scholarly journals Optimality conditions for fractional variational problems with Caputo-Fabrizio fractional derivatives

2017 ◽  
Vol 2017 (1) ◽  
Author(s):  
Jianke Zhang ◽  
Xiaojue Ma ◽  
Lifeng Li
Keyword(s):  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Fractional Variational Problems

scholarly journals New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 592
Author(s):  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Lagrange Function ◽  
Sufficient Conditions ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Final State ◽  
Fractional Variational Problems ◽  
Generalized Fractional Derivatives ◽  
Arbitrary Kernel

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.

scholarly journals Fractional Variational Problems Depending on Fractional Derivatives of Differentiable Functions with Application to Nonlinear Chaotic Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Matheus Jatkoske Lazo
Keyword(s):  
Dynamical Systems ◽  
Fractional Derivatives ◽  
Chaotic Systems ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Necessary Condition ◽  
Chaotic Dynamical Systems ◽  
Differentiable Functions ◽  
Fractional Variational Problems ◽  
Derivatives Of

We formulate a necessary condition for functionals with Lagrangians depending on fractional derivatives of differentiable functions to possess an extremum. The Euler-Lagrange equation we obtained generalizes previously known results in the literature and enables us to construct simple Lagrangians for nonlinear systems. As examples of application, we obtain Lagrangians for some chaotic dynamical systems.

scholarly journals Fractional Variational Problems of Euler-Lagrange Equations with Holonomic Constrained Systems

2016 ◽  
Vol 8 (3) ◽  
pp. 60 ◽  
Author(s):  
Eyad Hasan Hasan
Keyword(s):  
Variational Problem ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Constrained Systems ◽  
Lagrange Equations ◽  
Integer Order ◽  
Fractional Variational Problems ◽  
Fractional Variational Problem ◽  
Problem Of Lagrange

<p class="1Body">In this paper, we examined the fractional Euler-Lagrange equations for Holonomic constrained systems. The Euler-Lagrange equations are derived using the fractional variational problem of Lagrange. In addition, we achieved that the classical results were obtained are agreement when fractional derivatives are replaced with the integer order derivatives. Two physical examples are discussed to demonstrate the formalism.</p>

A Development in the Extended Ritz Method for Solving a General Class of Fractional Variational Problems

2019 ◽  
Vol 15 (2) ◽  
Author(s):  
Ali Lotfi
Keyword(s):  
General Class ◽  
Fractional Derivatives ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Polynomial Functions ◽  
Great Flexibility ◽  
Fractional Polynomial ◽  
Fractional Variational Problems ◽  
Free Parameters

Abstract In this paper, based on the idea of the extended Ritz method, we introduce an efficient approximate technique for solving a general class of fractional variational problems. In the discussed problem, the fractional derivatives are considered in the Caputo sense. First, we introduce a family of fractional polynomial functions with a free parameter in the exponent. With the aid of the presented fractional polynomials, we construct a family of functions with free parameters, which provides the extended Ritz method with a great flexibility in searching for the approximate solution of the problem. The approximate solutions satisfy all the initial and the boundary conditions of the problem. The convergence of the method is analytically studied and some test examples are included to demonstrate the superiority of the new technique over the ordinary Ritz method.

Comparison of Four Numerical Schemes for Isoperimetric Constraint Fractional Variational Problems With A-Operator

2015 ◽  
Author(s):  
Rajesh K. Pandey ◽  
Om P. Agrawal
Keyword(s):  
Fractional Derivatives ◽  
Approximate Solutions ◽  
Variational Problems ◽  
Numerical Schemes ◽  
Linear Quadratic ◽  
Step Size ◽  
Special Cases ◽  
Fractional Variational Problems ◽  
Isoperimetric Constraint ◽  
Fractional Orders

This paper presents a comparative study of four numerical schemes for a class of Isoperimetric Constraint Fractional Variational Problems (ICFVPs) defined in terms of an A-operator introduced recently. The A-operator is defined in a more general way which in special cases reduces to Riemann-Liouville, Caputo, Riesz-Riemann-Liouville and Riesz-Caputo, and several other fractional derivatives defined in the literature. Four different schemes, namely linear, quadratic, quadratic-linear and Bernsteins polynomials approximations, are used to obtain approximate solutions of an ICFVP. All four schemes work well, and when the number of terms approximating the solution are increased, the desired solution is achieved. Results for a modified power kernel in A-operator for different fractional orders are presented to demonstrate the effectiveness of the proposed schemes. The accuracy of the numerical schemes with respect to parameters such as fractional order α and step size h are analyzed and illustrated in detail through various figures and tables. Finally, comparative performances of the schemes are discussed.

Sign in / Sign up

Export Citation Format

Share Document