Construction of Analytical Solutions toFractional Differential Equations UsingHomotopy Analysis...

7/30/2019 Construction of Analytical Solutions toFractional Differential Equations UsingHomotopy Analysis Method

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Abstract In this paper, we present an algorithm of the

homotopy analysis method (HAM) to obtain symbolicapproximate solutions for linear and nonlinear differentialequations of fractional order. We show that the HAM isdifferent from all analytical methods; it provides us with asimple way to adjust and control the convergence region of theseries solution by introducing the auxiliary parameter , the

auxiliary function , the initial guess ( ) and the auxiliarylinear operator . Three examples, the fractional oscillationequation, the fractional Riccati equation and the fractionalLane-Emden equation, are tested using the modified algorithm.The obtained results show that the Adomain decompositionmethod, Variational iteration method and homotopyperturbation method are special cases of homotopy analysismethod. The modified algorithm can be widely implemented tosolve both ordinary and partial differential equations of fractional order.

Index Terms Adomian decomposition method, Caputoderivative, Fractional Lane-Emden equation, Fractionaloscillation equation, Fractional Riccati equation, Homotopy

analysis method.

I. INTRODUCTION Fractional differential equations have gained importance

and popularity during the past three decades or so, mainly dueto its demonstrated applications in numerous seeminglydiverse fields of science and engineering. For example, thenonlinear oscillation of earthquake can be modeled withfractional derivatives, and the fluid-dynamic traffic modelwith fractional derivatives can eliminate the deficiencyarising from the assumption of continuum traffic flow. The

differential equations with fractional order have recentlyproved to be valuable tools to the modeling of many physicalphenomena [1-8]. This is because of the fact that the realisticmodeling of a physical phenomenon does not depend only onthe instant time, but also on the history of the previous timewhich can also be successfully achieved by using fractional

1Department of Mathematics, Zarqa Private University, Zarqa 1311,Jordan, (email: [email protected])2Prince Abdullah Bin Ghazi Faculty of Science and IT, Al-Balqa'Applied University, Salt 19117, Jordan, (email: [email protected])3Department of Mathematics, Mutah University, P. O. Box 7, Al-Karak,Jordan, (email: [email protected])4Department of Mathematics, University of Jordan, Amman 11942,Jordan, (email: [email protected])*Corresponding author. Email addresses: [email protected];[email protected], Tel. +962-5-3532519, Fax. +962-5-3530462.

calculus. Most nonlinear fractional equations do not haveexact analytic solutions, so approximation and numericaltechniques must be used. The Adomain decompositionmethod (ADM) [9-14], the homotopy perturbation method(HPM) [15-25], the variational iteration method (VIM)[26-28] and other methods have been used to provideanalytical approximation to linear and nonlinear problems.

However, the convergence region of the correspondingresults is rather small as shown in [9-28]. The homotopyanalysis method (HAM) is proposed first by Liao [29-33] forsolving linear and nonlinear differential and integralequations. Different from perturbation techniques; the HAMdoesn't depend upon any small or large parameter. Thismethod has been successfully applied to solve many types of nonlinear differential equations, such as projectile motionwith the quadratic resistance law [34], Klein-Gordonequation [35], solitary waves with discontinuity [36], thegeneralized Hirota-Satsuma coupled KdV equation [37], heatradiation equations [38], MHD flows of an Oldroyd

8-constant fluid [39], Vakhnenko equation [40], unsteadyboundary-layer flows [41]. Recently, Song and Zhang [42]used the HAM to solve fractional KdV-Burgers-Kuramotoequation, Cang and his co-authors [43] constructed a seriessolution of non-linear Riccati differential equations withfractional order using HAM. They proved that the Adomiandecomposition method is a special case of HAM, and we canadjust and control the convergence region of solution seriesby choosing the auxiliary parameter closed to zero.

The objective of the present paper is to modify the HAM toprovide symbolic approximate solutions for linear andnonlinear differential equations of fractional order. Ourmodification is implemented on the fractional oscillationequation, the fractional Riccati equation and the fractionalLane-Emden equation. By choosing suitable values of theauxiliary parameter , the auxiliary function , the initialguess 0 ( ) and the auxiliary linear operator we canadjust and control the convergence region of solution series.Moreover, we illustrated for several examples that theAdomain decomposition, Variational iteration and homotopyperturbation solutions are special cases of homotopy analysissolution.

II. DEFINITIONS For the concept of fractional derivative we will adopt

Caputo s definition [7] which is a modification of theRiemann-Liouville definition and has the advantage of dealing properly with initial value problems in which the

Construction of Analytical Solutions toFractional Differential Equations Using

Homotopy Analysis Method

Ahmad El-Ajou1, Zaid Odibat

*2, Shaher Momani

3, Ahmad Alawneh

4

IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_01

(Advance online publication: 13 May 2010)

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initial conditions are given in terms of the field variables andtheir integer order which is the case in most physicalprocesses [4].

Definition 1 . A real function ( ), > 0 , is said to be in thespace , if there exists a real number , suchthat

=

1 , where

1 0,

and it is said

to be in the space iff , . Definition 2. The Riemann-Liouville fractional integraloperator of order > 0 , of a function , 1 ,is defined as:

= 1 10 ,> 0 , > 0 ,

(1)

0 = . (2)Properties of the operator

can be found in [5-8], we

mention only the following:For , 1, , 0 and 1

= + = ( ) , (3) = ( + 1)( + + 1) + . (4)

Definition 3. The fractional derivative of ( ) in theCaputo sense is defined as: = = 1 10 ( ) ,

(5)

for 1 < , , > 0 , 1 . Lemma 1. If 1 < , and , 1 ,then

= , (6)

= 0+ ! , > 0

1=0

. (7)

III. HOMOTOPY ANALYSIS METHODThe principles of the HAM and its applicability for various

kinds of differential equations are given in [29-43]. Forconvenience of the reader, we will present a review of theHAM [29, 38]. To achieve our goal, we consider thenonlinear differential equation.

= 0 ,

0 , (8)

where N is a nonlinear differential operator, and ( ) isunknown function of the independent variable .

A. Zeroth- order Deformation EquationLiao [29] constructs the so-called zeroth-order

deformation equation:

1 ; 0 = ; , (9)where 0,1 is an embedding parameter,

0 is an

auxiliary parameter, 0 is an auxiliary function,

isan auxiliary linear operator, is nonlinear differentialoperator, ; is an unknown function, and 0 is aninitial guess of ( ) , which satisfies the initial conditions. Itshould be emphasized that one has great freedom to choosethe initial guess 0 , the auxiliary linear operator , theauxiliary parameter and the auxiliary function . According to the auxiliary linear operator and the suitableinitial conditions, when = 0 , we have

; 0 = 0 , (10)

and when = 1 , since

0 and

0 , the

zeroth-order deformation equation (9) is equivalent to (8),hence

; 1 = . (11)

Thus, as increasing from 0 to 1 , the solution ; various from 0 to ( ). Define

=1

!;

=0

. (12)

Expanding ; in a Taylor series with respect to theembedding parameter , by using (10) and (12), we have:

; = 0 +=1

. (13)

Assume that the auxiliary parameter , the auxiliaryfunction , the initial approximation 0 ( ) and theauxiliary linear operator are properly chosen so that theseries (13) converges at = 1 . Then at = 1 , from (11), theseries solution (13) becomes

= 0 +=1

. (14)

B. High-order Deformation EquationDefine the vector:

= 0 , 1 , 2 ,, ( ) . (15)Differentiating equation (9) m-times with respect toembedding parameter , then setting = 0 and dividingthem by ! , we have, using (12), the so-called mth-order

deformation equation

1 = 1 ,= 1,2, , , (16)

IAENG International Journal of Applied Mathematics, 40:2, IJAM_40_2_01

(Advance online publication: 13 May 2010)

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where

1 = 1( 1)! 1 [ ; ]

1 =0 , (17)and

= 0 , 11 , > 1 . (18)The so-called mth-order deformation equation (16) is a linearwhich can be easily solved using Mathematica package.

IV. AN ALGORITHM OF HOMOTOPY ANALYSIS METHOD The HAM has been extended in [42,43] to solve some

fractional differential equations. They use the auxiliary linearoperator to be / or , where 0 < 1 . Also theyfix the auxiliary function to be 1 . In this section, wepresent an efficient algorithm of the HAM. This algorithmcan be established based on the assumptions that thenonlinear operator can involve fractional derivatives, theauxiliary function can be freely selected and theauxiliary linear operator can be considered as = ,

> 0 . To illustrate its basic ideas, we consider the followingfractional initial value problem

= 0 , 0 , (19)where is a nonlinear differential operator that mayinvolves fractional derivatives, where the highest order

derivative is , subject to the initial conditions

0 = , = 0,1,2, , 1 . (20)The so-called zeroth-order deformation equation can bedefined as

1 ; 0 = ; ,> 0 ,

(21)

subject to the initial conditions

0; = , = 0,1,2, , 1 . (22)Obviously, when = 0 , since 0 satisfies the initialconditions (20) and = , > 0 , we have

; 0 = 0 , (23)

and, the so-called mth-order deformation equation can beconstructed as

1 = 1 ,> 0 ,

(24)

subject to the initial conditions

( ) 0 = 0 , = 0,1,2, , 1. (25)Operating , > 0 on both sides of (24) gives themth-order deformation equation in the form:

= 1 + 1 . (26)V. EXAMPLES

In this section we employ our algorithm of the homotopyanalysis method to find out series solutions for somefractional initial value problems.

Example 1. Consider the composite fractional oscillationequation [18]

2

2 8 = 0 , 0 ,

1

Construction of Analytical Solutions toFractional Differential Equations UsingHomotopy Analysis...

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