APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES

APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES

N. Derdar

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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.

Keywords

ψ-Caputo fractional derivative, Approximation, Rectangle formula, Trapezoidal formula, Simpson’s formula, Error estimate.