Research Article  Open Access
O. Abdulaziz, I. Hashim, A. Saif, "Series Solutions of TimeFractional 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 TimeFractional PDEs by Homotopy Analysis Method
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 hyperbolictype 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) [1–4].
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) [5–14], the homotopyperturbation method (HPM) [15–19], variational iteration method (VIM) [12, 20–24], and the EXPfunction method [25]. Another analytical approach that can be applied to solve nonlinear differential equations is to employ the homotopy analysis method (HAM) [26–29]. Some of the recent applications of HAM can be found in [30–41]. 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 shallowwater 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 runup.
Very recently, Song and Zhang [44] applied HAM to solve fractional KdVBurgersKuramoto 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 BBMBurgers 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 twodimensional fractional heatlike PDEs subject to the Neumann boundary conditions. Jafari and Seifi [51] applied HAM to linear and nonlinear homogeneous fractional diffusionwave 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 RiemannLiouville 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. Homotopy Analysis Method (HAM)
3.1. The ZerothOrder 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 socalled zerothorder 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 zerothorder 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 thOrder Deformation Equation
Let us define the vectorFollowing Liao [26–29], differentiating (3.4) times with respect to the embedding parameter , then setting , and finally dividing them by , we have the socalled thorder 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 timefractional wavelike equations:We note that the heatlike 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.

4.2. Problem 2
In this example, we consider the following onedimensional linear inhomogeneous timefractional 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 timefractional hyperbolic equation:subject to the initial conditionsEquation (4.14) can be rewritten as follows:
From (3.4), construct the following zerothorder 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 highorder 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 fourterm 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, .

4.4. Problem 4
Consider the following nonlinear timefractional 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 3term 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, .

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 (STGL0112006) is gratefully acknowledged.
References
 K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 111 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1974. View at: Zentralblatt MATH  MathSciNet
 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A WileyInterscience Publication, John Wiley & Sons, New York, NY, USA, 1993. View at: Zentralblatt MATH  MathSciNet
 I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at: Zentralblatt MATH  MathSciNet
 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of NorthHolland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. View at: Zentralblatt MATH
 N. T. Shawagfeh, “Analytical approximate solutions for nonlinear fractional differential equations,” Applied Mathematics and Computation, vol. 131, no. 23, pp. 517–529, 2002. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. S. Ray and R. K. Bera, “Solution of an extraordinary differential equation by Adomian decomposition method,” Journal of Applied Mathematics, vol. 2004, no. 4, pp. 331–338, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. S. Ray and R. K. Bera, “An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 167, no. 1, pp. 561–571, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 O. Abdulaziz, I. Hashim, M. S. H. Chowdhury, and A. K. Zulkifle, “Assessment of decomposition method for linear and nonlinear fractional differential equations,” Far East Journal of Applied Mathematics, vol. 28, no. 1, pp. 95–112, 2007. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 O. Abdulaziz, I. Hashim, and E. S. Ismail, “Approximate analytical solutions to fractional modified KdV equations by decomposition method,” Far East Journal of Applied Mathematics, vol. 29, no. 3, pp. 455–468, 2007. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 I. Hashim and M. S. Selamat, “Simulation of timedependent enzyme kinetics by an explicit numericanalytic technique,” Far East Journal of Applied Mathematics, vol. 30, no. 1, pp. 115–124, 2008. View at: Google Scholar  Zentralblatt MATH  MathSciNet
 S. Momani, “Nonperturbative analytical solutions of the space and timefractional Burgers equations,” Chaos, Solitons & Fractals, vol. 28, no. 4, pp. 930–937, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. Momani and Z. Odibat, “Analytical approach to linear fractional partial differential equations arising in fluid mechanics,” Physics Letters A, vol. 355, no. 45, pp. 271–279, 2006. View at: Publisher Site  Google Scholar
 Q. Wang, “Numerical solutions for fractional KdVBurgers equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1048–1055, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Q. M. AlMdallal, “An efficient method for solving fractional SturmLiouville problems,” Chaos, Solitons & Fractals. In press. View at: Publisher Site  Google Scholar
 J.H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons & Fractals, vol. 26, no. 3, pp. 695–700, 2005. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 167–174, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 S. Momani and Z. Odibat, “Homotopy perturbation method for nonlinear partial differential equations of fractional order,” Physics Letters A, vol. 365, no. 56, pp. 345–350, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 Q. Wang, “Homotopy perturbation method for fractional KdVBurgers equation,” Chaos, Solitons & Fractals, vol. 35, no. 5, pp. 843–850, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Odibat, “Exact solitary solutions for variants of the KdV equations with fractional time derivatives,” Chaos, Solitons & Fractals. In press. View at: Publisher Site  Google Scholar
 Z. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,” International Journal of Nonlinear Science and Numerical Simulation, vol. 7, no. 1, pp. 27–34, 2006. View at: Google Scholar
 S. Momani and Z. Odibat, “Numerical comparison of methods for solving linear differential equations of fractional order,” Chaos, Solitons & Fractals, vol. 31, no. 5, pp. 1248–1255, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J.H. He and X.H. Wu, “Construction of solitary solution and compactonlike solution by variational iteration method,” Chaos, Solitons & Fractals, vol. 29, no. 1, pp. 108–113, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Y. Molliq R, M. S. M. Noorani, and I. Hashim, “Variational iteration method for fractional heat and wavelike equations,” Nonlinear Analysis: Real World Applications. In press. View at: Publisher Site  Google Scholar
 S. Abbasbandy, “An approximation solution of a nonlinear equation with RiemannLiouville's fractional derivatives by He's variational iteration method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 53–58, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X.H. Wu and J.H. He, “EXPfunction method and its application to nonlinear equations,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 903–910, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University, Shanghai, China, 1992.
 S.J. Liao, “An approximate solution technique not depending on small parameters: a special example,” International Journal of NonLinear Mechanics, vol. 30, no. 3, pp. 371–380, 1995. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S.J. Liao, “A kind of approximate solution technique which does not depend upon small parameters—II. An application in fluid mechanics,” International Journal of NonLinear Mechanics, vol. 32, no. 5, pp. 815–822, 1997. View at: Publisher Site  Google Scholar  MathSciNet
 S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, vol. 2 of CRC Series: Modern Mechanics and Mathematics, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. View at: Zentralblatt MATH  MathSciNet
 A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Solving systems of ODEs by homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 10, pp. 2060–2070, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Approximate analytical solutions of systems of PDEs by homotopy analysis method,” Computers & Mathematics with Applications, vol. 55, no. 12, pp. 2913–2923, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Approximate solutions of singular twopoint BVPs by modified homotopy analysis method,” Physics Letters A, vol. 372, no. 22, pp. 4062–4066, 2008. View at: Publisher Site  Google Scholar
 A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Modified homotopy analysis method for solving systems of secondorder BVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 430–442, 2009. View at: Publisher Site  Google Scholar
 A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “The homotopy analysis method for Cauchy reactiondiffusion problems,” Physics Letters A, vol. 372, no. 5, pp. 613–618, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 S. Abbasbandy, M. Yürüsoy, and M. Pakdemirli, “The analysis approach of boundary layer equations of powerlaw fluids of second grade,” Zeitschrift für Naturforschung, vol. 63a, pp. 564–570, 2008. View at: Google Scholar
 S. Abbasbandy, “Soliton solutions for the FitzhughNagumo equation with the homotopy analysis method,” Applied Mathematical Modelling, vol. 32, no. 12, pp. 2706–2714, 2008. View at: Google Scholar
 T. Hayat and Z. Abbas, “Channel flow of a Maxwell fluid with chemical reaction,” Zeitschrift für Angewandte Mathematik und Physik, vol. 59, no. 1, pp. 124–144, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 T. Hayat, Z. Abbas, and M. Sajid, “Heat and mass transfer analysis on the flow of a second grade fluid in the presence of chemical reaction,” Physics Letters A, vol. 372, no. 14, pp. 2400–2408, 2008. View at: Publisher Site  Google Scholar
 Z. Abbas and T. Hayat, “Radiation effects on MHD flow in a porous space,” International Journal of Heat and Mass Transfer, vol. 51, no. 56, pp. 1024–1033, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 M. M. Rashidi, G. Domairry, and S. Dinarvand, “Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 708–717, 2009. View at: Google Scholar
 G. Domairry and M. Fazeli, “Homotopy analysis method to determine the fin efficiency of convective straight fins with temperaturedependent thermal conductivity,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 2, pp. 489–499, 2009. View at: Publisher Site  Google Scholar
 S.J. Liao, “Notes on the homotopy analysis method: some definitions and theorems,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 983–997, 2009. View at: Publisher Site  Google Scholar
 Y. Wu and K. F. Cheung, “Explicit solution to the exact Riemann problem and application in nonlinear shallowwater equations,” International Journal for Numerical Methods in Fluids, vol. 57, no. 11, pp. 1649–1668, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 L. Song and H. Zhang, “Application of homotopy analysis method to fractional KdVBurgersKuramoto equation,” Physics Letters A, vol. 367, no. 12, pp. 88–94, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 J. Cang, Y. Tan, H. Xu, and S.J. Liao, “Series solutions of nonlinear Riccati differential equations with fractional order,” Chaos, Solitons & Fractals. In press. View at: Publisher Site  Google Scholar
 I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method for fractional IVPs,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 3, pp. 674–684, 2009. View at: Publisher Site  Google Scholar
 L. Song and H. Zhang, “Solving the fractional BBMBurgers equation using the homotopy analysis method,” Chaos, Solitons & Fractals. In press. View at: Publisher Site  Google Scholar
 A. S. Bataineh, A. K. Alomari, M. S. M. Noorani, I. Hashim, and R. Nazar, “Series solutions of systems of nonlinear fractional differential equations,” Acta Applicandae Mathematicae, vol. 105, no. 2, pp. 189–198, 2009. View at: Publisher Site  Google Scholar
 H. Xu and J. Cang, “Analysis of a time fractional wavelike equation with the homotopy analysis method,” Physics Letters A, vol. 372, no. 8, pp. 1250–1255, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 H. Xu, S.J. Liao, and X.C. You, “Analysis of nonlinear fractional partial differential equations with the homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1152–1156, 2009. View at: Publisher Site  Google Scholar
 H. Jafari and S. Seifi, “Homotopy analysis method for solving linear and nonlinear fractional diffusionwave equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 2006–2012, 2009. View at: Publisher Site  Google Scholar
 H. Jafari and S. Seifi, “Solving a system of nonlinear fractional partial differential equations using homotopy analysis method,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1962–1969, 2009. View at: Google Scholar
 M. Caputo, “Linear models of dissipation whose $Q$ is almost frequency independentII,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967. View at: Publisher Site  Google Scholar
 R. Gorenflo and F. Mainardi, “Fractional calculus: integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics (Udine, 1996), vol. 378 of CISM Courses and Lectures, pp. 223–276, Springer, Vienna, Austria, 1997. View at: Google Scholar  MathSciNet
Copyright
Copyright © 2008 O. Abdulaziz et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.