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2020, Volume 13, Issue 3: 485-501. Doi: 10.3934/dcdss.2020027
This issue Previous Article Models of fluid flowing in non-conventional media: New numerical analysis Next Article New approximate solutions to the nonlinear Klein-Gordon equations using perturbation iteration techniques

Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application

  • Department of Mathematics, College of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

* Permanent address: Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt. Email: Bahaa_gm@yahoo.com

Received:  March 2018
Revised:  May 2018
Published:  December 2019

The author is supported by Taibah University, Dean of Scientific Research

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    Abstract

  • In this paper, the generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's Derivatives are discussed. We consider the Hilfers generalized fractional derivative that in sense Atangana-Baleanu derivatives. We develop integration by parts formulas for the generalized fractional derivatives which are key to developing fractional variational calculus. It is shown that many derivatives used recently and their variational formulations can be obtained by setting different parameters to different values. The fractional Euler-Lagrange equations of fractional Lagrangians for constrained systems contains a fractional Hilfer-Atangana-Baleanu's derivatives with multi parameters are investigated. We also define fractional generalized momenta and provide fractional Hamiltonian formulations in terms of the new generalized derivatives. An example is presented to show applications of the formulations presented here. Some possible extensions of this research are also discussed. We present a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The calculus of variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler-Lagrange equations for the FOCP.

    Mathematics Subject Classification: 26A33, 34A08, 49J20. Secondary: 35R11, 49J15, 49K20, 45D10.

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