Central difference approximation of convection in Caputo fractional derivative two-point boundary value problems

Abstract

A rigorous numerical analysis is given for a fractional derivative two-point boundary value problem. The highest-order term of the differential operator is a Caputo fractional derivative of order δ∈(1,2). The second and higher-order derivatives of the solution of the problem are in general unbounded at one end of the interval, which creates difficulties for the analysis. The problem is discretized on an equidistant mesh of diameter h using a standard central difference approximation of the convection term, and nodal convergence of order O(hδ−1) is proved provided that h satisfies a stability condition that depends on δ and the convective coefficient. This condition may be restrictive when δ is near 1; when it is violated the solution may exhibit small oscillations, but these can be removed by a simple and inexpensive postprocessing technique. Numerical results are given to display the performance of the method.