%0 Journal Article
%T Solving Fractional Diffusion Equation via the Collocation Method Based on Fractional Legendre Functions
%A Muhammed Syam
%A Mohammed Al-Refai
%J Journal of Computational Methods in Physics
%D 2014
%R 10.1155/2014/381074
%X 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¨C14]. 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
%U http://www.hindawi.com/journals/jcmp/2014/381074/