Approximations to Caputo fractional derivative with arbitrary kernels and uniform meshes

Abstract

The main objective of this paper is to find numerical approximations of the Caputo fractional derivative for α > 0 with arbitrary kernels and uniform meshes. These numerical approximations are based on polynomial interpolation. Firstly, we derive three numerical formulas: the fractional rectangular formula (FRF), fractional trapezoidal formula (FTF) and fractional Simpson’s formula (FSF). In addition, error estimations for all these rules are analyzed. A test example from the literature is considered to validate the effectiveness of the presented formulas. It is observed that FRF, FTF and FSF yield convergence orders of approximately O(h), O(h²) and O(h³), respectively.