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

摘要

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.