Stable evaluations of fractional derivative of the Müntz-Legendre polynomials and application to fractional differential equations

The aim of this paper is to present efficient and stable methods to compute Caputo fractional derivative (CFD) of the Müntz-Legendre polynomials based on three-term recurrence relations and Gauss-Jacobi quadrature rules. This approach with collocation method at Chebyshev-Gauss-Lobatto points has been applied for solving linear and nonlinear fractional multi-order differential equations (FDEs) described in Caputo sense. The main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations. In this work, for the first time, we present the new rates of convergence for projection error which are more accurate than the rate presented by Shen and Wang in Shen and Wang (2016). Moreover, we present convergence rate for spectral collocation method for linear FDEs with initial value on a finite interval and endpoint singularities. Also, we propose an error analysis for Jacobi-Gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper. Finally, the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples.