Approximate Solution of Fractional Differential Equation by Quadratic Splines

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<2 and 0<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 alpha, the involved undetermined parameters in the quadratic spline interpolation function can be exactly resolved.

Automated summary

This article considers the approximate solutions using quadratic splines for a fractional differential equation with two Caputo fractional derivatives. Numerical computing schemes for the derivatives are derived based on quadratic spline interpolation function, and the recursion scheme and approximate solution are generated. The proposed method is validated through numerical examples and compared with L1-L2 numerical solutions.