Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations

In this study, we propose shifted fractional-order Jacobi orthogonal functions (SFJEs) based on the definition of the classical Jacobi polynomials. We derive a new formula that explicitly expresses any Caputo fractional-order derivatives of SEJFs in terms of the SFJEs themselves. We also propose a shifted fractional-order Jacobi tau technique based on the derived fractional-order derivative formula of SFJEs for solving Caputo type fractional differential equations (FDEs) of order nu (0 < nu < 1). A shifted fractional-order Jacobi pseudo-spectral approximation is investigated for solving the nonlinear initial value problem of fractional order nu. An extension of the fractional-order Jacobi pseudo-spectral method is given to solve systems of FDEs. We describe the advantages of using the spectral schemes based on SFJEs and we compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and efficiency of the proposed techniques. (C) 2015 Elsevier Inc. All rights reserved.