Convergence analysis of a finite difference scheme for a Riemann-Liouville fractional derivative two-point boundary value problem on an adaptive grid

A two-point boundary value problem whose highest order term is a Riemann-Liouville fractional derivative of order 2 - delta with O < delta < 1 is considered. Such problem is reformulated as a Volterra integral equation of the second kind. An integral discrete scheme is developed for this Volterra integral equation on an adaptive grid that is constructed adaptively from a knowledge of the exact solution. It is shown from a rigorous priori error analysis that the discrete solutions are uniformly convergent with respect to the parameter delta. Besides, in order to establish the parameter of the Volterra integral equation, we construct a nonlinear optimization problem, which is solved by the Nelder-Mead simplex method. Numerical results are given to demonstrate the performance of presented method. (C) 2020 Elsevier B.V. All rights reserved.