A finite difference method for a two-point boundary value problem with a Caputo fractional derivative

A two-point boundary value problem whose highest order term is a Caputo fractional derivative of order δ∈(1, 2) is considered. Al-Refai's comparison principle is improved and modified to fit our problem. Sharp a priori bounds on derivatives of the solution u of the boundary value problem are established, showing that u″(x) may be unbounded at the interval endpoint x=0. These bounds and a discrete comparison principle are used to prove pointwise convergence of a finite difference method for the problem, where the convective term is discretized using simple upwinding to yield stability on coarse meshes for all values of δ. Numerical results are presented to illustrate the performance of the method.