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On Shifted Jacobi Spectral Approximations For Solving Fractional Differential Equations

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2013

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Elsevier Science

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Abstract

In this paper, a new formula of Caputo fractional-order derivatives of shifted Jacobi polynomials of any degree in terms of shifted Jacobi polynomials themselves is proved. We discuss a direct solution technique for linear multi-order fractional differential equations (FDEs) subject to nonhomogeneous initial conditions using a shifted Jacobi tau approximation. A quadrature shifted Jacobi tau (Q-SJT) approximation is introduced for the solution of linear multi-order FDEs with variable coefficients. We also propose a shifted Jacobi collocation technique for solving nonlinear multi-order fractional initial value. problems. The advantages of using the proposed techniques are discussed and we compare them with other existing methods. We investigate some illustrative examples of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques. (C) 2013 Elsevier Inc. All rights reserved.

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Keywords

Multi-Term Fractional Differential Equations, Nonlinear Fractional Initial Value Problems, Spectral Methods, Shifted Jacobi Polynomials, Jacobi-Gauss-Lobatto Quadrature, Caputo Derivative

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Source

Applied Mathematics and Computation

Volume

219

Issue

15

Start Page

8042

End Page

8056