Approximate Solution of Fractional Differential Equation by Quadratic Splines

Abstract

In this article, we consider approximate solutions by quadratic splines for a fractional differential equation with two Caputo fractional derivatives, the orders of which satisfy 1<α<2 and 0<β<1. Numerical computing schemes of the two fractional derivatives based on quadratic spline interpolation function are derived. Then, the recursion scheme for numerical solutions and the quadratic spline approximate solution are generated. Two numerical examples are used to check the proposed method. Additionally, comparisons with the L1–L2 numerical solutions are conducted. For the considered fractional differential equation with the leading order α, the involved undetermined parameters in the quadratic spline interpolation function can be exactly resolved.