APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES

Year 2026, Volume: 16 Issue: 1, 48 - 72, 08.01.2026

Nedjemeddine Derdar

Abstract

The main objective of this paper is to find numerical approximations of the Caputo fractional derivative for $\alpha>0$ 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 $O(h)$, $O(h^{2})$ and $O(h^{3})$, respectively.

Keywords

$\psi$-Caputo fractional derivative , Approximation , Rectangle formula , Trapezoidal formula , Simpson's formula , Error estimate

Thanks

The author are very grateful to the anonymous reviewers for their useful comments that led to improvements of the original manuscript.

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