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

Franco Rubio-López; Dennis Quispe-Sánchez; Obidio Rubio; Franco Rubio-López; Dennis Quispe-Sánchez; Obidio Rubio

doi:10.3934/math.2026445

AIMS Mathematics

2026, Volume 11, Issue 4: 10831-10856. doi: 10.3934/math.2026445

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Research article

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

1.
Research Group on Complex Systems and Scientific Computing, Department of Mathematics, National University of Trujillo, Trujillo 13011, Peru
2.
Departamento de Ciencias, Universidad Privada del Norte, UPN, Sede San Isidro, Av. El Ejercito 920, Trujillo, Peru

Received: 15 February 2026 Revised: 01 April 2026 Accepted: 08 April 2026 Published: 20 April 2026
MSC : 26A33, 53B20

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.
- curvature of curve,
- fractional curvature,
- Caputo's fractional derivative,
- geometric interpretation,
- Frenet-Serret equations
Citation: Franco Rubio-López, Dennis Quispe-Sánchez, Obidio Rubio. Geometric interpretation of Caputo's fractional derivative and its relation to initial value problems of integer and fractional order[J]. AIMS Mathematics, 2026, 11(4): 10831-10856. doi: 10.3934/math.2026445

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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.

References

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