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Research Article | Open Access

Geometric interpretation of Caputo's fractional derivative and its relation to initial value problems of integer and fractional order

Franco Rubio-López1( ), Dennis Quispe-Sánchez2, Obidio Rubio1
Research Group on Complex Systems and Scientific Computing, Department of Mathematics, National University of Trujillo, Trujillo 13011, Peru
Departamento de Ciencias, Universidad Privada del Norte, UPN, Sede San Isidro, Av. El Ejercito 920, Trujillo, Peru
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Abstract

In this paper, we give a new geometric interpretation to Caputo's fractional derivative, based on the fundamental theorem of plane curves and the fractional curvature of plane curves introduced by Rubio-López et al. in 2023. Furthermore, our results are related to initial value problems of integer and fractional order. Finally, some examples related to differential geometry and viscoelasticity theory are given.

Keywords

curvature of curvefractional curvatureCaputo's fractional derivativegeometric interpretationFrenet-Serret equations
CLC number: 26A33, 53B20

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AIMS Mathematics
Volume 11 Issue 4,
April 2026
Pages 10831-10856
DOI: 10.3934/math.2026445

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Rubio-López F, Quispe-Sánchez D, Rubio O. Geometric interpretation of Caputo's fractional derivative and its relation to initial value problems of integer and fractional order. AIMS Mathematics, 2026, 11(4): 10831-10856. https://doi.org/10.3934/math.2026445

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Received: 15 February 2026
Revised: 01 April 2026
Accepted: 08 April 2026
Published: 20 April 2026
©2026 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)