Optimality conditions for fractional variational problems with free terminal time

This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided.

Download Full-text

Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics

Abstract and Applied Analysis ◽

10.1155/2012/871912 ◽

2012 ◽

Vol 2012 ◽

pp. 1-24 ◽

Cited By ~ 28

Author(s):

Tatiana Odzijewicz ◽

Agnieszka B. Malinowska ◽

Delfim F. M. Torres

Keyword(s):

Fractional Integral ◽

Necessary Optimality Conditions ◽

Variational Problems ◽

Fractional Integrals ◽

Natural Boundary ◽

Free Boundary Value Problems ◽

Fractional Variational Problems ◽

Generalized Fractional Integral ◽

Fractional Calculus Of Variations ◽

Natural Boundary Conditions

We study fractional variational problems in terms of a generalized fractional integral with Lagrangians depending on classical derivatives, generalized fractional integrals and derivatives. We obtain necessary optimality conditions for the basic and isoperimetric problems, as well as natural boundary conditions for free-boundary value problems. The fractional action-like variational approach (FALVA) is extended and some applications to physics discussed.

Download Full-text

Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability

Journal of Function Spaces ◽

10.1155/2018/2740678 ◽

2018 ◽

Vol 2018 ◽

pp. 1-15 ◽

Cited By ~ 1

Author(s):

Jianke Zhang ◽

Gaofeng Wang ◽

Xiaobin Zhi ◽

Chang Zhou

Keyword(s):

Fractional Derivative ◽

Lagrange Function ◽

Variational Problems ◽

Sufficient Optimality Conditions ◽

Lagrange Equations ◽

Hukuhara Difference ◽

Generalized Hukuhara Difference ◽

Fractional Variational Problems ◽

Fractional Calculus Of Variations ◽

Necessary And Sufficient

We study in this paper the Atangana-Baleanu fractional derivative of fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-Atangana-Baleanu fractional differentiability, we prove the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function. The new kernel of gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.

Download Full-text

New Numerical Approach for Fractional Variational Problems Using Shifted Legendre Orthonormal Polynomials

Journal of Optimization Theory and Applications ◽

10.1007/s10957-016-0886-1 ◽

2016 ◽

Vol 174 (1) ◽

pp. 295-320 ◽

Cited By ~ 32

Author(s):

Samer S. Ezz-Eldien ◽

Ramy M. Hafez ◽

Ali H. Bhrawy ◽

Dumitru Baleanu ◽

Ahmed A. El-Kalaawy

Keyword(s):

Variational Problems ◽

Numerical Approach ◽

Orthonormal Polynomials ◽

Fractional Variational Problems

Download Full-text

An Euler-Lagrange Equation only Depending on Derivatives of Caputo for Fractional Variational Problems with Classical Derivatives

Statistics Optimization & Information Computing ◽

10.19139/soic-2310-5070-865 ◽

2020 ◽

Vol 8 (2) ◽

pp. 590-601

Author(s):

Melani Barrios ◽

Gabriela Reyero

Keyword(s):

Exact Solution ◽

Variational Problem ◽

Lagrange Equation ◽

Integral Form ◽

Variational Problems ◽

The Other ◽

Isoperimetric Problems ◽

Caputo Derivatives ◽

Fractional Variational Problems ◽

Derivatives Of

In this paper we present advances in fractional variational problems with a Lagrangian depending on Caputofractional and classical derivatives. New formulations of the fractional Euler-Lagrange equation are shown for the basic and isoperimetric problems, one in an integral form, and the other that depends only on the Caputo derivatives. The advantage is that Caputo derivatives are more appropriate for modeling problems than the Riemann-Liouville derivatives and makes the calculations easier to solve because, in some cases, its behavior is similar to the behavior of classical derivatives. Finally, anew exact solution for a particular variational problem is obtained.

Download Full-text