) = 119905 sin119909 which is easily verified

Also problem (21a) and (21b) is solved in [40] by usingthe VIM By beginning with 1199060 = 0 the following approxi-mations can be obtained

1199061 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199062 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199063 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909 minus 1199051+120572

Γ (2 + 120572)

cos119909

+

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+3120572

Γ (2 + 3120572)

cos119909

(24)

As in (23) canceling the noise terms and keeping the non-noise terms yield the exact solution of (21a) and (21b)

According to the NIM and by (15) and (16) we obtain

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909 (25)

Therefore the initial value problem (21a) and (21b) is equiva-lent to the integral equation

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909 minus 119868120572119905[119906119909]

(26)

Let119873(119906) = minus119868120572119905[119906119909] we can obtain the following first few

components of the new iterative solution for (21a) and (21b)

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199061 (119909 119905) = minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199062 (119909 119905) = minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909

(27)

The solution in series form is given by

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909

minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909 + sdot sdot sdot

(28)

Canceling the noise terms and keeping the non-noise termsin (28) yield the exact solution of (21a) and (21b)

From (23) (24) and (28) it is clear that the threemethodsare the same in solving (21a) and (21b)

Example 2 Consider the following one-dimensional linearinhomogeneous fractional Burgers equation

119863120572

119905119906 + 119906119909 minus 119906119909119909

=

21199052minus120572

Γ (3 minus 120572)

+ 2119909 minus 2 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(29a)

subject to the initial condition

119906 (119909 0) = 1199092 (29b)

Abstract and Applied Analysis 5

Problem (29a) and (29b) is solved in [40] by using theADM the first few components of solution are as follows

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(30)

Therefore the exact solution is given by 119906(119909 119905) = 1199092 + 1199052

Also problem (29a) and (29b) is solved in [40] by usingthe VIM with 1199060 = 119909

2 the following approximations can beobtained

1199060 (119909 119905) = sdot sdot sdot = 119906119899 (119909 119905) = 1199092+ 1199052 (31)

The exact solution 119906(119909 119905) = 1199092 + 1199052 follows immediatelyAccording to the NIM by (15) and (16) we obtain

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) (32)

Therefore the initial value problem (29a) and (29b) is equiv-alent to the integral equation

119906 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus 119868120572

119905[119906119909 minus 119906119909119909]

(33)

Let 119873(119906) = minus119868120572119905[119906119909 minus 119906119909119909] we obtain the following first few

components of the new iterative solution for (29a) and (29b)

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(34)

Therefore the exact solution 119906(119909 119905) = 1199092 + 1199052 follows imme-diately

Example 3 Consider the following one-dimensional linearinhomogeneous fractional Klein-Gordon equation

119863120572

119905119906 minus 119906119909119909 + 119906

= 61199093119905 + (119909

3minus 6119909) 119905

3 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(35a)

subject to the initial conditions

119906 (119909 0) = 0 119906119905 (119909 0) = 0 (35b)

In view of the ADM [40] the first few components ofsolution for (35a) and (35b) are derived as follows

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(36)

The solution in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(37)

In view of the VIM [40] with 1199060 = 0 the followingapproximations for (35a) and (35b) are obtained

1199061 (119909 119905) = (120572 minus 1) [61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

1199062 (119909 119905)

= 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

minus (120572 minus 1)2

times [6 (1199093minus 6119909)

1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

+ sdot sdot sdot

(38)

According to the NIM by (15) and (16) we can obtain

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

(39)

Therefore the initial value problem (35a) and (35b) is equiv-alent to the integral equation

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 119868120572

119905[119906119909119909 minus 119906]

(40)

6 Abstract and Applied Analysis

Let119873(119906) = 119868120572119905[119906119909119909 minus 119906] we can obtain the following first few

components of the new iterative solution for (35a) and (35b)

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(41)

The solution for (35a) and (35b) in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(42)

From (37) (38) and (42) the ADM the VIM and theNIM give the same solution for the classical Klein-Gordonequation (35a) and (35b) in the case 120572 = 2 which is given by

119906 (119909 119905) = 11990931199053+ (1199093minus 6119909)

61199055

Γ (6)

+ 36119909

1199055

Γ (6)

minus 36119909

61199057

Γ (8)

minus 61199093 1199055

Γ (6)

minus (1199093minus 6119909)

61199057

Γ (8)

+ sdot sdot sdot

(43)

Canceling the noise terms and keeping the non-noise terms in(43) yield the exact solution of (35a) and (35b) for the specialcase 120572 = 2 which is given by 119906(119909 119905) = 11990931199053 which is easilyverified

42 Nonlinear Problems For nonlinear equations in generalthere exists no method that yields the exact solution andtherefore only approximate solutions can be derived In thissubsection we use the NIM to provide approximate solutionsfor two kinds of nonlinear time-fractional partial differentialequations

Example 4 Consider the following one-dimensional nonlin-ear homogeneous time-fractional KdV equation

119863120572

119905119906 + 6119906119906119909 + 119906119909119909119909 = 0 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(44a)

subject to the initial condition

119906 (119909 0) =

1

2

sec ℎ2 (1199092

) (44b)

The time-fractional KdV equation (44a) and (44b) issolved in [11 40] by using the ADM The solution in seriesform is found as

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(45)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

)

1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(46)

Also problem (44a) and (44b) is solved in [40] by usingtheVIMwith1199060 = (12)sec ℎ

2(1199092) the first few approximate

solutions are1199061 (119909 119905) = 1198910 (119909)

1199062 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

1199063 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

minus 61198911 (119909) 1198911015840

1(119909)

Γ (1 + 2120572) 1199053120572

Γ(1 + 120572)2Γ (1 + 3120572)

(47)

According to the NIM by (15) and (9) we can obtain

1199060 (119909 119905) =

1

2

sec ℎ2 (1199092

) (48)

Therefore the initial value problem (44a) and (44b) is equiv-alent to the integral equation

119906 (119909 119905) =

1

2

sec ℎ2 (1199092

) minus 119868120572

119905[6119906119906119909 + 119906119909119909119909]

(49)

Let 119873(119906) = minus119868120572119905[6119906119906119909 + 119906119909119909119909] we obtain the following first

few components of the new iterative solution for (44a) and(44b)

1199060 (119909 119905) = 1198910 (119909)

1199061 (119909 119905) = 1198911 (119909)

119905120572

Γ (1 + 120572)

1199062 (119909 119905) = 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

(50)

Abstract and Applied Analysis 7

The solution in series form is given by

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(51)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

) 1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(52)

Now the approximate solution for (44a) and (44b)obtained by the NIM in (51) is the same solution obtained byboth the ADM in (45) and the VIM in (47) Therefore all thethree methods provide the same approximate solution for thetime-fractional KdV equation

Example 5 In this example we consider the one-dimensionalnonlinear homogeneous time-fractional Boussinesq-likeequation

119863120572

119905119906 + (119906

2)119909119909minus (1199062)119909119909119909119909

= 0 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(53a)

subject to the initial conditions

119906 (119909 0) =

4

3

sinh2 (1199094

) 119906119905 (119909 0) = minus

1

3

sinh(1199092

)

(53b)

The time-fractional Boussinesq-like equation (53a) and(53b) is solved by the ADM in [40] The fourth-order termapproximate solution in series form is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(54)

Also problem (53a) and (53b) is solved by the VIM in[40] The fourth-order term approximate solution with 1199060 =(43)sinh2(1199094) minus (13) sinh(1199092) sdot 119905 is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+ (120572 minus 1) [

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

]

+ (120572 minus 1)2[

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

]

+ (120572 minus 1)3[

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

]

(55)

According to the NIM by (15) and (9) we can obtain thefollowing first approximations

1199060 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

1199061 (119909 119905) =

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

1199062 (119909 119905) =

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

1199063 (119909 119905) =

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(56)

The fourth-order term approximate solution in series form isgiven by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

8 Abstract and Applied Analysis

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(57)

where

119873(119906) = minus119868120572

119905[(1199062)119909119909minus (1199062)119909119909119909119909] (58)

It is interesting to point out that for the case of 120572 = 2 theapproximate solution

119906 (119909 119905) =

2

3

[cosh (1199092

)(1 +

1

22

1199052

2

+

1

24

1199054

4

+ sdot sdot sdot ) minus 1]

minus

2

3

sinh(1199092

)[

1

2

119905 +

1

23

1199053

3

+

1

25

1199055

5

+ sdot sdot sdot ]

(59)

follows immediately upon replacing 120572 by 2 in the decompo-sition solution (54) the variational iteration solution (55) orthe new iteration solution (57) which converges to the exactsolution of the Boussinesq-like equation (53a) and (53b)when 120572 = 2 119906(119909 119905) = (43)sinh2((119909 minus 119905)4)

The advantage of the NIM in the nonlinear problemsExamples 4 and 5 is that there is no need to calculate Ado-mianrsquos polynomials as done in the ADM which means thatthe first method is simple and easy of the procedure of calcu-lations over the second Also for more details about the solu-tions of the fractional differential equations by the VIM andsome new asymptotic methods you can see [41ndash43]

5 Conclusion

NIM has been known as a powerful tool for solving manyfunctional equations such as ordinary partial differentialequations integral equations integrodifferential equationsand somany other equations In this work we have presenteda general framework of the NIM for the analytical treatmentof fractional partial differential equations in fluid mechanicsThe present work shows the validity and great potential of theNIM for solving linear and nonlinear fractional partial differ-ential equations All of the examples show that the results ofthe NIM are in excellent agreement with those obtained byboth the ADM and the VIM without calculating Adomianrsquospolynomials for the nonlinear problems as in the ADM Thebasic idea described in this work is expected to be furtheremployed to solve other similar linear and nonlinear prob-lems in fractional calculus

Conflict of Interests

The author declared that there is no conflict of interestsregarding the publication of this paper

References

[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsapplicationsrdquo in Proceedings of the International Conference onVibrating Engineering 98 Dalian China 1988

[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 no 2 pp 86ndash90 1999

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[5] K Al-Khaled and S Momani ldquoAn approximate solution fora fractional diffusion-wave equation using the decompositionmethodrdquoAppliedMathematics and Computation vol 165 no 2pp 473ndash483 2005

[6] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[7] A Hanyga ldquoMultidimensional solutions of time-fractional dif-fusion-wave equationsrdquoTheRoyal Society of London A vol 458no 2020 pp 933ndash957 2002

[8] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquo The ANZIAM Journal vol46 no 3 pp 317ndash330 2005

[9] F Huang and F Liu ldquoThe fundamental solution of the space-time fractional advection-dispersion equationrdquo Journal ofApplied Mathematics amp Computing vol 18 no 1-2 pp 339ndash3502005

[10] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005

[11] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

[12] L Debnath and D D Bhatta ldquoSolutions to few linear fractionalinhomogeneous partial differential equations in fluid mechan-icsrdquo Fractional Calculus amp Applied Analysis vol 7 no 1 pp 21ndash36 2004

[13] VDaftardar-Gejji andH Jafari ldquoAn iterativemethod for solvingnonlinear functional equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 316 no 2 pp 753ndash763 2006

[14] A A Hemeda ldquoNew iterative method application to the nth-order integro-differential equationsrdquo Information B vol 16 no6 pp 3841ndash3852 2013

[15] S Bhalekar and V Daftardar-Gejji ldquoNew iterative methodapplication to partial differential equationsrdquoAppliedMathemat-ics and Computation vol 203 no 2 pp 778ndash783 2008

[16] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer 1994

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

[18] A A Hemeda ldquoVariational iteration method for solving non-linear coupled equations in 2-dimensional space in fluid mech-anicsrdquo International Journal of Contemporary MathematicalSciences vol 7 no 37 pp 1839ndash1852 2012

[19] G Adomian ldquoA review of the decompositionmethod in appliedmathematicsrdquo Journal of Mathematical Analysis and Applica-tions vol 135 no 2 pp 501ndash544 1988

Abstract and Applied Analysis 9

[20] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008

[21] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[22] J-H He ldquoVariational principle for nano thin film lubricationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 4 no 3 pp 313ndash314 2003

[23] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[24] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[25] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[26] A A Hemeda ldquoFormulation and solution of nth-order deriva-tive fuzzy integro-differential equation using new iterativemethodwith a reliable algorithmrdquo Journal of AppliedMathemat-ics vol 2012 Article ID 325473 17 pages 2012

[27] S Bhalekar andVDaftardar-Gejji ldquoSolving evolution equationsusing a new iterative methodrdquo Numerical Methods for PartialDifferential Equations vol 26 no 4 pp 906ndash916 2010

[28] A A Hemeda ldquoVariational iteration method for solving waveequationrdquo Computers ampMathematics with Applications vol 56no 8 pp 1948ndash1953 2008

[29] A A Hemeda ldquoVariational iteration method for solving non-linear partial differential equationsrdquo Chaos Solitons and Frac-tals vol 39 no 3 pp 1297ndash1303 2009

[30] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[31] A A Hemeda ldquoHomotopy perturbation method for solvingpartial differential equations of fractional orderrdquo InternationalJournal of Mathematical Analysis vol 6 no 49ndash52 pp 2431ndash2448 2012

[32] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency indepedentmdashpart IIrdquoGeophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[33] A Y Luchko and R Groreo The Initial Value Problem forSome Fractional Differential Equations with Caputo DerivativePreprint series A80-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin 1998

[34] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[35] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[36] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] A J Jerri Introduction to Integral Equations with ApplicationsWiley-Interscience 2nd edition 1999

[39] A A Hemeda ldquoNew iterative method an application forsolving fractional physical differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 617010 9 pages 2013

[40] Z Odibat and S Momani ldquoThe variational iteration methodan efficient scheme for handling fractional partial differentialequations in fluid mechanicsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2199ndash2208 2009

[41] G-CWu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013

[42] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 pp 59ndash75 2011

[43] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

Then

(1199061 + 1199062 + sdot sdot sdot + 119906119899+1)

= 119873 (1199060 + 1199061 + sdot sdot sdot + 119906119899) 119899 = 1 2

119906 = 119891 +

infin

sum

119894=0

119906119894

(10)

The n-term approximate solution of (5) and (6) is given by

119906 (119909) = 1199060 + 1199061 + sdot sdot sdot + 119906119899minus1 (11)

Remark 6 If 119873 is a contraction that is 119873(119909) minus 119873(119910) le119896119909 minus 119910 0 lt 119896 lt 1 then

1003817100381710038171003817119906119899+1

1003817100381710038171003817le 119896119899+1 10038171003817100381710038171199060

1003817100381710038171003817 119899 = 0 1 2 (12)

Proof From (9) we have

1199060 = 11989110038171003817100381710038171199061

1003817100381710038171003817=1003817100381710038171003817119873 (1199060)

1003817100381710038171003817le 11989610038171003817100381710038171199060

1003817100381710038171003817

10038171003817100381710038171199062

1003817100381710038171003817=1003817100381710038171003817119873 (1199060 + 1199061) minus 119873 (1199060)

1003817100381710038171003817le 11989610038171003817100381710038171199061

1003817100381710038171003817le 1198962 10038171003817100381710038171199060

1003817100381710038171003817

10038171003817100381710038171199063

1003817100381710038171003817=1003817100381710038171003817119873 (1199060 + 1199061 + 1199062) minus 119873 (1199060 + 1199061)

1003817100381710038171003817

le 11989610038171003817100381710038171199062

1003817100381710038171003817le 1198963 10038171003817100381710038171199060

1003817100381710038171003817

1003817100381710038171003817119906119899+1

1003817100381710038171003817=1003817100381710038171003817119873 (1199060 + sdot sdot sdot + 119906119899) minus 119873 (1199060 + sdot sdot sdot + 119906119899minus1)

1003817100381710038171003817

le 1198961003817100381710038171003817119906119899

1003817100381710038171003817le 119896119899+1 10038171003817100381710038171199060

1003817100381710038171003817 119899 = 0 1 2

(13)

and the series suminfin119894=0119906119894 absolutely and uniformly converges to

a solution of (5) [37] which is unique in view of the Banachfixed point theorem [38] For more details about NIM see[39]

31 Reliable Algorithm To illustrate the basic idea of the reli-able algorithm we consider the general fractional partial dif-ferential equation of arbitrary fractional order

120597120572119906 (119909 119905)

120597119905120572

= 119860 (119906 120597119906) + 119861 (119909 119905) 119898 minus 1 lt 120572 le 119898 (14a)

with the initial conditions

120597119896119906 (119909 0)

120597119905119896

= ℎ119896 (119909) 119896 = 0 1 2 119898 minus 1 119898 isin 119873

(14b)

where119860 is a nonlinear function of 119906 120597119906 (partial derivatives of119906with respect to 119909 and 119905) and119861 is the source function In viewof the fractional calculus and the properties of the fractionalintegral operators the initial value problem (14a) and (14b) isequivalent to the following fractional integral equation

119906 (119909 119905) =

119898minus1

sum

119896=0

ℎ119896 (119909)

119905119896

119896

+ 119868120572

119905119861 + 119868120572

119905119860 = 119891 + 119873 (119906) (15)

where 119891 = sum119898minus1119896=0ℎ119896(119909)(119905

119896119896) + 119868

120572

119905119861 and119873(119906) = 119868120572

119905119860 We get

the solution of (15) by employing the recurrence relation (9)

Remark 7 When the general functional equation (5) is linearthe recurrence relation (9) can be simplified in the form

1199060 = 119891

119906119899+1 = 119873 (119906119899) 119899 = 0 1 2

(16)

Proof From the properties of integration in case119873 is an inte-gral operator we have

119906119899+1 = 119873 (1199060 + sdot sdot sdot + 119906119899minus1 + 119906119899) minus 119873 (1199060 + sdot sdot sdot + 119906119899minus1)

= 119868120572

119909[1199060 + sdot sdot sdot + 119906119899minus1 + 119906119899] minus 119868

120572

119909[1199060 + sdot sdot sdot + 119906119899minus1]

= 119868120572

119909[1199060] + sdot sdot sdot + 119868

120572

119909[119906119899minus1] + 119868

120572

119909[119906119899]

minus 119868120572

119909[1199060] minus sdot sdot sdot minus 119868

120572

119909[119906119899minus1]

= 119868120572

119909[119906119899] = 119873 (119906119899) 119899 = 0 1 2

(17)

In case 119873 is a differential operator we obtain the sameresult

32 Convergence ofNIM Nowwe analyze the convergence ofthe NIM for solving any general functional equation (5) Let119890 = 119906lowastminus119906 where 119906lowast is the exact solution 119906 is the approximate

solution and 119890 is the error in the solution of (5) obviously 119890satisfies (5) that is

119890 (119909) = 119891 (119909) + 119873 (119890 (119909)) (18)

and the recurrence relation (9) becomes

1198900 = 119891 1198901 = 119873 (1198900)

119890119899+1 = 119873 (1198900 + 1198901 + sdot sdot sdot + 119890119899)

minus 119873 (1198900 + 1198901 + sdot sdot sdot + 119890119899minus1) 119899 = 1 2

(19)

If 119873(119909) minus 119873(119910) le 119896119909 minus 119910 0 lt 119896 lt 1 then

1198900 = 11989110038171003817100381710038171198901

1003817100381710038171003817=1003817100381710038171003817119873 (1198900)

1003817100381710038171003817le 11989610038171003817100381710038171198900

1003817100381710038171003817

10038171003817100381710038171198902

1003817100381710038171003817=1003817100381710038171003817119873 (1198900 + 1198901) minus 119873 (1198900)

1003817100381710038171003817

le 11989610038171003817100381710038171198901

1003817100381710038171003817le 1198962 10038171003817100381710038171198900

1003817100381710038171003817

10038171003817100381710038171198903

1003817100381710038171003817=1003817100381710038171003817119873 (1198900 + 1198901 + 1198902) minus 119873 (1198900 + 1198901)

1003817100381710038171003817

le 11989610038171003817100381710038171198902

1003817100381710038171003817le 1198963 10038171003817100381710038171198900

1003817100381710038171003817

1003817100381710038171003817119890119899+1

1003817100381710038171003817=1003817100381710038171003817119873 (1198900 + sdot sdot sdot + 119890119899) minus 119873 (1198900 + sdot sdot sdot + 119890119899minus1)

1003817100381710038171003817

le 1198961003817100381710038171003817119890119899

1003817100381710038171003817le 119896119899+1 10038171003817100381710038171198900

1003817100381710038171003817 119899 = 0 1 2

(20)

Thus 119890119899+1 rarr 0 as 119899 rarr infin which proves the convergence ofthe NIM for solving the general functional equation (5)

Remark 8 For linear problems we get the solution of (5) byemploying the recurrence relation (16) in place of the recur-rence relation (9)

4 Abstract and Applied Analysis

4 Numerical Examples

41 Linear Problems To incorporate the above discussionthree linear fractional partial differential equations will bestudiedTheNIM is used to obtain the exact solution of theseproblems

Example 1 Consider the following one-dimensional linearinhomogeneous fractional wave equation

119863120572

119909119906 + 119906119909 =

1199051+120572

Γ (2 minus 120572)

sin119909 + 119905 cos119909

119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(21a)

subject to the initial condition

119906 (119909 0) = 0 (21b)

Problem (21a) and (21b) is solved in [40] by using theADM the first few components of solution are as follows

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199061 (119909 119905) = minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199062 (119909 119905) = minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909

(22)

The solution for (21a) and (21b) in series form is given by

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909 minus 1199051+120572

Γ (2 + 120572)

cos119909

+

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+3120572

Γ (2 + 3120572)

cos119909 + sdot sdot sdot

(23)

Canceling the noise terms and keeping the nonnoise termsin (23) yield the exact solution of (21a) and (21b) given by119906(119909 119905) = 119905 sin119909 which is easily verified

Also problem (21a) and (21b) is solved in [40] by usingthe VIM By beginning with 1199060 = 0 the following approxi-mations can be obtained

1199061 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199062 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199063 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909 minus 1199051+120572

Γ (2 + 120572)

cos119909

+

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+3120572

Γ (2 + 3120572)

cos119909

(24)

As in (23) canceling the noise terms and keeping the non-noise terms yield the exact solution of (21a) and (21b)

According to the NIM and by (15) and (16) we obtain

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909 (25)

Therefore the initial value problem (21a) and (21b) is equiva-lent to the integral equation

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909 minus 119868120572119905[119906119909]

(26)

Let119873(119906) = minus119868120572119905[119906119909] we can obtain the following first few

components of the new iterative solution for (21a) and (21b)

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199061 (119909 119905) = minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199062 (119909 119905) = minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909

(27)

The solution in series form is given by

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909

minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909 + sdot sdot sdot

(28)

Canceling the noise terms and keeping the non-noise termsin (28) yield the exact solution of (21a) and (21b)

From (23) (24) and (28) it is clear that the threemethodsare the same in solving (21a) and (21b)

Example 2 Consider the following one-dimensional linearinhomogeneous fractional Burgers equation

119863120572

119905119906 + 119906119909 minus 119906119909119909

=

21199052minus120572

Γ (3 minus 120572)

+ 2119909 minus 2 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(29a)

subject to the initial condition

119906 (119909 0) = 1199092 (29b)

Abstract and Applied Analysis 5

Problem (29a) and (29b) is solved in [40] by using theADM the first few components of solution are as follows

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(30)

Therefore the exact solution is given by 119906(119909 119905) = 1199092 + 1199052

Also problem (29a) and (29b) is solved in [40] by usingthe VIM with 1199060 = 119909

2 the following approximations can beobtained

1199060 (119909 119905) = sdot sdot sdot = 119906119899 (119909 119905) = 1199092+ 1199052 (31)

The exact solution 119906(119909 119905) = 1199092 + 1199052 follows immediatelyAccording to the NIM by (15) and (16) we obtain

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) (32)

Therefore the initial value problem (29a) and (29b) is equiv-alent to the integral equation

119906 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus 119868120572

119905[119906119909 minus 119906119909119909]

(33)

Let 119873(119906) = minus119868120572119905[119906119909 minus 119906119909119909] we obtain the following first few

components of the new iterative solution for (29a) and (29b)

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(34)

Therefore the exact solution 119906(119909 119905) = 1199092 + 1199052 follows imme-diately

Example 3 Consider the following one-dimensional linearinhomogeneous fractional Klein-Gordon equation

119863120572

119905119906 minus 119906119909119909 + 119906

= 61199093119905 + (119909

3minus 6119909) 119905

3 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(35a)

subject to the initial conditions

119906 (119909 0) = 0 119906119905 (119909 0) = 0 (35b)

In view of the ADM [40] the first few components ofsolution for (35a) and (35b) are derived as follows

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(36)

The solution in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(37)

In view of the VIM [40] with 1199060 = 0 the followingapproximations for (35a) and (35b) are obtained

1199061 (119909 119905) = (120572 minus 1) [61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

1199062 (119909 119905)

= 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

minus (120572 minus 1)2

times [6 (1199093minus 6119909)

1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

+ sdot sdot sdot

(38)

According to the NIM by (15) and (16) we can obtain

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

(39)

Therefore the initial value problem (35a) and (35b) is equiv-alent to the integral equation

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 119868120572

119905[119906119909119909 minus 119906]

(40)

6 Abstract and Applied Analysis

Let119873(119906) = 119868120572119905[119906119909119909 minus 119906] we can obtain the following first few

components of the new iterative solution for (35a) and (35b)

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(41)

The solution for (35a) and (35b) in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(42)

From (37) (38) and (42) the ADM the VIM and theNIM give the same solution for the classical Klein-Gordonequation (35a) and (35b) in the case 120572 = 2 which is given by

119906 (119909 119905) = 11990931199053+ (1199093minus 6119909)

61199055

Γ (6)

+ 36119909

1199055

Γ (6)

minus 36119909

61199057

Γ (8)

minus 61199093 1199055

Γ (6)

minus (1199093minus 6119909)

61199057

Γ (8)

+ sdot sdot sdot

(43)

Canceling the noise terms and keeping the non-noise terms in(43) yield the exact solution of (35a) and (35b) for the specialcase 120572 = 2 which is given by 119906(119909 119905) = 11990931199053 which is easilyverified

42 Nonlinear Problems For nonlinear equations in generalthere exists no method that yields the exact solution andtherefore only approximate solutions can be derived In thissubsection we use the NIM to provide approximate solutionsfor two kinds of nonlinear time-fractional partial differentialequations

Example 4 Consider the following one-dimensional nonlin-ear homogeneous time-fractional KdV equation

119863120572

119905119906 + 6119906119906119909 + 119906119909119909119909 = 0 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(44a)

subject to the initial condition

119906 (119909 0) =

1

2

sec ℎ2 (1199092

) (44b)

The time-fractional KdV equation (44a) and (44b) issolved in [11 40] by using the ADM The solution in seriesform is found as

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(45)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

)

1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(46)

Also problem (44a) and (44b) is solved in [40] by usingtheVIMwith1199060 = (12)sec ℎ

2(1199092) the first few approximate

solutions are1199061 (119909 119905) = 1198910 (119909)

1199062 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

1199063 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

minus 61198911 (119909) 1198911015840

1(119909)

Γ (1 + 2120572) 1199053120572

Γ(1 + 120572)2Γ (1 + 3120572)

(47)

According to the NIM by (15) and (9) we can obtain

1199060 (119909 119905) =

1

2

sec ℎ2 (1199092

) (48)

Therefore the initial value problem (44a) and (44b) is equiv-alent to the integral equation

119906 (119909 119905) =

1

2

sec ℎ2 (1199092

) minus 119868120572

119905[6119906119906119909 + 119906119909119909119909]

(49)

Let 119873(119906) = minus119868120572119905[6119906119906119909 + 119906119909119909119909] we obtain the following first

few components of the new iterative solution for (44a) and(44b)

1199060 (119909 119905) = 1198910 (119909)

1199061 (119909 119905) = 1198911 (119909)

119905120572

Γ (1 + 120572)

1199062 (119909 119905) = 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

(50)

Abstract and Applied Analysis 7

The solution in series form is given by

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(51)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

) 1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(52)

Now the approximate solution for (44a) and (44b)obtained by the NIM in (51) is the same solution obtained byboth the ADM in (45) and the VIM in (47) Therefore all thethree methods provide the same approximate solution for thetime-fractional KdV equation

Example 5 In this example we consider the one-dimensionalnonlinear homogeneous time-fractional Boussinesq-likeequation

119863120572

119905119906 + (119906

2)119909119909minus (1199062)119909119909119909119909

= 0 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(53a)

subject to the initial conditions

119906 (119909 0) =

4

3

sinh2 (1199094

) 119906119905 (119909 0) = minus

1

3

sinh(1199092

)

(53b)

The time-fractional Boussinesq-like equation (53a) and(53b) is solved by the ADM in [40] The fourth-order termapproximate solution in series form is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(54)

Also problem (53a) and (53b) is solved by the VIM in[40] The fourth-order term approximate solution with 1199060 =(43)sinh2(1199094) minus (13) sinh(1199092) sdot 119905 is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+ (120572 minus 1) [

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

]

+ (120572 minus 1)2[

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

]

+ (120572 minus 1)3[

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

]

(55)

According to the NIM by (15) and (9) we can obtain thefollowing first approximations

1199060 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

1199061 (119909 119905) =

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

1199062 (119909 119905) =

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

1199063 (119909 119905) =

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(56)

The fourth-order term approximate solution in series form isgiven by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

8 Abstract and Applied Analysis

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(57)

where

119873(119906) = minus119868120572

119905[(1199062)119909119909minus (1199062)119909119909119909119909] (58)

It is interesting to point out that for the case of 120572 = 2 theapproximate solution

119906 (119909 119905) =

2

3

[cosh (1199092

)(1 +

1

22

1199052

2

+

1

24

1199054

4

+ sdot sdot sdot ) minus 1]

minus

2

3

sinh(1199092

)[

1

2

119905 +

1

23

1199053

3

+

1

25

1199055

5

+ sdot sdot sdot ]

(59)

follows immediately upon replacing 120572 by 2 in the decompo-sition solution (54) the variational iteration solution (55) orthe new iteration solution (57) which converges to the exactsolution of the Boussinesq-like equation (53a) and (53b)when 120572 = 2 119906(119909 119905) = (43)sinh2((119909 minus 119905)4)

The advantage of the NIM in the nonlinear problemsExamples 4 and 5 is that there is no need to calculate Ado-mianrsquos polynomials as done in the ADM which means thatthe first method is simple and easy of the procedure of calcu-lations over the second Also for more details about the solu-tions of the fractional differential equations by the VIM andsome new asymptotic methods you can see [41ndash43]

5 Conclusion

NIM has been known as a powerful tool for solving manyfunctional equations such as ordinary partial differentialequations integral equations integrodifferential equationsand somany other equations In this work we have presenteda general framework of the NIM for the analytical treatmentof fractional partial differential equations in fluid mechanicsThe present work shows the validity and great potential of theNIM for solving linear and nonlinear fractional partial differ-ential equations All of the examples show that the results ofthe NIM are in excellent agreement with those obtained byboth the ADM and the VIM without calculating Adomianrsquospolynomials for the nonlinear problems as in the ADM Thebasic idea described in this work is expected to be furtheremployed to solve other similar linear and nonlinear prob-lems in fractional calculus

Conflict of Interests

The author declared that there is no conflict of interestsregarding the publication of this paper

References

[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsapplicationsrdquo in Proceedings of the International Conference onVibrating Engineering 98 Dalian China 1988

[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 no 2 pp 86ndash90 1999

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[5] K Al-Khaled and S Momani ldquoAn approximate solution fora fractional diffusion-wave equation using the decompositionmethodrdquoAppliedMathematics and Computation vol 165 no 2pp 473ndash483 2005

[6] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[7] A Hanyga ldquoMultidimensional solutions of time-fractional dif-fusion-wave equationsrdquoTheRoyal Society of London A vol 458no 2020 pp 933ndash957 2002

[8] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquo The ANZIAM Journal vol46 no 3 pp 317ndash330 2005

[9] F Huang and F Liu ldquoThe fundamental solution of the space-time fractional advection-dispersion equationrdquo Journal ofApplied Mathematics amp Computing vol 18 no 1-2 pp 339ndash3502005

[10] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005

[11] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

[12] L Debnath and D D Bhatta ldquoSolutions to few linear fractionalinhomogeneous partial differential equations in fluid mechan-icsrdquo Fractional Calculus amp Applied Analysis vol 7 no 1 pp 21ndash36 2004

[13] VDaftardar-Gejji andH Jafari ldquoAn iterativemethod for solvingnonlinear functional equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 316 no 2 pp 753ndash763 2006

[14] A A Hemeda ldquoNew iterative method application to the nth-order integro-differential equationsrdquo Information B vol 16 no6 pp 3841ndash3852 2013

[15] S Bhalekar and V Daftardar-Gejji ldquoNew iterative methodapplication to partial differential equationsrdquoAppliedMathemat-ics and Computation vol 203 no 2 pp 778ndash783 2008

[16] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer 1994

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

[18] A A Hemeda ldquoVariational iteration method for solving non-linear coupled equations in 2-dimensional space in fluid mech-anicsrdquo International Journal of Contemporary MathematicalSciences vol 7 no 37 pp 1839ndash1852 2012

[19] G Adomian ldquoA review of the decompositionmethod in appliedmathematicsrdquo Journal of Mathematical Analysis and Applica-tions vol 135 no 2 pp 501ndash544 1988

Abstract and Applied Analysis 9

[20] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008

[21] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[22] J-H He ldquoVariational principle for nano thin film lubricationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 4 no 3 pp 313ndash314 2003

[23] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[24] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[25] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[26] A A Hemeda ldquoFormulation and solution of nth-order deriva-tive fuzzy integro-differential equation using new iterativemethodwith a reliable algorithmrdquo Journal of AppliedMathemat-ics vol 2012 Article ID 325473 17 pages 2012

[27] S Bhalekar andVDaftardar-Gejji ldquoSolving evolution equationsusing a new iterative methodrdquo Numerical Methods for PartialDifferential Equations vol 26 no 4 pp 906ndash916 2010

[28] A A Hemeda ldquoVariational iteration method for solving waveequationrdquo Computers ampMathematics with Applications vol 56no 8 pp 1948ndash1953 2008

[29] A A Hemeda ldquoVariational iteration method for solving non-linear partial differential equationsrdquo Chaos Solitons and Frac-tals vol 39 no 3 pp 1297ndash1303 2009

[30] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[31] A A Hemeda ldquoHomotopy perturbation method for solvingpartial differential equations of fractional orderrdquo InternationalJournal of Mathematical Analysis vol 6 no 49ndash52 pp 2431ndash2448 2012

[32] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency indepedentmdashpart IIrdquoGeophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[33] A Y Luchko and R Groreo The Initial Value Problem forSome Fractional Differential Equations with Caputo DerivativePreprint series A80-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin 1998

[34] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[35] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[36] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] A J Jerri Introduction to Integral Equations with ApplicationsWiley-Interscience 2nd edition 1999

[39] A A Hemeda ldquoNew iterative method an application forsolving fractional physical differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 617010 9 pages 2013

[40] Z Odibat and S Momani ldquoThe variational iteration methodan efficient scheme for handling fractional partial differentialequations in fluid mechanicsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2199ndash2208 2009

[41] G-CWu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013

[42] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 pp 59ndash75 2011

[43] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

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Differential EquationsInternational Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

4 Numerical Examples

41 Linear Problems To incorporate the above discussionthree linear fractional partial differential equations will bestudiedTheNIM is used to obtain the exact solution of theseproblems

Example 1 Consider the following one-dimensional linearinhomogeneous fractional wave equation

119863120572

119909119906 + 119906119909 =

1199051+120572

Γ (2 minus 120572)

sin119909 + 119905 cos119909

119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(21a)

subject to the initial condition

119906 (119909 0) = 0 (21b)

Problem (21a) and (21b) is solved in [40] by using theADM the first few components of solution are as follows

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199061 (119909 119905) = minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199062 (119909 119905) = minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909

(22)

The solution for (21a) and (21b) in series form is given by

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909 minus 1199051+120572

Γ (2 + 120572)

cos119909

+

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+3120572

Γ (2 + 3120572)

cos119909 + sdot sdot sdot

(23)

Canceling the noise terms and keeping the nonnoise termsin (23) yield the exact solution of (21a) and (21b) given by119906(119909 119905) = 119905 sin119909 which is easily verified

Also problem (21a) and (21b) is solved in [40] by usingthe VIM By beginning with 1199060 = 0 the following approxi-mations can be obtained

1199061 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199062 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199063 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909 minus 1199051+120572

Γ (2 + 120572)

cos119909

+

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+3120572

Γ (2 + 3120572)

cos119909

(24)

As in (23) canceling the noise terms and keeping the non-noise terms yield the exact solution of (21a) and (21b)

According to the NIM and by (15) and (16) we obtain

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909 (25)

Therefore the initial value problem (21a) and (21b) is equiva-lent to the integral equation

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909 minus 119868120572119905[119906119909]

(26)

Let119873(119906) = minus119868120572119905[119906119909] we can obtain the following first few

components of the new iterative solution for (21a) and (21b)

1199060 (119909 119905) = 119905 sin119909 +1199051+120572

Γ (2 + 120572)

cos119909

1199061 (119909 119905) = minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

1199062 (119909 119905) = minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909

(27)

The solution in series form is given by

119906 (119909 119905) = 119905 sin119909 + 1199051+120572

Γ (2 + 120572)

cos119909

minus

1199051+120572

Γ (2 + 120572)

cos119909 + 1199051+2120572

Γ (2 + 2120572)

sin119909

minus

1199051+2120572

Γ (2 + 2120572)

sin119909 minus 1199051+3120572

Γ (2 + 3120572)

cos119909 + sdot sdot sdot

(28)

Canceling the noise terms and keeping the non-noise termsin (28) yield the exact solution of (21a) and (21b)

From (23) (24) and (28) it is clear that the threemethodsare the same in solving (21a) and (21b)

Example 2 Consider the following one-dimensional linearinhomogeneous fractional Burgers equation

119863120572

119905119906 + 119906119909 minus 119906119909119909

=

21199052minus120572

Γ (3 minus 120572)

+ 2119909 minus 2 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(29a)

subject to the initial condition

119906 (119909 0) = 1199092 (29b)

Abstract and Applied Analysis 5

Problem (29a) and (29b) is solved in [40] by using theADM the first few components of solution are as follows

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(30)

Therefore the exact solution is given by 119906(119909 119905) = 1199092 + 1199052

Also problem (29a) and (29b) is solved in [40] by usingthe VIM with 1199060 = 119909

2 the following approximations can beobtained

1199060 (119909 119905) = sdot sdot sdot = 119906119899 (119909 119905) = 1199092+ 1199052 (31)

The exact solution 119906(119909 119905) = 1199092 + 1199052 follows immediatelyAccording to the NIM by (15) and (16) we obtain

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) (32)

Therefore the initial value problem (29a) and (29b) is equiv-alent to the integral equation

119906 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus 119868120572

119905[119906119909 minus 119906119909119909]

(33)

Let 119873(119906) = minus119868120572119905[119906119909 minus 119906119909119909] we obtain the following first few

components of the new iterative solution for (29a) and (29b)

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(34)

Therefore the exact solution 119906(119909 119905) = 1199092 + 1199052 follows imme-diately

Example 3 Consider the following one-dimensional linearinhomogeneous fractional Klein-Gordon equation

119863120572

119905119906 minus 119906119909119909 + 119906

= 61199093119905 + (119909

3minus 6119909) 119905

3 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(35a)

subject to the initial conditions

119906 (119909 0) = 0 119906119905 (119909 0) = 0 (35b)

In view of the ADM [40] the first few components ofsolution for (35a) and (35b) are derived as follows

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(36)

The solution in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(37)

In view of the VIM [40] with 1199060 = 0 the followingapproximations for (35a) and (35b) are obtained

1199061 (119909 119905) = (120572 minus 1) [61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

1199062 (119909 119905)

= 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

minus (120572 minus 1)2

times [6 (1199093minus 6119909)

1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

+ sdot sdot sdot

(38)

According to the NIM by (15) and (16) we can obtain

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

(39)

Therefore the initial value problem (35a) and (35b) is equiv-alent to the integral equation

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 119868120572

119905[119906119909119909 minus 119906]

(40)

6 Abstract and Applied Analysis

Let119873(119906) = 119868120572119905[119906119909119909 minus 119906] we can obtain the following first few

components of the new iterative solution for (35a) and (35b)

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(41)

The solution for (35a) and (35b) in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(42)

From (37) (38) and (42) the ADM the VIM and theNIM give the same solution for the classical Klein-Gordonequation (35a) and (35b) in the case 120572 = 2 which is given by

119906 (119909 119905) = 11990931199053+ (1199093minus 6119909)

61199055

Γ (6)

+ 36119909

1199055

Γ (6)

minus 36119909

61199057

Γ (8)

minus 61199093 1199055

Γ (6)

minus (1199093minus 6119909)

61199057

Γ (8)

+ sdot sdot sdot

(43)

Canceling the noise terms and keeping the non-noise terms in(43) yield the exact solution of (35a) and (35b) for the specialcase 120572 = 2 which is given by 119906(119909 119905) = 11990931199053 which is easilyverified

42 Nonlinear Problems For nonlinear equations in generalthere exists no method that yields the exact solution andtherefore only approximate solutions can be derived In thissubsection we use the NIM to provide approximate solutionsfor two kinds of nonlinear time-fractional partial differentialequations

Example 4 Consider the following one-dimensional nonlin-ear homogeneous time-fractional KdV equation

119863120572

119905119906 + 6119906119906119909 + 119906119909119909119909 = 0 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(44a)

subject to the initial condition

119906 (119909 0) =

1

2

sec ℎ2 (1199092

) (44b)

The time-fractional KdV equation (44a) and (44b) issolved in [11 40] by using the ADM The solution in seriesform is found as

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(45)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

)

1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(46)

Also problem (44a) and (44b) is solved in [40] by usingtheVIMwith1199060 = (12)sec ℎ

2(1199092) the first few approximate

solutions are1199061 (119909 119905) = 1198910 (119909)

1199062 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

1199063 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

minus 61198911 (119909) 1198911015840

1(119909)

Γ (1 + 2120572) 1199053120572

Γ(1 + 120572)2Γ (1 + 3120572)

(47)

According to the NIM by (15) and (9) we can obtain

1199060 (119909 119905) =

1

2

sec ℎ2 (1199092

) (48)

Therefore the initial value problem (44a) and (44b) is equiv-alent to the integral equation

119906 (119909 119905) =

1

2

sec ℎ2 (1199092

) minus 119868120572

119905[6119906119906119909 + 119906119909119909119909]

(49)

Let 119873(119906) = minus119868120572119905[6119906119906119909 + 119906119909119909119909] we obtain the following first

few components of the new iterative solution for (44a) and(44b)

1199060 (119909 119905) = 1198910 (119909)

1199061 (119909 119905) = 1198911 (119909)

119905120572

Γ (1 + 120572)

1199062 (119909 119905) = 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

(50)

Abstract and Applied Analysis 7

The solution in series form is given by

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(51)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

) 1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(52)

Now the approximate solution for (44a) and (44b)obtained by the NIM in (51) is the same solution obtained byboth the ADM in (45) and the VIM in (47) Therefore all thethree methods provide the same approximate solution for thetime-fractional KdV equation

Example 5 In this example we consider the one-dimensionalnonlinear homogeneous time-fractional Boussinesq-likeequation

119863120572

119905119906 + (119906

2)119909119909minus (1199062)119909119909119909119909

= 0 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(53a)

subject to the initial conditions

119906 (119909 0) =

4

3

sinh2 (1199094

) 119906119905 (119909 0) = minus

1

3

sinh(1199092

)

(53b)

The time-fractional Boussinesq-like equation (53a) and(53b) is solved by the ADM in [40] The fourth-order termapproximate solution in series form is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(54)

Also problem (53a) and (53b) is solved by the VIM in[40] The fourth-order term approximate solution with 1199060 =(43)sinh2(1199094) minus (13) sinh(1199092) sdot 119905 is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+ (120572 minus 1) [

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

]

+ (120572 minus 1)2[

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

]

+ (120572 minus 1)3[

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

]

(55)

According to the NIM by (15) and (9) we can obtain thefollowing first approximations

1199060 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

1199061 (119909 119905) =

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

1199062 (119909 119905) =

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

1199063 (119909 119905) =

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(56)

The fourth-order term approximate solution in series form isgiven by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

8 Abstract and Applied Analysis

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(57)

where

119873(119906) = minus119868120572

119905[(1199062)119909119909minus (1199062)119909119909119909119909] (58)

It is interesting to point out that for the case of 120572 = 2 theapproximate solution

119906 (119909 119905) =

2

3

[cosh (1199092

)(1 +

1

22

1199052

2

+

1

24

1199054

4

+ sdot sdot sdot ) minus 1]

minus

2

3

sinh(1199092

)[

1

2

119905 +

1

23

1199053

3

+

1

25

1199055

5

+ sdot sdot sdot ]

(59)

follows immediately upon replacing 120572 by 2 in the decompo-sition solution (54) the variational iteration solution (55) orthe new iteration solution (57) which converges to the exactsolution of the Boussinesq-like equation (53a) and (53b)when 120572 = 2 119906(119909 119905) = (43)sinh2((119909 minus 119905)4)

The advantage of the NIM in the nonlinear problemsExamples 4 and 5 is that there is no need to calculate Ado-mianrsquos polynomials as done in the ADM which means thatthe first method is simple and easy of the procedure of calcu-lations over the second Also for more details about the solu-tions of the fractional differential equations by the VIM andsome new asymptotic methods you can see [41ndash43]

5 Conclusion

NIM has been known as a powerful tool for solving manyfunctional equations such as ordinary partial differentialequations integral equations integrodifferential equationsand somany other equations In this work we have presenteda general framework of the NIM for the analytical treatmentof fractional partial differential equations in fluid mechanicsThe present work shows the validity and great potential of theNIM for solving linear and nonlinear fractional partial differ-ential equations All of the examples show that the results ofthe NIM are in excellent agreement with those obtained byboth the ADM and the VIM without calculating Adomianrsquospolynomials for the nonlinear problems as in the ADM Thebasic idea described in this work is expected to be furtheremployed to solve other similar linear and nonlinear prob-lems in fractional calculus

Conflict of Interests

The author declared that there is no conflict of interestsregarding the publication of this paper

References

[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsapplicationsrdquo in Proceedings of the International Conference onVibrating Engineering 98 Dalian China 1988

[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 no 2 pp 86ndash90 1999

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[5] K Al-Khaled and S Momani ldquoAn approximate solution fora fractional diffusion-wave equation using the decompositionmethodrdquoAppliedMathematics and Computation vol 165 no 2pp 473ndash483 2005

[6] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[7] A Hanyga ldquoMultidimensional solutions of time-fractional dif-fusion-wave equationsrdquoTheRoyal Society of London A vol 458no 2020 pp 933ndash957 2002

[8] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquo The ANZIAM Journal vol46 no 3 pp 317ndash330 2005

[9] F Huang and F Liu ldquoThe fundamental solution of the space-time fractional advection-dispersion equationrdquo Journal ofApplied Mathematics amp Computing vol 18 no 1-2 pp 339ndash3502005

[10] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005

[11] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

[12] L Debnath and D D Bhatta ldquoSolutions to few linear fractionalinhomogeneous partial differential equations in fluid mechan-icsrdquo Fractional Calculus amp Applied Analysis vol 7 no 1 pp 21ndash36 2004

[13] VDaftardar-Gejji andH Jafari ldquoAn iterativemethod for solvingnonlinear functional equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 316 no 2 pp 753ndash763 2006

[14] A A Hemeda ldquoNew iterative method application to the nth-order integro-differential equationsrdquo Information B vol 16 no6 pp 3841ndash3852 2013

[15] S Bhalekar and V Daftardar-Gejji ldquoNew iterative methodapplication to partial differential equationsrdquoAppliedMathemat-ics and Computation vol 203 no 2 pp 778ndash783 2008

[16] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer 1994

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

[18] A A Hemeda ldquoVariational iteration method for solving non-linear coupled equations in 2-dimensional space in fluid mech-anicsrdquo International Journal of Contemporary MathematicalSciences vol 7 no 37 pp 1839ndash1852 2012

[19] G Adomian ldquoA review of the decompositionmethod in appliedmathematicsrdquo Journal of Mathematical Analysis and Applica-tions vol 135 no 2 pp 501ndash544 1988

Abstract and Applied Analysis 9

[20] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008

[21] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[22] J-H He ldquoVariational principle for nano thin film lubricationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 4 no 3 pp 313ndash314 2003

[23] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[24] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[25] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[26] A A Hemeda ldquoFormulation and solution of nth-order deriva-tive fuzzy integro-differential equation using new iterativemethodwith a reliable algorithmrdquo Journal of AppliedMathemat-ics vol 2012 Article ID 325473 17 pages 2012

[27] S Bhalekar andVDaftardar-Gejji ldquoSolving evolution equationsusing a new iterative methodrdquo Numerical Methods for PartialDifferential Equations vol 26 no 4 pp 906ndash916 2010

[28] A A Hemeda ldquoVariational iteration method for solving waveequationrdquo Computers ampMathematics with Applications vol 56no 8 pp 1948ndash1953 2008

[29] A A Hemeda ldquoVariational iteration method for solving non-linear partial differential equationsrdquo Chaos Solitons and Frac-tals vol 39 no 3 pp 1297ndash1303 2009

[30] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[31] A A Hemeda ldquoHomotopy perturbation method for solvingpartial differential equations of fractional orderrdquo InternationalJournal of Mathematical Analysis vol 6 no 49ndash52 pp 2431ndash2448 2012

[32] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency indepedentmdashpart IIrdquoGeophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[33] A Y Luchko and R Groreo The Initial Value Problem forSome Fractional Differential Equations with Caputo DerivativePreprint series A80-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin 1998

[34] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[35] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[36] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] A J Jerri Introduction to Integral Equations with ApplicationsWiley-Interscience 2nd edition 1999

[39] A A Hemeda ldquoNew iterative method an application forsolving fractional physical differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 617010 9 pages 2013

[40] Z Odibat and S Momani ldquoThe variational iteration methodan efficient scheme for handling fractional partial differentialequations in fluid mechanicsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2199ndash2208 2009

[41] G-CWu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013

[42] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 pp 59ndash75 2011

[43] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

Problem (29a) and (29b) is solved in [40] by using theADM the first few components of solution are as follows

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(30)

Therefore the exact solution is given by 119906(119909 119905) = 1199092 + 1199052

Also problem (29a) and (29b) is solved in [40] by usingthe VIM with 1199060 = 119909

2 the following approximations can beobtained

1199060 (119909 119905) = sdot sdot sdot = 119906119899 (119909 119905) = 1199092+ 1199052 (31)

The exact solution 119906(119909 119905) = 1199092 + 1199052 follows immediatelyAccording to the NIM by (15) and (16) we obtain

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) (32)

Therefore the initial value problem (29a) and (29b) is equiv-alent to the integral equation

119906 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus 119868120572

119905[119906119909 minus 119906119909119909]

(33)

Let 119873(119906) = minus119868120572119905[119906119909 minus 119906119909119909] we obtain the following first few

components of the new iterative solution for (29a) and (29b)

1199060 (119909 119905) = 1199092+ 1199052+

119905120572

Γ (1 + 120572)

(2119909 minus 2)

1199061 (119909 119905) = minus

119905120572

Γ (1 + 120572)

(2119909 minus 2) minus

21199052120572

Γ (1 + 2120572)

1199062 (119909 119905) =

21199052120572

Γ (1 + 2120572)

1199063 (119909 119905) = 0

(34)

Therefore the exact solution 119906(119909 119905) = 1199092 + 1199052 follows imme-diately

Example 3 Consider the following one-dimensional linearinhomogeneous fractional Klein-Gordon equation

119863120572

119905119906 minus 119906119909119909 + 119906

= 61199093119905 + (119909

3minus 6119909) 119905

3 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(35a)

subject to the initial conditions

119906 (119909 0) = 0 119906119905 (119909 0) = 0 (35b)

In view of the ADM [40] the first few components ofsolution for (35a) and (35b) are derived as follows

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(36)

The solution in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(37)

In view of the VIM [40] with 1199060 = 0 the followingapproximations for (35a) and (35b) are obtained

1199061 (119909 119905) = (120572 minus 1) [61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

1199062 (119909 119905)

= 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

minus (120572 minus 1)2

times [6 (1199093minus 6119909)

1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

]

+ sdot sdot sdot

(38)

According to the NIM by (15) and (16) we can obtain

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

(39)

Therefore the initial value problem (35a) and (35b) is equiv-alent to the integral equation

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 119868120572

119905[119906119909119909 minus 119906]

(40)

6 Abstract and Applied Analysis

Let119873(119906) = 119868120572119905[119906119909119909 minus 119906] we can obtain the following first few

components of the new iterative solution for (35a) and (35b)

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(41)

The solution for (35a) and (35b) in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(42)

From (37) (38) and (42) the ADM the VIM and theNIM give the same solution for the classical Klein-Gordonequation (35a) and (35b) in the case 120572 = 2 which is given by

119906 (119909 119905) = 11990931199053+ (1199093minus 6119909)

61199055

Γ (6)

+ 36119909

1199055

Γ (6)

minus 36119909

61199057

Γ (8)

minus 61199093 1199055

Γ (6)

minus (1199093minus 6119909)

61199057

Γ (8)

+ sdot sdot sdot

(43)

Canceling the noise terms and keeping the non-noise terms in(43) yield the exact solution of (35a) and (35b) for the specialcase 120572 = 2 which is given by 119906(119909 119905) = 11990931199053 which is easilyverified

42 Nonlinear Problems For nonlinear equations in generalthere exists no method that yields the exact solution andtherefore only approximate solutions can be derived In thissubsection we use the NIM to provide approximate solutionsfor two kinds of nonlinear time-fractional partial differentialequations

Example 4 Consider the following one-dimensional nonlin-ear homogeneous time-fractional KdV equation

119863120572

119905119906 + 6119906119906119909 + 119906119909119909119909 = 0 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(44a)

subject to the initial condition

119906 (119909 0) =

1

2

sec ℎ2 (1199092

) (44b)

The time-fractional KdV equation (44a) and (44b) issolved in [11 40] by using the ADM The solution in seriesform is found as

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(45)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

)

1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(46)

Also problem (44a) and (44b) is solved in [40] by usingtheVIMwith1199060 = (12)sec ℎ

2(1199092) the first few approximate

solutions are1199061 (119909 119905) = 1198910 (119909)

1199062 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

1199063 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

minus 61198911 (119909) 1198911015840

1(119909)

Γ (1 + 2120572) 1199053120572

Γ(1 + 120572)2Γ (1 + 3120572)

(47)

According to the NIM by (15) and (9) we can obtain

1199060 (119909 119905) =

1

2

sec ℎ2 (1199092

) (48)

Therefore the initial value problem (44a) and (44b) is equiv-alent to the integral equation

119906 (119909 119905) =

1

2

sec ℎ2 (1199092

) minus 119868120572

119905[6119906119906119909 + 119906119909119909119909]

(49)

Let 119873(119906) = minus119868120572119905[6119906119906119909 + 119906119909119909119909] we obtain the following first

few components of the new iterative solution for (44a) and(44b)

1199060 (119909 119905) = 1198910 (119909)

1199061 (119909 119905) = 1198911 (119909)

119905120572

Γ (1 + 120572)

1199062 (119909 119905) = 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

(50)

Abstract and Applied Analysis 7

The solution in series form is given by

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(51)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

) 1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(52)

Now the approximate solution for (44a) and (44b)obtained by the NIM in (51) is the same solution obtained byboth the ADM in (45) and the VIM in (47) Therefore all thethree methods provide the same approximate solution for thetime-fractional KdV equation

Example 5 In this example we consider the one-dimensionalnonlinear homogeneous time-fractional Boussinesq-likeequation

119863120572

119905119906 + (119906

2)119909119909minus (1199062)119909119909119909119909

= 0 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(53a)

subject to the initial conditions

119906 (119909 0) =

4

3

sinh2 (1199094

) 119906119905 (119909 0) = minus

1

3

sinh(1199092

)

(53b)

The time-fractional Boussinesq-like equation (53a) and(53b) is solved by the ADM in [40] The fourth-order termapproximate solution in series form is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(54)

Also problem (53a) and (53b) is solved by the VIM in[40] The fourth-order term approximate solution with 1199060 =(43)sinh2(1199094) minus (13) sinh(1199092) sdot 119905 is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+ (120572 minus 1) [

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

]

+ (120572 minus 1)2[

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

]

+ (120572 minus 1)3[

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

]

(55)

According to the NIM by (15) and (9) we can obtain thefollowing first approximations

1199060 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

1199061 (119909 119905) =

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

1199062 (119909 119905) =

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

1199063 (119909 119905) =

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(56)

The fourth-order term approximate solution in series form isgiven by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

8 Abstract and Applied Analysis

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(57)

where

119873(119906) = minus119868120572

119905[(1199062)119909119909minus (1199062)119909119909119909119909] (58)

It is interesting to point out that for the case of 120572 = 2 theapproximate solution

119906 (119909 119905) =

2

3

[cosh (1199092

)(1 +

1

22

1199052

2

+

1

24

1199054

4

+ sdot sdot sdot ) minus 1]

minus

2

3

sinh(1199092

)[

1

2

119905 +

1

23

1199053

3

+

1

25

1199055

5

+ sdot sdot sdot ]

(59)

follows immediately upon replacing 120572 by 2 in the decompo-sition solution (54) the variational iteration solution (55) orthe new iteration solution (57) which converges to the exactsolution of the Boussinesq-like equation (53a) and (53b)when 120572 = 2 119906(119909 119905) = (43)sinh2((119909 minus 119905)4)

The advantage of the NIM in the nonlinear problemsExamples 4 and 5 is that there is no need to calculate Ado-mianrsquos polynomials as done in the ADM which means thatthe first method is simple and easy of the procedure of calcu-lations over the second Also for more details about the solu-tions of the fractional differential equations by the VIM andsome new asymptotic methods you can see [41ndash43]

5 Conclusion

NIM has been known as a powerful tool for solving manyfunctional equations such as ordinary partial differentialequations integral equations integrodifferential equationsand somany other equations In this work we have presenteda general framework of the NIM for the analytical treatmentof fractional partial differential equations in fluid mechanicsThe present work shows the validity and great potential of theNIM for solving linear and nonlinear fractional partial differ-ential equations All of the examples show that the results ofthe NIM are in excellent agreement with those obtained byboth the ADM and the VIM without calculating Adomianrsquospolynomials for the nonlinear problems as in the ADM Thebasic idea described in this work is expected to be furtheremployed to solve other similar linear and nonlinear prob-lems in fractional calculus

Conflict of Interests

The author declared that there is no conflict of interestsregarding the publication of this paper

References

[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsapplicationsrdquo in Proceedings of the International Conference onVibrating Engineering 98 Dalian China 1988

[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 no 2 pp 86ndash90 1999

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[5] K Al-Khaled and S Momani ldquoAn approximate solution fora fractional diffusion-wave equation using the decompositionmethodrdquoAppliedMathematics and Computation vol 165 no 2pp 473ndash483 2005

[6] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[7] A Hanyga ldquoMultidimensional solutions of time-fractional dif-fusion-wave equationsrdquoTheRoyal Society of London A vol 458no 2020 pp 933ndash957 2002

[8] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquo The ANZIAM Journal vol46 no 3 pp 317ndash330 2005

[9] F Huang and F Liu ldquoThe fundamental solution of the space-time fractional advection-dispersion equationrdquo Journal ofApplied Mathematics amp Computing vol 18 no 1-2 pp 339ndash3502005

[10] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005

[11] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

[12] L Debnath and D D Bhatta ldquoSolutions to few linear fractionalinhomogeneous partial differential equations in fluid mechan-icsrdquo Fractional Calculus amp Applied Analysis vol 7 no 1 pp 21ndash36 2004

[13] VDaftardar-Gejji andH Jafari ldquoAn iterativemethod for solvingnonlinear functional equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 316 no 2 pp 753ndash763 2006

[14] A A Hemeda ldquoNew iterative method application to the nth-order integro-differential equationsrdquo Information B vol 16 no6 pp 3841ndash3852 2013

[15] S Bhalekar and V Daftardar-Gejji ldquoNew iterative methodapplication to partial differential equationsrdquoAppliedMathemat-ics and Computation vol 203 no 2 pp 778ndash783 2008

[16] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer 1994

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

[18] A A Hemeda ldquoVariational iteration method for solving non-linear coupled equations in 2-dimensional space in fluid mech-anicsrdquo International Journal of Contemporary MathematicalSciences vol 7 no 37 pp 1839ndash1852 2012

[19] G Adomian ldquoA review of the decompositionmethod in appliedmathematicsrdquo Journal of Mathematical Analysis and Applica-tions vol 135 no 2 pp 501ndash544 1988

Abstract and Applied Analysis 9

[20] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008

[21] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[22] J-H He ldquoVariational principle for nano thin film lubricationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 4 no 3 pp 313ndash314 2003

[23] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[24] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[25] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[26] A A Hemeda ldquoFormulation and solution of nth-order deriva-tive fuzzy integro-differential equation using new iterativemethodwith a reliable algorithmrdquo Journal of AppliedMathemat-ics vol 2012 Article ID 325473 17 pages 2012

[27] S Bhalekar andVDaftardar-Gejji ldquoSolving evolution equationsusing a new iterative methodrdquo Numerical Methods for PartialDifferential Equations vol 26 no 4 pp 906ndash916 2010

[28] A A Hemeda ldquoVariational iteration method for solving waveequationrdquo Computers ampMathematics with Applications vol 56no 8 pp 1948ndash1953 2008

[29] A A Hemeda ldquoVariational iteration method for solving non-linear partial differential equationsrdquo Chaos Solitons and Frac-tals vol 39 no 3 pp 1297ndash1303 2009

[30] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[31] A A Hemeda ldquoHomotopy perturbation method for solvingpartial differential equations of fractional orderrdquo InternationalJournal of Mathematical Analysis vol 6 no 49ndash52 pp 2431ndash2448 2012

[32] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency indepedentmdashpart IIrdquoGeophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[33] A Y Luchko and R Groreo The Initial Value Problem forSome Fractional Differential Equations with Caputo DerivativePreprint series A80-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin 1998

[34] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[35] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[36] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] A J Jerri Introduction to Integral Equations with ApplicationsWiley-Interscience 2nd edition 1999

[39] A A Hemeda ldquoNew iterative method an application forsolving fractional physical differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 617010 9 pages 2013

[40] Z Odibat and S Momani ldquoThe variational iteration methodan efficient scheme for handling fractional partial differentialequations in fluid mechanicsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2199ndash2208 2009

[41] G-CWu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013

[42] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 pp 59ndash75 2011

[43] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

Let119873(119906) = 119868120572119905[119906119909119909 minus 119906] we can obtain the following first few

components of the new iterative solution for (35a) and (35b)

1199060 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

1199061 (119909 119905) = 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

(41)

The solution for (35a) and (35b) in series form is given by

119906 (119909 119905) = 61199093 1199051+120572

Γ (2 + 120572)

+ 6 (1199093minus 6119909)

1199053+120572

Γ (4 + 120572)

+ 36119909

1199051+2120572

Γ (2 + 2120572)

+ 36119909

1199053+2120572

Γ (4 + 2120572)

minus 61199093 1199051+2120572

Γ (2 + 2120572)

minus 6 (1199093minus 6119909)

1199053+2120572

Γ (4 + 2120572)

+ sdot sdot sdot

(42)

From (37) (38) and (42) the ADM the VIM and theNIM give the same solution for the classical Klein-Gordonequation (35a) and (35b) in the case 120572 = 2 which is given by

119906 (119909 119905) = 11990931199053+ (1199093minus 6119909)

61199055

Γ (6)

+ 36119909

1199055

Γ (6)

minus 36119909

61199057

Γ (8)

minus 61199093 1199055

Γ (6)

minus (1199093minus 6119909)

61199057

Γ (8)

+ sdot sdot sdot

(43)

Canceling the noise terms and keeping the non-noise terms in(43) yield the exact solution of (35a) and (35b) for the specialcase 120572 = 2 which is given by 119906(119909 119905) = 11990931199053 which is easilyverified

42 Nonlinear Problems For nonlinear equations in generalthere exists no method that yields the exact solution andtherefore only approximate solutions can be derived In thissubsection we use the NIM to provide approximate solutionsfor two kinds of nonlinear time-fractional partial differentialequations

Example 4 Consider the following one-dimensional nonlin-ear homogeneous time-fractional KdV equation

119863120572

119905119906 + 6119906119906119909 + 119906119909119909119909 = 0 119905 gt 0 119909 isin 119877 0 lt 120572 le 1

(44a)

subject to the initial condition

119906 (119909 0) =

1

2

sec ℎ2 (1199092

) (44b)

The time-fractional KdV equation (44a) and (44b) issolved in [11 40] by using the ADM The solution in seriesform is found as

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(45)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

)

1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(46)

Also problem (44a) and (44b) is solved in [40] by usingtheVIMwith1199060 = (12)sec ℎ

2(1199092) the first few approximate

solutions are1199061 (119909 119905) = 1198910 (119909)

1199062 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

1199063 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

minus 61198911 (119909) 1198911015840

1(119909)

Γ (1 + 2120572) 1199053120572

Γ(1 + 120572)2Γ (1 + 3120572)

(47)

According to the NIM by (15) and (9) we can obtain

1199060 (119909 119905) =

1

2

sec ℎ2 (1199092

) (48)

Therefore the initial value problem (44a) and (44b) is equiv-alent to the integral equation

119906 (119909 119905) =

1

2

sec ℎ2 (1199092

) minus 119868120572

119905[6119906119906119909 + 119906119909119909119909]

(49)

Let 119873(119906) = minus119868120572119905[6119906119906119909 + 119906119909119909119909] we obtain the following first

few components of the new iterative solution for (44a) and(44b)

1199060 (119909 119905) = 1198910 (119909)

1199061 (119909 119905) = 1198911 (119909)

119905120572

Γ (1 + 120572)

1199062 (119909 119905) = 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

(50)

Abstract and Applied Analysis 7

The solution in series form is given by

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(51)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

) 1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(52)

Now the approximate solution for (44a) and (44b)obtained by the NIM in (51) is the same solution obtained byboth the ADM in (45) and the VIM in (47) Therefore all thethree methods provide the same approximate solution for thetime-fractional KdV equation

Example 5 In this example we consider the one-dimensionalnonlinear homogeneous time-fractional Boussinesq-likeequation

119863120572

119905119906 + (119906

2)119909119909minus (1199062)119909119909119909119909

= 0 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(53a)

subject to the initial conditions

119906 (119909 0) =

4

3

sinh2 (1199094

) 119906119905 (119909 0) = minus

1

3

sinh(1199092

)

(53b)

The time-fractional Boussinesq-like equation (53a) and(53b) is solved by the ADM in [40] The fourth-order termapproximate solution in series form is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(54)

Also problem (53a) and (53b) is solved by the VIM in[40] The fourth-order term approximate solution with 1199060 =(43)sinh2(1199094) minus (13) sinh(1199092) sdot 119905 is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+ (120572 minus 1) [

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

]

+ (120572 minus 1)2[

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

]

+ (120572 minus 1)3[

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

]

(55)

According to the NIM by (15) and (9) we can obtain thefollowing first approximations

1199060 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

1199061 (119909 119905) =

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

1199062 (119909 119905) =

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

1199063 (119909 119905) =

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(56)

The fourth-order term approximate solution in series form isgiven by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

8 Abstract and Applied Analysis

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(57)

where

119873(119906) = minus119868120572

119905[(1199062)119909119909minus (1199062)119909119909119909119909] (58)

It is interesting to point out that for the case of 120572 = 2 theapproximate solution

119906 (119909 119905) =

2

3

[cosh (1199092

)(1 +

1

22

1199052

2

+

1

24

1199054

4

+ sdot sdot sdot ) minus 1]

minus

2

3

sinh(1199092

)[

1

2

119905 +

1

23

1199053

3

+

1

25

1199055

5

+ sdot sdot sdot ]

(59)

follows immediately upon replacing 120572 by 2 in the decompo-sition solution (54) the variational iteration solution (55) orthe new iteration solution (57) which converges to the exactsolution of the Boussinesq-like equation (53a) and (53b)when 120572 = 2 119906(119909 119905) = (43)sinh2((119909 minus 119905)4)

The advantage of the NIM in the nonlinear problemsExamples 4 and 5 is that there is no need to calculate Ado-mianrsquos polynomials as done in the ADM which means thatthe first method is simple and easy of the procedure of calcu-lations over the second Also for more details about the solu-tions of the fractional differential equations by the VIM andsome new asymptotic methods you can see [41ndash43]

5 Conclusion

NIM has been known as a powerful tool for solving manyfunctional equations such as ordinary partial differentialequations integral equations integrodifferential equationsand somany other equations In this work we have presenteda general framework of the NIM for the analytical treatmentof fractional partial differential equations in fluid mechanicsThe present work shows the validity and great potential of theNIM for solving linear and nonlinear fractional partial differ-ential equations All of the examples show that the results ofthe NIM are in excellent agreement with those obtained byboth the ADM and the VIM without calculating Adomianrsquospolynomials for the nonlinear problems as in the ADM Thebasic idea described in this work is expected to be furtheremployed to solve other similar linear and nonlinear prob-lems in fractional calculus

Conflict of Interests

The author declared that there is no conflict of interestsregarding the publication of this paper

References

[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsapplicationsrdquo in Proceedings of the International Conference onVibrating Engineering 98 Dalian China 1988

[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 no 2 pp 86ndash90 1999

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[5] K Al-Khaled and S Momani ldquoAn approximate solution fora fractional diffusion-wave equation using the decompositionmethodrdquoAppliedMathematics and Computation vol 165 no 2pp 473ndash483 2005

[6] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[7] A Hanyga ldquoMultidimensional solutions of time-fractional dif-fusion-wave equationsrdquoTheRoyal Society of London A vol 458no 2020 pp 933ndash957 2002

[8] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquo The ANZIAM Journal vol46 no 3 pp 317ndash330 2005

[9] F Huang and F Liu ldquoThe fundamental solution of the space-time fractional advection-dispersion equationrdquo Journal ofApplied Mathematics amp Computing vol 18 no 1-2 pp 339ndash3502005

[10] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005

[11] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

[12] L Debnath and D D Bhatta ldquoSolutions to few linear fractionalinhomogeneous partial differential equations in fluid mechan-icsrdquo Fractional Calculus amp Applied Analysis vol 7 no 1 pp 21ndash36 2004

[13] VDaftardar-Gejji andH Jafari ldquoAn iterativemethod for solvingnonlinear functional equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 316 no 2 pp 753ndash763 2006

[14] A A Hemeda ldquoNew iterative method application to the nth-order integro-differential equationsrdquo Information B vol 16 no6 pp 3841ndash3852 2013

[15] S Bhalekar and V Daftardar-Gejji ldquoNew iterative methodapplication to partial differential equationsrdquoAppliedMathemat-ics and Computation vol 203 no 2 pp 778ndash783 2008

[16] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer 1994

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

[18] A A Hemeda ldquoVariational iteration method for solving non-linear coupled equations in 2-dimensional space in fluid mech-anicsrdquo International Journal of Contemporary MathematicalSciences vol 7 no 37 pp 1839ndash1852 2012

[19] G Adomian ldquoA review of the decompositionmethod in appliedmathematicsrdquo Journal of Mathematical Analysis and Applica-tions vol 135 no 2 pp 501ndash544 1988

Abstract and Applied Analysis 9

[20] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008

[21] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[22] J-H He ldquoVariational principle for nano thin film lubricationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 4 no 3 pp 313ndash314 2003

[23] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[24] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[25] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[26] A A Hemeda ldquoFormulation and solution of nth-order deriva-tive fuzzy integro-differential equation using new iterativemethodwith a reliable algorithmrdquo Journal of AppliedMathemat-ics vol 2012 Article ID 325473 17 pages 2012

[27] S Bhalekar andVDaftardar-Gejji ldquoSolving evolution equationsusing a new iterative methodrdquo Numerical Methods for PartialDifferential Equations vol 26 no 4 pp 906ndash916 2010

[28] A A Hemeda ldquoVariational iteration method for solving waveequationrdquo Computers ampMathematics with Applications vol 56no 8 pp 1948ndash1953 2008

[29] A A Hemeda ldquoVariational iteration method for solving non-linear partial differential equationsrdquo Chaos Solitons and Frac-tals vol 39 no 3 pp 1297ndash1303 2009

[30] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[31] A A Hemeda ldquoHomotopy perturbation method for solvingpartial differential equations of fractional orderrdquo InternationalJournal of Mathematical Analysis vol 6 no 49ndash52 pp 2431ndash2448 2012

[32] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency indepedentmdashpart IIrdquoGeophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[33] A Y Luchko and R Groreo The Initial Value Problem forSome Fractional Differential Equations with Caputo DerivativePreprint series A80-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin 1998

[34] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[35] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[36] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] A J Jerri Introduction to Integral Equations with ApplicationsWiley-Interscience 2nd edition 1999

[39] A A Hemeda ldquoNew iterative method an application forsolving fractional physical differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 617010 9 pages 2013

[40] Z Odibat and S Momani ldquoThe variational iteration methodan efficient scheme for handling fractional partial differentialequations in fluid mechanicsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2199ndash2208 2009

[41] G-CWu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013

[42] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 pp 59ndash75 2011

[43] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

The solution in series form is given by

119906 (119909 119905) = 1198910 (119909) + 1198911 (119909)

119905120572

Γ (1 + 120572)

+ 1198912 (119909)

1199052120572

Γ (1 + 2120572)

+ 1198913 (119909)

1199053120572

Γ (1 + 3120572)

+ sdot sdot sdot

(51)

where

1198910 (119909) =

1

2

sec ℎ2 (1199092

) 1198911 (119909) = minus611989101198911015840

0minus 119891101584010158401015840

0

1198912 (119909) = minus611989101198911015840

1minus 61198911119891

1015840

0minus 119891101584010158401015840

1

1198913 (119909) = minus611989111198911015840

1

Γ (1 + 2120572)

Γ(1 + 120572)2

(52)

Now the approximate solution for (44a) and (44b)obtained by the NIM in (51) is the same solution obtained byboth the ADM in (45) and the VIM in (47) Therefore all thethree methods provide the same approximate solution for thetime-fractional KdV equation

Example 5 In this example we consider the one-dimensionalnonlinear homogeneous time-fractional Boussinesq-likeequation

119863120572

119905119906 + (119906

2)119909119909minus (1199062)119909119909119909119909

= 0 119905 gt 0 119909 isin 119877 1 lt 120572 le 2

(53a)

subject to the initial conditions

119906 (119909 0) =

4

3

sinh2 (1199094

) 119906119905 (119909 0) = minus

1

3

sinh(1199092

)

(53b)

The time-fractional Boussinesq-like equation (53a) and(53b) is solved by the ADM in [40] The fourth-order termapproximate solution in series form is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(54)

Also problem (53a) and (53b) is solved by the VIM in[40] The fourth-order term approximate solution with 1199060 =(43)sinh2(1199094) minus (13) sinh(1199092) sdot 119905 is given by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+ (120572 minus 1) [

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

]

+ (120572 minus 1)2[

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

]

+ (120572 minus 1)3[

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

]

(55)

According to the NIM by (15) and (9) we can obtain thefollowing first approximations

1199060 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

1199061 (119909 119905) =

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

1199062 (119909 119905) =

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

minus

1

24(3)

sinh(1199092

)

1199051+2120572

Γ (2 + 2120572)

1199063 (119909 119905) =

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(56)

The fourth-order term approximate solution in series form isgiven by

119906 (119909 119905) =

4

3

sinh2 (1199094

) minus

1

3

sinh(1199092

) sdot 119905

+

1

2 (3)

cosh (1199092

)

119905120572

Γ (1 + 120572)

minus

1

22(3)

sinh(1199092

)

1199051+120572

Γ (2 + 120572)

+

1

23(3)

cosh (1199092

)

1199052120572

Γ (1 + 2120572)

8 Abstract and Applied Analysis

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(57)

where

119873(119906) = minus119868120572

119905[(1199062)119909119909minus (1199062)119909119909119909119909] (58)

It is interesting to point out that for the case of 120572 = 2 theapproximate solution

119906 (119909 119905) =

2

3

[cosh (1199092

)(1 +

1

22

1199052

2

+

1

24

1199054

4

+ sdot sdot sdot ) minus 1]

minus

2

3

sinh(1199092

)[

1

2

119905 +

1

23

1199053

3

+

1

25

1199055

5

+ sdot sdot sdot ]

(59)

follows immediately upon replacing 120572 by 2 in the decompo-sition solution (54) the variational iteration solution (55) orthe new iteration solution (57) which converges to the exactsolution of the Boussinesq-like equation (53a) and (53b)when 120572 = 2 119906(119909 119905) = (43)sinh2((119909 minus 119905)4)

The advantage of the NIM in the nonlinear problemsExamples 4 and 5 is that there is no need to calculate Ado-mianrsquos polynomials as done in the ADM which means thatthe first method is simple and easy of the procedure of calcu-lations over the second Also for more details about the solu-tions of the fractional differential equations by the VIM andsome new asymptotic methods you can see [41ndash43]

5 Conclusion

NIM has been known as a powerful tool for solving manyfunctional equations such as ordinary partial differentialequations integral equations integrodifferential equationsand somany other equations In this work we have presenteda general framework of the NIM for the analytical treatmentof fractional partial differential equations in fluid mechanicsThe present work shows the validity and great potential of theNIM for solving linear and nonlinear fractional partial differ-ential equations All of the examples show that the results ofthe NIM are in excellent agreement with those obtained byboth the ADM and the VIM without calculating Adomianrsquospolynomials for the nonlinear problems as in the ADM Thebasic idea described in this work is expected to be furtheremployed to solve other similar linear and nonlinear prob-lems in fractional calculus

Conflict of Interests

The author declared that there is no conflict of interestsregarding the publication of this paper

References

[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsapplicationsrdquo in Proceedings of the International Conference onVibrating Engineering 98 Dalian China 1988

[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 no 2 pp 86ndash90 1999

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[5] K Al-Khaled and S Momani ldquoAn approximate solution fora fractional diffusion-wave equation using the decompositionmethodrdquoAppliedMathematics and Computation vol 165 no 2pp 473ndash483 2005

[6] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[7] A Hanyga ldquoMultidimensional solutions of time-fractional dif-fusion-wave equationsrdquoTheRoyal Society of London A vol 458no 2020 pp 933ndash957 2002

[8] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquo The ANZIAM Journal vol46 no 3 pp 317ndash330 2005

[9] F Huang and F Liu ldquoThe fundamental solution of the space-time fractional advection-dispersion equationrdquo Journal ofApplied Mathematics amp Computing vol 18 no 1-2 pp 339ndash3502005

[10] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005

[11] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

[12] L Debnath and D D Bhatta ldquoSolutions to few linear fractionalinhomogeneous partial differential equations in fluid mechan-icsrdquo Fractional Calculus amp Applied Analysis vol 7 no 1 pp 21ndash36 2004

[13] VDaftardar-Gejji andH Jafari ldquoAn iterativemethod for solvingnonlinear functional equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 316 no 2 pp 753ndash763 2006

[14] A A Hemeda ldquoNew iterative method application to the nth-order integro-differential equationsrdquo Information B vol 16 no6 pp 3841ndash3852 2013

[15] S Bhalekar and V Daftardar-Gejji ldquoNew iterative methodapplication to partial differential equationsrdquoAppliedMathemat-ics and Computation vol 203 no 2 pp 778ndash783 2008

[16] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer 1994

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

[18] A A Hemeda ldquoVariational iteration method for solving non-linear coupled equations in 2-dimensional space in fluid mech-anicsrdquo International Journal of Contemporary MathematicalSciences vol 7 no 37 pp 1839ndash1852 2012

[19] G Adomian ldquoA review of the decompositionmethod in appliedmathematicsrdquo Journal of Mathematical Analysis and Applica-tions vol 135 no 2 pp 501ndash544 1988

Abstract and Applied Analysis 9

[20] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008

[21] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[22] J-H He ldquoVariational principle for nano thin film lubricationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 4 no 3 pp 313ndash314 2003

[23] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[24] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[25] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[26] A A Hemeda ldquoFormulation and solution of nth-order deriva-tive fuzzy integro-differential equation using new iterativemethodwith a reliable algorithmrdquo Journal of AppliedMathemat-ics vol 2012 Article ID 325473 17 pages 2012

[27] S Bhalekar andVDaftardar-Gejji ldquoSolving evolution equationsusing a new iterative methodrdquo Numerical Methods for PartialDifferential Equations vol 26 no 4 pp 906ndash916 2010

[28] A A Hemeda ldquoVariational iteration method for solving waveequationrdquo Computers ampMathematics with Applications vol 56no 8 pp 1948ndash1953 2008

[29] A A Hemeda ldquoVariational iteration method for solving non-linear partial differential equationsrdquo Chaos Solitons and Frac-tals vol 39 no 3 pp 1297ndash1303 2009

[30] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[31] A A Hemeda ldquoHomotopy perturbation method for solvingpartial differential equations of fractional orderrdquo InternationalJournal of Mathematical Analysis vol 6 no 49ndash52 pp 2431ndash2448 2012

[32] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency indepedentmdashpart IIrdquoGeophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[33] A Y Luchko and R Groreo The Initial Value Problem forSome Fractional Differential Equations with Caputo DerivativePreprint series A80-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin 1998

[34] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[35] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[36] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] A J Jerri Introduction to Integral Equations with ApplicationsWiley-Interscience 2nd edition 1999

[39] A A Hemeda ldquoNew iterative method an application forsolving fractional physical differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 617010 9 pages 2013

[40] Z Odibat and S Momani ldquoThe variational iteration methodan efficient scheme for handling fractional partial differentialequations in fluid mechanicsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2199ndash2208 2009

[41] G-CWu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013

[42] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 pp 59ndash75 2011

[43] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

minus

1

24(3)

sinh(1199092

) sdot

1199051+2120572

Γ (2 + 2120572)

+

1

25(3)

cosh (1199092

)

1199053120572

Γ (1 + 3120572)

minus

1

26(3)

sinh(1199092

)

1199051+3120572

Γ (2 + 3120572)

(57)

where

119873(119906) = minus119868120572

119905[(1199062)119909119909minus (1199062)119909119909119909119909] (58)

It is interesting to point out that for the case of 120572 = 2 theapproximate solution

119906 (119909 119905) =

2

3

[cosh (1199092

)(1 +

1

22

1199052

2

+

1

24

1199054

4

+ sdot sdot sdot ) minus 1]

minus

2

3

sinh(1199092

)[

1

2

119905 +

1

23

1199053

3

+

1

25

1199055

5

+ sdot sdot sdot ]

(59)

follows immediately upon replacing 120572 by 2 in the decompo-sition solution (54) the variational iteration solution (55) orthe new iteration solution (57) which converges to the exactsolution of the Boussinesq-like equation (53a) and (53b)when 120572 = 2 119906(119909 119905) = (43)sinh2((119909 minus 119905)4)

The advantage of the NIM in the nonlinear problemsExamples 4 and 5 is that there is no need to calculate Ado-mianrsquos polynomials as done in the ADM which means thatthe first method is simple and easy of the procedure of calcu-lations over the second Also for more details about the solu-tions of the fractional differential equations by the VIM andsome new asymptotic methods you can see [41ndash43]

5 Conclusion

NIM has been known as a powerful tool for solving manyfunctional equations such as ordinary partial differentialequations integral equations integrodifferential equationsand somany other equations In this work we have presenteda general framework of the NIM for the analytical treatmentof fractional partial differential equations in fluid mechanicsThe present work shows the validity and great potential of theNIM for solving linear and nonlinear fractional partial differ-ential equations All of the examples show that the results ofthe NIM are in excellent agreement with those obtained byboth the ADM and the VIM without calculating Adomianrsquospolynomials for the nonlinear problems as in the ADM Thebasic idea described in this work is expected to be furtheremployed to solve other similar linear and nonlinear prob-lems in fractional calculus

Conflict of Interests

The author declared that there is no conflict of interestsregarding the publication of this paper

References

[1] J HHe ldquoNonlinear oscillationwith fractional derivative and itsapplicationsrdquo in Proceedings of the International Conference onVibrating Engineering 98 Dalian China 1988

[2] J H He ldquoSome applications of nonlinear fractional differentialequations and their approximationsrdquo Bulletin of Science Tech-nology amp Society vol 15 no 2 pp 86ndash90 1999

[3] J-H He ldquoApproximate analytical solution for seepage flowwithfractional derivatives in porous mediardquo Computer Methods inApplied Mechanics and Engineering vol 167 no 1-2 pp 57ndash681998

[4] I Podlubny Fractional Differential Equations Academic PressNew York NY USA 1999

[5] K Al-Khaled and S Momani ldquoAn approximate solution fora fractional diffusion-wave equation using the decompositionmethodrdquoAppliedMathematics and Computation vol 165 no 2pp 473ndash483 2005

[6] F Mainardi Y Luchko and G Pagnini ldquoThe fundamental solu-tion of the space-time fractional diffusion equationrdquo FractionalCalculus amp Applied Analysis vol 4 no 2 pp 153ndash192 2001

[7] A Hanyga ldquoMultidimensional solutions of time-fractional dif-fusion-wave equationsrdquoTheRoyal Society of London A vol 458no 2020 pp 933ndash957 2002

[8] F Huang and F Liu ldquoThe time fractional diffusion equation andthe advection-dispersion equationrdquo The ANZIAM Journal vol46 no 3 pp 317ndash330 2005

[9] F Huang and F Liu ldquoThe fundamental solution of the space-time fractional advection-dispersion equationrdquo Journal ofApplied Mathematics amp Computing vol 18 no 1-2 pp 339ndash3502005

[10] S Momani ldquoAnalytic and approximate solutions of the space-and time-fractional telegraph equationsrdquo Applied Mathematicsand Computation vol 170 no 2 pp 1126ndash1134 2005

[11] S Momani ldquoAn explicit and numerical solutions of the frac-tional KdV equationrdquo Mathematics and Computers in Simula-tion vol 70 no 2 pp 110ndash118 2005

[12] L Debnath and D D Bhatta ldquoSolutions to few linear fractionalinhomogeneous partial differential equations in fluid mechan-icsrdquo Fractional Calculus amp Applied Analysis vol 7 no 1 pp 21ndash36 2004

[13] VDaftardar-Gejji andH Jafari ldquoAn iterativemethod for solvingnonlinear functional equationsrdquo Journal of Mathematical Anal-ysis and Applications vol 316 no 2 pp 753ndash763 2006

[14] A A Hemeda ldquoNew iterative method application to the nth-order integro-differential equationsrdquo Information B vol 16 no6 pp 3841ndash3852 2013

[15] S Bhalekar and V Daftardar-Gejji ldquoNew iterative methodapplication to partial differential equationsrdquoAppliedMathemat-ics and Computation vol 203 no 2 pp 778ndash783 2008

[16] G Adomian Solving Frontier Problems of Physics The Decom-position Method Kluwer 1994

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

[18] A A Hemeda ldquoVariational iteration method for solving non-linear coupled equations in 2-dimensional space in fluid mech-anicsrdquo International Journal of Contemporary MathematicalSciences vol 7 no 37 pp 1839ndash1852 2012

[19] G Adomian ldquoA review of the decompositionmethod in appliedmathematicsrdquo Journal of Mathematical Analysis and Applica-tions vol 135 no 2 pp 501ndash544 1988

Abstract and Applied Analysis 9

[20] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008

[21] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[22] J-H He ldquoVariational principle for nano thin film lubricationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 4 no 3 pp 313ndash314 2003

[23] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[24] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[25] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[26] A A Hemeda ldquoFormulation and solution of nth-order deriva-tive fuzzy integro-differential equation using new iterativemethodwith a reliable algorithmrdquo Journal of AppliedMathemat-ics vol 2012 Article ID 325473 17 pages 2012

[27] S Bhalekar andVDaftardar-Gejji ldquoSolving evolution equationsusing a new iterative methodrdquo Numerical Methods for PartialDifferential Equations vol 26 no 4 pp 906ndash916 2010

[28] A A Hemeda ldquoVariational iteration method for solving waveequationrdquo Computers ampMathematics with Applications vol 56no 8 pp 1948ndash1953 2008

[29] A A Hemeda ldquoVariational iteration method for solving non-linear partial differential equationsrdquo Chaos Solitons and Frac-tals vol 39 no 3 pp 1297ndash1303 2009

[30] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[31] A A Hemeda ldquoHomotopy perturbation method for solvingpartial differential equations of fractional orderrdquo InternationalJournal of Mathematical Analysis vol 6 no 49ndash52 pp 2431ndash2448 2012

[32] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency indepedentmdashpart IIrdquoGeophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[33] A Y Luchko and R Groreo The Initial Value Problem forSome Fractional Differential Equations with Caputo DerivativePreprint series A80-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin 1998

[34] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[35] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[36] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] A J Jerri Introduction to Integral Equations with ApplicationsWiley-Interscience 2nd edition 1999

[39] A A Hemeda ldquoNew iterative method an application forsolving fractional physical differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 617010 9 pages 2013

[40] Z Odibat and S Momani ldquoThe variational iteration methodan efficient scheme for handling fractional partial differentialequations in fluid mechanicsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2199ndash2208 2009

[41] G-CWu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013

[42] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 pp 59ndash75 2011

[43] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

[20] T Ozis and A Yıldırım ldquoComparison between Adomianrsquosmethod and Hersquos homotopy perturbation methodrdquo Computersamp Mathematics with Applications vol 56 no 5 pp 1216ndash12242008

[21] S Momani and S Abuasad ldquoApplication of Hersquos variationaliteration method to Helmholtz equationrdquo Chaos Solitons ampFractals vol 27 no 5 pp 1119ndash1123 2006

[22] J-H He ldquoVariational principle for nano thin film lubricationrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 4 no 3 pp 313ndash314 2003

[23] J-H He ldquoVariational principles for some nonlinear partialdifferential equations with variable coefficientsrdquoChaos Solitonsamp Fractals vol 19 no 4 pp 847ndash851 2004

[24] J-H He ldquoVariational iteration methodmdashsome recent resultsand new interpretationsrdquo Journal of Computational and AppliedMathematics vol 207 no 1 pp 3ndash17 2007

[25] J-H He and X-H Wu ldquoVariational iteration method newdevelopment and applicationsrdquo Computers ampMathematics withApplications vol 54 no 7-8 pp 881ndash894 2007

[26] A A Hemeda ldquoFormulation and solution of nth-order deriva-tive fuzzy integro-differential equation using new iterativemethodwith a reliable algorithmrdquo Journal of AppliedMathemat-ics vol 2012 Article ID 325473 17 pages 2012

[27] S Bhalekar andVDaftardar-Gejji ldquoSolving evolution equationsusing a new iterative methodrdquo Numerical Methods for PartialDifferential Equations vol 26 no 4 pp 906ndash916 2010

[28] A A Hemeda ldquoVariational iteration method for solving waveequationrdquo Computers ampMathematics with Applications vol 56no 8 pp 1948ndash1953 2008

[29] A A Hemeda ldquoVariational iteration method for solving non-linear partial differential equationsrdquo Chaos Solitons and Frac-tals vol 39 no 3 pp 1297ndash1303 2009

[30] A A Hemeda ldquoHomotopy perturbation method for solvingsystems of nonlinear coupled equationsrdquo Applied MathematicalSciences vol 6 no 93ndash96 pp 4787ndash4800 2012

[31] A A Hemeda ldquoHomotopy perturbation method for solvingpartial differential equations of fractional orderrdquo InternationalJournal of Mathematical Analysis vol 6 no 49ndash52 pp 2431ndash2448 2012

[32] M Caputo ldquoLinearmodels of dissipationwhoseQ is almost fre-quency indepedentmdashpart IIrdquoGeophysical Journal Internationalvol 13 no 5 pp 529ndash539 1967

[33] A Y Luchko and R Groreo The Initial Value Problem forSome Fractional Differential Equations with Caputo DerivativePreprint series A80-98 Fachbreich Mathematik und Infor-matik Freic Universitat Berlin 1998

[34] K S Miller and B Ross An Introduction to the Fractional Cal-culus and Fractional Differential Equations John Wiley amp SonsNew York NY USA 1993

[35] K B Oldham and J SpanierThe Fractional Calculus AcademicPress New York NY USA 1974

[36] I Podlubny ldquoGeometric and physical interpretation of frac-tional integration and fractional differentiationrdquo FractionalCalculus amp Applied Analysis vol 5 no 4 pp 367ndash386 2002

[37] Y Cherruault ldquoConvergence of AdomianrsquosmethodrdquoKybernetesvol 18 no 2 pp 31ndash38 1989

[38] A J Jerri Introduction to Integral Equations with ApplicationsWiley-Interscience 2nd edition 1999

[39] A A Hemeda ldquoNew iterative method an application forsolving fractional physical differential equationsrdquo Abstract andApplied Analysis vol 2013 Article ID 617010 9 pages 2013

[40] Z Odibat and S Momani ldquoThe variational iteration methodan efficient scheme for handling fractional partial differentialequations in fluid mechanicsrdquo Computers amp Mathematics withApplications vol 58 no 11-12 pp 2199ndash2208 2009

[41] G-CWu and D Baleanu ldquoVariational iteration method for theBurgersrsquo flow with fractional derivativesmdashnew Lagrange multi-pliersrdquo Applied Mathematical Modelling vol 37 no 9 pp 6183ndash6190 2013

[42] G C Wu ldquoNew trends in the variational iteration methodrdquoCommunications in Fractional Calculus vol 2 pp 59ndash75 2011

[43] J-H He ldquoAsymptotic methods for solitary solutions and com-pactonsrdquo Abstract and Applied Analysis vol 2012 Article ID916793 130 pages 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of