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On shifted Jacobi spectral approximations for solving fractional differential equations

Published: 01 April 2013 Publication History

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.

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        cover image Applied Mathematics and Computation
        Applied Mathematics and Computation  Volume 219, Issue 15
        April, 2013
        496 pages

        Publisher

        Elsevier Science Inc.

        United States

        Publication History

        Published: 01 April 2013

        Author Tags

        1. Caputo derivative
        2. Jacobi-Gauss-Lobatto quadrature
        3. Multi-term fractional differential equations
        4. Nonlinear fractional initial value problems
        5. Shifted Jacobi polynomials
        6. Spectral methods

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