APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES

Nedjemeddine Derdar

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 $\alpha>0$ 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

Thanks

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

References

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Details

Primary Language

English

Subjects

Numerical Solution of Differential and Integral Equations, Ordinary Differential Equations, Difference Equations and Dynamical Systems, Real and Complex Functions (Incl. Several Variables)

Journal Section

Research Article

Authors

Nedjemeddine Derdar ^* This is me
0000-0002-5771-4798
Algeria

Publication Date

January 8, 2026

Submission Date

December 20, 2024

Acceptance Date

May 4, 2025

Published in Issue

Year 2026 Volume: 16 Number: 1

IZ

https://izlik.org/JA26CY46JJ

Cite

RIS / Bibtex

APA

Derdar, N. (2026). APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES. TWMS Journal of Applied and Engineering Mathematics, 16(1), 48-72. https://izlik.org/JA26CY46JJ

AMA

1.Derdar N. APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES. JAEM. 2026;16(1):48-72. https://izlik.org/JA26CY46JJ

Chicago

Derdar, Nedjemeddine. 2026. “APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES”. TWMS Journal of Applied and Engineering Mathematics 16 (1): 48-72. https://izlik.org/JA26CY46JJ.

EndNote

Derdar N (January 1, 2026) APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES. TWMS Journal of Applied and Engineering Mathematics 16 1 48–72.

IEEE

[1]N. Derdar, “APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES”, JAEM, vol. 16, no. 1, pp. 48–72, Jan. 2026, [Online]. Available: https://izlik.org/JA26CY46JJ

ISNAD

Derdar, Nedjemeddine. “APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES”. TWMS Journal of Applied and Engineering Mathematics 16/1 (January 1, 2026): 48-72. https://izlik.org/JA26CY46JJ.

JAMA

1.Derdar N. APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES. JAEM. 2026;16:48–72.

MLA

Derdar, Nedjemeddine. “APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 1, Jan. 2026, pp. 48-72, https://izlik.org/JA26CY46JJ.

Vancouver

1.Nedjemeddine Derdar. APPROXIMATIONS TO CAPUTO FRACTIONAL DERIVATIVE WITH ARBITRARY KERNELS AND UNIFORM MESHES. JAEM [Internet]. 2026 Jan. 1;16(1):48-72. Available from: https://izlik.org/JA26CY46JJ