Variational problems of variable fractional order involving arbitrary kernels

Ricardo Almeida; Ricardo Almeida

doi:10.3934/math.20221028

AIMS Mathematics

2022, Volume 7, Issue 10: 18690-18707. doi: 10.3934/math.20221028

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Research article

Variational problems of variable fractional order involving arbitrary kernels

Ricardo Almeida ^,

Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal

Received: 12 July 2022 Revised: 01 August 2022 Accepted: 11 August 2022 Published: 23 August 2022
MSC : 26A33, 49L99

The aim of this work is to study several problems of the calculus of variations, where the dynamics of the state function is given by a generalized fractional derivative. This derivative combines two well-known concepts: fractional derivative with respect to another function and fractional derivative of variable order. We present the Euler–Lagrange equation, which is a necessary condition that every optimal solution of the problem must satisfy. Other problems are also studied: with integral and holonomic constraints, with higher order derivatives, and the Herglotz variational problem.
- fractional calculus,
- calculus of variations,
- variable-order fractional derivative
Citation: Ricardo Almeida. Variational problems of variable fractional order involving arbitrary kernels[J]. AIMS Mathematics, 2022, 7(10): 18690-18707. doi: 10.3934/math.20221028

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Abstract

The aim of this work is to study several problems of the calculus of variations, where the dynamics of the state function is given by a generalized fractional derivative. This derivative combines two well-known concepts: fractional derivative with respect to another function and fractional derivative of variable order. We present the Euler–Lagrange equation, which is a necessary condition that every optimal solution of the problem must satisfy. Other problems are also studied: with integral and holonomic constraints, with higher order derivatives, and the Herglotz variational problem.

References

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