Convergence analysis of a second-order scheme for fractional differential equation with integral boundary conditions

A fractional differential equation with integral boundary conditions at both the boundary points is considered in this paper. The highest order derivative appears in the fractional differential equation is a Caputo derivative of order a, where 1 < alpha < 2. The Caputo derivative is approximated by a spline method and trapezoidal rule is used to approximate the integral-type boundary conditions. We have proved that the proposed method is of second-order convergent. We have applied the proposed numerical scheme to solve semilinear fractional differential equations. Numerical experiments have been carried out in favor of the scheme.