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 (SFJFs) based on the definition of the classical Jacobi polynomials. We derive a new formula that explicitly expresses any Caputo fractional-order derivatives of SFJFs in terms of the SFJFs themselves. We also propose a shifted fractional-order Jacobi tau technique based on the derived fractional-order derivative formula of SFJFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). A shifted fractional-order Jacobi pseudo-spectral approximation is investigated for solving the nonlinear initial value problem of fractional order ν. 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 SFJFs 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.