An extension of the spectral Tau method for numerical solution of multi-order fractional differential equations with convergence analysis

The main purpose of this paper is to provide an efficient numerical approach for the fractional differential equations (FDEs) based on a spectral Tau method. An extension of the operational approach of the Tau method with the orthogonal polynomial bases is proposed to convert FDEs to its matrix–vector multiplication representation. The fractional derivatives are described in the Caputo sense. The spectral rate of convergence for the proposed method is established in the L2L2 norm. We tested our procedure on several examples and observed that the obtained numerical results confirm the theoretical prediction of the exponential rate of convergence.