A formulation of the fractional Legendre functions is constructed to solve the generalized time-fractional diffusion equation. The fractional derivative is described in the Caputo sense. The method is based on the collection Legendre and path following methods. Analysis for the presented method is given and numerical results are presented. 1. Introduction We consider the generalized time-fractional diffusion equation of the form with initial and boundary conditions where , , , , , , and . For , the fractional diffusion equation is reduced to a conventional diffusion-reaction equation which is well studied, so we focus on . Some existence and uniqueness results of Problem (1)-(2) were established in [1]. In recent years, great interests were devoted to the analytical and numerical treatments of fractional differential equations (FDEs). Usually, FDEs appear as generalizations to existing models with integer derivative and they also present new models for some physical problems [2, 3]. In general, FDEs do not possess exact solutions in closed forms, and, therefore, numerical methods such as the variational iteration (VIM) [4, 5], the homotopy analysis method (HAM) [6, 7], and the Adomian decomposition method (ADM) [8, 9] have been implemented for several types of FDEs. Also, the maximum principle and the method of lower and upper solutions have been extended to deal with FDEs and obtain analytical and numerical results [10, 11]. The Tau method, the pseudospectral method, and the wavelet method based on the Legendre polynomials have been implemented for several types of FDEs [12–14]. Kazem et al. [12] have constructed the Legendre functions of fractional order and discussed some of their properties. The resulting Legendre function operational and product matrices, together with the Tau method, have been implemented to solve linear and nonlinear fractional differential equations. The effectiveness of the approach has been examined through several examples. In [13], a fractional diffusion equation is considered, where the fractional derivative of order refers to the spatial variable . The Legendre pseudospectral method is implemented to solve the problem, where the solution is expanded with regular Legendre polynomials. As a result, a system of linear equation has been obtained and integrated using the finite difference method. However, in solving fractional differential equations of order using series expansions, it is common and more efficient to expand the solution with fractional functions of the form . Rawashdeh [14] has implemented the Legendre wavelets
References
[1]
Y. Luchko, “Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1766–1772, 2010.
[2]
J. He, “Some applications of nonlinear fractional differential equations and their approximations,” Bulletin of Science Technology & Society, vol. 15, pp. 86–90, 1999.
[3]
K. Moaddy, S. Momani, and I. Hashim, “The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1209–1216, 2011.
[4]
M. Inc, “The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method,” Journal of Mathematical Analysis and Applications, vol. 345, no. 1, pp. 476–484, 2008.
[5]
K. Al-Khaled and S. Momani, “An approximate solution for a fractional diffusion-wave equation using the decomposition method,” Applied Mathematics and Computation, vol. 165, no. 2, pp. 473–483, 2005.
[6]
N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,” Physics Letters A, vol. 371, no. 1-2, pp. 26–33, 2007.
[7]
L. Song and H. Zhang, “Application of homotopy analysis method to fractional KdV-Burgers-KURamoto equation,” Physics Letters A, vol. 367, no. 1-2, pp. 88–94, 2007.
[8]
H. Jafari and V. Daftardar-Gejji, “Solving linear and nonlinear fractional diffusion and wave equations by Adomian decomposition,” Applied Mathematics and Computation, vol. 180, no. 2, pp. 488–497, 2006.
[9]
C. Yang and J. Hou, “An approximate solution of nonlinear fractional differential equation by Laplace transform and Adomian polynomials,” Journal of Information and Computational Science, vol. 10, no. 1, pp. 213–222, 2013.
[10]
M. Al-Refai and M. Ali Hajji, “Monotone iterative sequences for nonlinear boundary value problems of fractional order,” Nonlinear Analysis A: Theory, Methods and Applications, vol. 74, no. 11, pp. 3531–3539, 2011.
[11]
M. Syam and M. Al-Refai, “Positive solutions and monotone iterative sequences for a class of higher order boundary value problems,” Journal of Fractional Calculus and Applications, vol. 4, no. 14, pp. 1–13, 2013.
[12]
S. Kazem, S. Abbasbandy, and S. Kumar, “Fractional-order Legendre functions for solving fractional-order differential equations,” Applied Mathematical Modelling: Simulation and Computation for Engineering and Environmental Systems, vol. 37, no. 7, pp. 5498–5510, 2013.
[13]
N. H. Sweilam, M. M. Khader, and A. M. S. Mahdy, “An efficient numerical method for solving the fractional diffusion equation,” Journal of Applied Mathematics and Bioinformatics, vol. 12, no. 1, pp. 1–12, 2011.
[14]
E. A. Rawashdeh, “Legendre wavelets method for fractional integro-differential equations,” Applied Mathematical Sciences, vol. 5, no. 49–52, pp. 2467–2474, 2011.
[15]
Q. M. Al-Mdallal, M. I. Syam, and M. N. Anwar, “A collocation-shooting method for solving fractional boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3814–3822, 2010.
[16]
I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
[17]
Y. Zhang and H. Ding, “Finite difference method for solving the time fractional diffusion equation,” Communications in Computer and Information Science, vol. 325, no. 3, pp. 115–123, 2012.
[18]
Y. Zhang, Z. Sun, and H. Liao, “Finite difference methods for the time fractional diffusion equation on non-uniform meshes,” Journal of Computational Physics, vol. 265, pp. 195–210, 2014.
[19]
M. I. Syam and S. M. Al-Sharo, “Collocation-continuation technique for solving nonlinear ordinary boundary value problems,” Computers & Mathematics with Applications, vol. 37, no. 10, pp. 11–17, 1999.
[20]
M. I. Syam, H. Siyyam, and I. Al-Subaihi, “Tau-Path following method for solving the Riccati equation with fractional order,” Journal of Computational Methods in Physics, vol. 2014, Article ID 207916, 6 pages, 2014.
