Discrete Chebyshev Polynomials for Solving Fractional Variational Problems

In ‎the current study, a‎ general formulation of the discrete Chebyshev polynomials is given. ‎The operational matrix of fractional integration for these discrete polynomials is also derived. ‎Then,‎ a numerical scheme based on the discrete Chebyshev polynomials and their operational matrix has been developed to solve fractional variational problems‎. In this method, the need for using Lagrange multiplier during the solution procedure is eliminated.‎ The performance of the proposed scheme is validated through some illustrative examples. ‎Moreover, ‎the obtained numerical results ‎‎‎‎were compared to the previously acquired results by the classical Chebyshev polynomials. Finally, a comparison for the required CPU time is presented, which indicates more efficiency and less complexity of the proposed method.

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

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.

Efficiency for Vector Variational Quotient Problems with Curvilinear Integrals on Riemannian Manifolds via Geodesic Quasiinvexity

In the paper, we analyze the necessary efficiency conditions for scalar, vectorial and vector fractional variational problems using curvilinear integrals as objectives and we establish sufficient conditions of efficiency to the above variational problems. The efficiency sufficient conditions use of notions of the geodesic invex set and of (strictly, monotonic) ( ρ , b)-geodesic quasiinvex functions.

Modified wavelet method for solving fractional variational problems

In this article, a newly modified Bessel wavelet method for solving fractional variational problems is considered. The modified operational matrix of integration based on Bessel wavelet functions is proposed for solving the problems. In the process of computing this matrix, we have tried to provide a high-accuracy operational matrix. We also introduce the pseudo-operational matrix of derivative and the dual operational matrix with the coefficient. Also, we investigate the error analysis of the computational method. In the examples section, the behavior of the approximate solutions with respect to various parameters involved in the construction method is tested to illustrate the efficiency and accuracy of the proposed method.

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

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.

Duality for a class of second order symmetric nondifferentiable fractional variational problems

The present work frames a pair of symmetric dual problems for second order nondifferentiable fractional variational problems over cone constraints with the help of support functions. Weak, strong and converse duality theorems are derived under second order F-convexity assumptions. By removing time dependency, static case of the problem is obtained. Suitable numerical example is constructed.

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

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.