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On shifted Jacobi spectral approximations for solving fractional differential equations. (English) Zbl 1291.65207

Summary: 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.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A08 Fractional ordinary differential equations
34A30 Linear ordinary differential equations and systems
34A34 Nonlinear ordinary differential equations and systems
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations

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