The fractional Chebyshev collocation method for the numerical solution of fractional differential equations with Riemann-Liouville derivatives

The topic of numerical methods for solving fractional differential equations (FDEs) with Riemann-Liouville (RL) derivatives has not received extensive attention compared to the ones for solving FDEs with Caputo derivatives. There is, also, not a sophisticated method to approximate fractional-order derivatives of a function in the sense of RL. In this paper, a new representation of FDEs with fractional-order initial conditions is given, which can be solved in a proposed spectral collocation framework. For this purpose, a new operational matrix of left-sided RL fractional differentiation is constructed to approximate the left-sided RL derivative operator at Chebyshev-Gauss-Lobatto points. In numerical examples, the advantages of using the proposed operational matrix in calculating fractional derivatives of a function or solving FDEs with RL derivatives are illustrated.