Research Article | Open Access

Volume 2008 |Article ID 686512 | https://doi.org/10.1155/2008/686512

O. Abdulaziz, I. Hashim, A. Saif, "Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method", International Journal of Differential Equations, vol. 2008, Article ID 686512, 16 pages, 2008. https://doi.org/10.1155/2008/686512

# Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method

Accepted30 Oct 2008
Published29 Jan 2009

#### Abstract

The homotopy analysis method (HAM) is applied to solve linear and nonlinear fractional partial differential equations (fPDEs). The fractional derivatives are described by Caputo's sense. Series solutions of the fPDEs are obtained. A convergence theorem for the series solution is also given. The test examples, which include a variable coefficient, inhomogeneous and hyperbolic-type equations, demonstrate the capability of HAM for nonlinear fPDEs.

#### 1. Introduction

Fractional calculus has been given considerable popularity and importance during the past three decades, due mainly to its applications in numerous fields of science and engineering. For example, phenomena in the areas of fluid flow, rheology, electrical networks, probability and statistics, control theory of dynamical systems, electrochemistry of corrosion, chemical physics, optics and signal processing, and so on can be successfully modelled by linear or nonlinear fractional differential equations (fDEs) [14].

Finding accurate methods for solving nonlinear differential equations has become important. Some of the analytical methods for nonlinear differential equations are the Adomian decomposition method (ADM) [514], the homotopy-perturbation method (HPM) [1519], variational iteration method (VIM) [12, 2024], and the EXP-function method [25]. Another analytical approach that can be applied to solve nonlinear differential equations is to employ the homotopy analysis method (HAM) [2629]. Some of the recent applications of HAM can be found in [3041]. An account of the recent developments of HAM was given by Liao [42]. HAM has been successfully applied into engineering fields. The method has been applied to give an explicit solution for the Riemann problem of the nonlinear shallow-water equations [43]. The obtained Riemann solver has been implemented into a numerical model to simulate long waves, such as storm surge or tsunami, propagation and run-up.

Very recently, Song and Zhang [44] applied HAM to solve fractional KdV-Burgers-Kuramoto equation. Cang et al. [45] solved nonlinear Riccati differential equations of fractional order using HAM. Hashim et al. [46] employed HAM to solve fractional initial value problems (fIVPs) for ordinary differential equations. In [47], the applicability of the HAM was extended to construct numerical solution for the fractional BBM-Burgers equation. The HAM solutions for systems of nonlinear fractional differential equations were presented by Bataineh et al. [48].

A specific linear, nonhomogeneous time fractional partial differential equation (fPDE) with variable coefficients was first transformed to two fractional ordinary differential equations which were then solved by HAM in [49]. Recently, Xu et al. [50] applied the HAM to linear, homogeneous one- and two-dimensional fractional heat-like PDEs subject to the Neumann boundary conditions. Jafari and Seifi [51] applied HAM to linear and nonlinear homogeneous fractional diffusion-wave equations. Very recently, the HAM was shown to be capable of solving linear and nonlinear systems of fPDEs [52].

In this paper, we shall consider linear and nonlinear fPDEs of the formsubject to the initial conditionswhere is an integer, is a linear/nonlinear function, and is a fractional differential operator. We shall demonstrate the applicability of HAM to fPDEs through several linear and nonlinear test examples.

#### 2. Preliminaries

The fractional derivative is defined in the Caputo sense as in [53],Here is the usual integer differential operator of order and is the Riemann-Liouville fractional integral operator of order , defined by Some of the properties of the operator which we will need in our work, are as follows [2, 3]:(1),(2),(3). Caputo’s fractional derivative has a useful property [54]The operator form of the nonlinear fPDEs (1.1) can be written as follows:subject to the initial conditionswhere is a linear operator which might include other fractional derivatives of order less that , is a nonlinear operator which also might include other fractional derivatives of order less that and is a known analytic function.

Applying the operator , the inverse operator of , to both sides of (2.4) with considering the initial conditions (2.5) according to (2.3), we obtain

#### 3.1. The Zeroth-Order Deformation Equation

Let denote an auxiliary linear operator, is an initial approximation of which satisfies the initial conditions (2.5). Note that, in this paper, the auxiliary linear operator is not the same linear operator of (2.4).

Note that the original equation (1.1) contains the linear operator . So, it is straightforward for us to choose the auxiliary linear operatorAccording to (2.6), we can choose the initial approximation to be

For simplicity, let us define, according to (2.4), the nonlinear operatorHence, in the frame of HAM [29], we can construct the so-called zeroth-order deformationsubject to the following initial conditions:where is the embedding parameter, is an auxiliary parameter, and is an unknown function on the independent variables , and .

When , since satisfies all the initial conditions (2.5), and is a solution of , we have obviouslyand when , the zeroth-order deformation equations (3.4) and (3.5) are equivalent to the original equations (2.4) and (2.5), providedUsing the parameter , we expand in Taylor series as follows:where

Assume that the auxiliary linear operator , the initial guess and the auxiliary parameter are properly chosen such that the series (3.8) is convergent at . Thus, due to (3.7) we have

#### 3.2. The th-Order Deformation Equation

Let us define the vectorFollowing Liao [2629], differentiating (3.4) times with respect to the embedding parameter , then setting , and finally dividing them by , we have the so-called th-order deformation equationsubject to the initial conditionswhereSubstituting (3.3) into (3.14), and since is a linear operator, can be given by

According to (3.1), we can apply the operator to both sides of (3.12) to obtainUsing the property (2.3) and the initial conditions (1.1), we have

Finally, for the purpose of computation, we will approximate the HAM solution (3.10) by the following truncated series:

#### 3.3. Convergence Theorem

Theorem 3.1. As long as the series converges, where is governed by (3.12) under the definitions (3.14) and (3.15), it must be a solution of (2.4).

Proof. If the series is convergent, we can writeand it holdsFrom (3.12) and by using (3.15), it yieldsSince , thenSubstituting (3.16) into the above equation and simplifying it, due to the convergence of the series and since is a linear operator, yieldNow, expanding the nonlinear term by using the general Taylor theorem at yieldsSetting in the above equation and using (3.8), we obtainThenFrom the initial conditions (3.5) and (3.13), it holds that
Thus, is satisfied and also must be the exact solution for (2.4).

#### 4. Test Examples

In this section, we shall illustrate the applicability of HAM to several linear and nonlinear fPDEs.

#### 4.1. Problem 1

Let us consider the following linear time-fractional wave-like equations:We note that the heat-like counterpart of (4.1) was solved by HAM in [50] without direct comparison with the result by the ADM. According to (3.2), we can choose the initial guess to beFrom (3.18), we haveConsequently, the first few terms of HAM series solutions are as follows:and so on. Hence, the HAM series solution isSince we choose the initial guess to be the same initial guess used by ADM [12], we can notice that when , the above expression gives the same solution given by ADM. Table 1 shows the HAM approximation solutions for (4.1)-(4.2) when , , and with and . It is to be noted that the first four terms of the HAM series were used to evaluate the approximate solutions in Table 1.

 𝑡 𝑥 𝛼 = 1 . 5 𝛼 = 1 . 7 5 𝛼 = 2 . 0 ℏ = − 1 ℏ = − 1 . 0 4 5 ℏ = − 1 ℏ = − 1 . 0 4 5 ℏ = − 1 ℏ = − 1 . 0 4 5 Exact 0.2 0.25 0.26284062 0.26284062 0.26266989 0.26266991 0.26258350 0.26258350 0.26258350 0.50 0.55136246 0.55136250 0.55067959 0.55067964 0.55033400 0.55033402 0.55033400 0.75 0.86556554 0.86556562 0.86402909 0.86402919 0.86325150 0.86325156 0.86325150 1.0 1.20544985 1.20544999 1.20271839 1.20271856 1.20133600 1.20133610 1.20133600 0.4 0.25 0.27697114 0.27697109 0.27615668 0.27615669 0.27567202 0.27567205 0.27567202 0.50 0.60788455 0.60788437 0.60462675 0.60462679 0.60268808 0.60268820 0.60268808 0.75 0.99274025 0.99273983 0.98541019 0.98541027 0.98104818 0.98104846 0.98104818 1.0 1.43153821 1.43153748 1.41850702 1.41850715 1.41075232 1.41075282 1.41075232 0.6 0.25 0.29309481 0.29309501 0.29109009 0.29108997 0.28979084 0.28979084 0.28979084 0.50 0.67237924 0.67238006 0.66436039 0.66435990 0.65916338 0.65916339 0.65916339 0.75 1.13785328 1.13785513 1.11981088 1.11980978 1.10811762 1.10811764 1.10811764 1.0 1.68951694 1.68952024 1.65744156 1.65743961 1.63665355 1.63665358 1.63665358

#### 4.2. Problem 2

In this example, we consider the following one-dimensional linear inhomogeneous time-fractional equation:subject to the initial conditionIn Section 3, we chose the initial guess to contain the initial conditions and the source term . In this example, due to the appearance of noise terms and also to get the exact solution, we will modify the way we choose the initial guess. The initial guess is set to contain only the initial condition (4.8), and the source term, , will be added to . The other terms are obtained the same as described in Section 3.

Hence, the initial guess is given byand according to (3.18), we haveThe terms of the HAM solution series can be given byand so on. Hence, the HAM series solution isTaking in (4.12), we obtain the exact solution,

#### 4.3. Problem 3

Consider the following nonlinear time-fractional hyperbolic equation:subject to the initial conditionsEquation (4.14) can be rewritten as follows:

From (3.4), construct the following zeroth-order deformation:subject to the following initial conditions:whereThe auxiliary linear operator can be chosen as follows:with the propertywhile, the initial guess isAgain from (3.12), the high-order deformation equation can be given bysubject to the initial conditionswhereThen, can be given byAccordingly, the governing equation is as follows:

Consequently, the first few terms of HAM series solutions are given byand so on. Hence, the HAM series solution isThe four-term HAM approximate solutions for (4.14)-(4.15), when , , and with and , are shown in Table 2. Notice that the HAM approximate solution when with is in good agreement with the exact solution, .

 𝛼 = 1 . 5 𝛼 = 1 . 7 5 𝛼 = 2 . 0 𝑡 𝑥 ℏ = − 1 ℏ = − 1 . 0 9 6 6 ℏ = − 1 ℏ = − 1 . 0 9 6 6 ℏ = − 1 ℏ = − 1 . 0 9 6 6 Exact 0.2 0.25 0.060049 0.060199 0.048780 0.048788 0.043403 0.043404 0.043403 0.50 0.240195 0.240795 0.195119 0.195152 0.173610 0.173617 0.173611 0.75 0.540439 0.541789 0.439018 0.439092 0.390623 0.390638 0.390625 1.0 0.960781 0.963180 0.780477 0.780607 0.694441 0.694468 0.694444 0.4 0.25 0.074536 0.076976 0.045434 0.045821 0.031853 0.031904 0.031888 0.50 0.298142 0.307903 0.181736 0.183285 0.127412 0.127616 0.127551 0.75 0.670821 0.692783 0.408907 0.412392 0.286677 0.287137 0.286990 1.0 1.192570 1.231614 0.726946 0.733141 0.509649 0.510465 0.510204 0.6 0.25 0.091469 0.099717 0.045907 0.048048 0.023920 0.024414 0.024414 0.50 0.365878 0.398867 0.183626 0.192191 0.095680 0.097656 0.097656 0.75 0.823225 0.897450 0.413159 0.432430 0.215280 0.219727 0.219727 1.0 1.463511 1.595467 0.734504 0.768765 0.382720 0.390626 0.390625

#### 4.4. Problem 4

Consider the following nonlinear time-fractional Fisher’s equation:for subject to the initial conditionAccording to (3.2), we can choose the initial guess to beand according to (3.18), we have

Consequently, the first few terms of HAM series solutions are as follows:and so on. Hence, the HAM series solution isTable 3 shows the 3-term HAM approximate solutions for (4.30)-(4.31), , when , and with and . We notice that the HAM approximate solution when with is in good agreement with the exact solution, .

 𝛼 = 0 . 5 𝛼 = 0 . 7 5 𝛼 = 1 . 0 𝑡 𝑥 ℏ = − 1 ℏ = − 1 . 0 5 1 3 ℏ = − 1 ℏ = − 1 . 0 5 1 3 ℏ = − 1 ℏ = − 1 . 0 5 1 3 Exact 0.1 0.25 0.946129 0.984066 0.488195 0.496911 0.317948 0.319612 0.316042 0.50 0.843908 0.883590 0.405740 0.414926 0.250500 0.252292 0.250000 0.75 0.715013 0.752540 0.324453 0.333180 0.190964 0.192689 0.191689 1.0 0.576466 0.609185 0.249683 0.257315 0.140979 0.142501 0.142537 0.2 0.25 1.475318 1.551785 0.791250 0.816533 0.481199 0.488421 0.461284 0.50 1.359827 1.439679 0.690146 0.716646 0.396941 0.404576 0.387456 0.75 1.180981 1.256421 0.574405 0.599498 0.315266 0.322535 0.316042 1.0 0.970078 1.035807 0.456647 0.478543 0.241175 0.247540 0.250000 0.3 0.25 1.967448 2.082543 1.124230 1.171123 0.681440 0.698116 0.604195 0.50 1.845234 1.965336 1.009482 1.058531 0.581860 0.599391 0.534447 0.75 1.622908 1.736324 0.859509 0.905896 0.475833 0.492464 0.461284 1.0 1.345505 1.444292 0.695479 0.735922 0.372917 0.387446 0.387456

#### 5. Conclusions

In this work, the homotopy analysis method (HAM) was implemented to derive exact and approximate analytical solutions for both linear and nonlinear partial differential equations of fractional order. The convergence region of the series solution obtained by HAM can be controlled and adjusted by the auxiliary parameter . We give some examples to show the efficiency and accuracy of the suggested method. It was also demonstrated that the Adomian decomposition method (ADM) is a special case of HAM for the first and second test examples.

#### Acknowledgment

The financial support received from the Academy of Sciences Malaysia under tSAGA Grant no. P24c (STGL-011-2006) is gratefully acknowledged.

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