Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels

Fátima Cruz,Ricardo Almeida,Natália Martins

doi:10.3934/math.2021315

Fátima Cruz, Ricardo Almeida + Show 1 more

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https://doi.org/10.3934/math.2021315

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Journal: AIMS Mathematics	Publication Date: Jan 1, 2021
Citations: 4	License type: cc-by

Affiliation: University of Aveiro

Abstract

<abstract> In this work we study necessary and sufficient optimality conditions for variational problems dealing with a new fractional derivative. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with arbitrary kernels. After proving a fractional integration by parts formula, we obtain the Euler-Lagrange equation and natural boundary conditions for the fundamental variational problem. Also, fractional variational problems with integral and holonomic constraints are considered. We end with some examples to exemplify our results. </abstract>

Highlights

The fractional calculus is an old subject and presents an extension of ordinary calculus [12, 14]
In this work we study necessary and sufficient optimality conditions for variational problems dealing with a new fractional derivative
One possible way to avoid such issue is to consider a more general class of fractional operators, like, for example, fractional integrals and derivatives with arbitrary kernels [3, 12] or other types of general fractional derivatives [16,17,18,19,20]. Another possible approach to fractional calculus is, instead of fixing the fractional order α, the introduction of a new function that acts like a distribution of the orders of differentiation [9, 10]

Summary

Introduction

The fractional calculus is an old subject and presents an extension of ordinary calculus [12, 14]. One possible way to avoid such issue is to consider a more general class of fractional operators, like, for example, fractional integrals and derivatives with arbitrary kernels [3, 12] or other types of general fractional derivatives [16,17,18,19,20] Another possible approach to fractional calculus is, instead of fixing the fractional order α, the introduction of a new function that acts like a distribution of the orders of differentiation [9, 10]. The main purpose of this paper is to prove optimality conditions for variational problems that depend on distributed-order fractional derivatives with arbitrary kernels. We finalize the paper with some illustrative examples and concluding remarks

Preliminaries

AIMS Mathematics

Main results

Necessary optimality conditions

Sufficient optimality conditions

Illustrative examples

Conclusions and future work