An efficient numerical method for a two-point boundary value problem with a Caputo fractional derivative

In this paper a two-point boundary value problem with a Caputo fractional derivative is considered. By using a shooting method based on the secant iterative method, the boundary value problem is turned into an initial value problem. Then the initial value problem is transformed into an equivalent integral-differential equation with a weakly singular kernel. An integral discretization scheme on the uniform mesh is developed to approximate the integral-differential equation. By applying the truncation error estimate techniques and a discrete analogue of Gronwall's inequality, it is proved that the numerical scheme is first-order convergent in the discrete maximum norm. Numerical experiments verify the theoretical results.