The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation
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RESEARCH
Open Access
The operational matrix formulation of the Jacobi tau approximation for space fractional diffusion equation Eid H Doha1 , Ali H Bhrawy2,3* , Dumitru Baleanu4,5,6 and Samer S Ezz-Eldien7 *
Correspondence: [email protected] 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia 3 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt Full list of author information is available at the end of the article
Abstract In this article, an accurate and efficient numerical method is presented for solving the space-fractional order diffusion equation (SFDE). Jacobi polynomials are used to approximate the solution of the equation as a base of the tau spectral method which is based on the Jacobi operational matrices of fractional derivative and integration. The main advantage of this method is based upon reducing the nonlinear partial differential equation into a system of algebraic equations in the expansion coefficient of the solution. In order to test the accuracy and efficiency of our method, the solutions of the examples presented are introduced in the form of tables to make a comparison with those obtained by other methods and with the exact solutions easy. Keywords: multi-term fractional differential equations; fractional diffusion equations; tau method; shifted Jacobi polynomials; operational matrix; Caputo derivative
1 Introduction Due to their accuracy, fractional partial differential equations (PDEs) in the description of nonlinear phenomena in engineering, physics, viscoelasticity, fluid mechanics, biology, and other areas of science have got a lot of attention in recent years [–]. There are many advantages of fractional derivatives, one of them is that it can be seen as a set of ordinary derivatives that give the fractional derivatives the ability to describe what integer-order derivatives cannot []. One may refer for the historical development of fractional operators to [, ]. The tau spectral method is one of the most important methods that have been used to find numerical solutions of differential equations. Expressing the solution as an expansion of certain orthogonal polynomials and then choosing the coefficients in the expansion in order to satisfy the differential equation as accurately as possible is the main idea of the spectral methods. In a lot of papers there have been proposals for solving the multi-term fractional differential equations such as the Haar wavelet [, ], the homotopy analysis method [, ], the Legendre wavelet method [], the homotopy-perturbation method [, ], and the variational iteration method []. Spectral methods have been used to introduce approximate solutions for the fractional differential equations based on collocation and tau methods, see [, ]. Moreover, spectral methods are applied with the help of the operational matrix © 2014 Doha et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecomm
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