Fractional Variational Problems Research Articles

A hybrid technique for solving fractional delay variational problems by the shifted Legendre polynomials

This study presents a practical method for solving fractional order delay variational problems. The fractional derivative is given in the Caputo sense. The suggested approach is based on the Laplace transform and the shifted Legendre polynomials by approximating the candidate function by the shifted Legendre series with unknown coefficients yet to be determined. The proposed method converts the fractional order delay variational problem into a set of (n ＋ 1) algebraic equations, where the solution to the resultant equation provides us the unknown coefficients of the terminated series that have been utilized to approximate the solution to the considered variational problem. Illustrative examples are given to show that the recommended approach is applicable and accurate for solving such kinds of problems.

Calculus of variations with higher order Caputo fractional derivatives

In this work, we consider fractional variational problems depending on higher order fractional derivatives. We obtain optimality conditions for such problems and we present and discuss some examples. We conclude with possible research directions.

Towards Solving Fractional Order Delay Variational Problems Using Euler Polynomial Operational Matrices

Towards Solving Fractional Order Delay Variational Problems Using Euler Polynomial Operational Matrices

Approximation schemes for a quadratic type generalized isoperimetric constraint fractional variational problems

Approximation schemes for a quadratic type generalized isoperimetric constraint fractional variational problems

On Sufficiency Conditions for Some Robust Variational Control Problems

We study the sufficient optimality conditions for a class of fractional variational control problems involving data uncertainty in the cost functional. Concretely, by using the parametric technique, we prove the sufficiency of the robust necessary optimality conditions by considering convexity, quasi-convexity, strictly quasi-convexity, and/or monotonic quasi-convexity assumptions of the involved functionals.

Lyapunov Inequalities for Two Dimensional Fractional Boundary-Value Problems with Mixed Fractional Derivatives

We consider two types of partial fractional differential equations in two dimensions with mixed fractional derivatives. Appropriate Lyapunov-type inequalities are proved, and applications to the certain eigenvalue problems are presented. Moreover, some connections with the fractional variational problems are highlighted.

Analysis of a Fractional Variational Problem Associated with Cantilever Beams Subjected to a Uniformly Distributed Load

In this paper, we investigate the existence and uniqueness of minimizers of a fractional variational problem generalized from the energy functional associated with a cantilever beam under a uniformly distributed load. We apply the fractional Euler–Lagrange condition to formulate the minimization problem as a boundary value problem and obtain existence and uniqueness results in both L2 and L∞ settings. Additionally, we characterize the continuous dependence of the minimizers on varying loads in the energy functional. Moreover, an approximate solution is derived via the homotopy perturbation method, which is numerically demonstrated in various examples. The results show that the deformations are larger for smaller orders of the fractional derivative.

On efficiency and duality for a class of nonconvex nondifferentiable multiobjective fractional variational control problems

In this paper, we consider the class of nondifferentiable multiobjective fractional variational control problems involving the nondifferentiable terms in the numerators and in the denominators. Under univexity and generalized univexity hypotheses, we prove optimality conditions and various duality results for such nondifferentiable multiobjective fractional variational control problems. The results established in the paper generalize many similar results established earlier in the literature for such nondifferentiable multiobjective fractional variational control problems.

Necessary and sufficient optimality conditions for fractional interval-valued variational problems

In this paper a special kind of variational programming problem involving fractional interval-valued objective function is considered. For such type of problem, insights into LU optimal solutions have been discussed. Using the LU optimal concept, we established optimality conditions for the considered problem. Further, We formulated a Mond-Weir dual problem and discussed appropriate duality theorems for the relationship between dual and primal problems.

FRACTIONAL VARIATIONAL PROBLEM INVOLVING INDEFINITE INTEGRALS AND NONSINGULAR KERNELS

This paper deals with a class of fractional variational problems involving Atangana–Baleanu (AB) derivative. The problem under consideration is a graceful combination of AB derivative with indefinite integral. A proposed numerical technique based on the shifted Vieta–Lucas polynomials is utilized for obtaining the solutions to the given problem. Our numerical process is principally based on the operational matrix (OM) of fractional integral in the AB sense in order to approximate solutions of the problem under study. By utilizing the OM along with the Lagrange multiplier method, the given problems are converted into a problem easier to solve. The resultant problem can then be solved using standard methods. Convergence of the presented method is investigated. In order to ensure and display the efficiency and accuracy of the proposed technique, some examples are given.

Robust Efficiency Conditions in Multiple-Objective Fractional Variational Control Problems

The aim of this study is to investigate multi-dimensional vector variational problems considering data uncertainty in each of the objective functional and constraints. We establish the robust necessary and sufficient efficiency conditions such that any robust feasible solution could be the robust weakly efficient solution for the problems under consideration. Emphatically, we present robust efficiency conditions for multi-dimensional scalar, vector, and vector fractional variational problems by using the notion of a convex functional.

Herglotz Variational Problems Involving Distributed-Order Fractional Derivatives with Arbitrary Smooth Kernels

In this paper, we consider Herglotz-type variational problems dealing with fractional derivatives of distributed-order with respect to another function. We prove necessary optimality conditions for the Herglotz fractional variational problem with and without time delay, with higher-order derivatives, and with several independent variables. Since the Herglotz-type variational problem is a generalization of the classical variational problem, our main results generalize several results from the fractional calculus of variations. To illustrate the theoretical developments included in this paper, we provide some examples.

Rigid plate submerged in a Newtonian fluid and fractional differential equation problems via Caputo fractional derivative

The study of fractional variational problems with derivatives in the sense of Caputo is a recent subject in Newtonian fluids. In this article, the homotopy perturbation method (HPM), the variational iteration method (VIM), and the Akbari–Ganji method (AGM) were used to solve linear fractional differential equations order in fluid mechanics and fractional optimal control problems in the sense of Caputo. Achieving an exact solution to these equations has been one of the researchers’ goals; according to the research results, the proposed methods provide an exact solution. Comparing the results obtained by the proposed methods with those obtained by the Shehu method reveals that the present methods are very effective and convenient in applied fields. Also, the VIM and AGM methods are more proper than HPM in solving these equations. The solutions obtained by the proposed methods agree with the solutions available in the literature.

A Modified Numerical Method Based on Bernstein Wavelets for Numerical Assessment of Fractional Variational and Optimal Control Problems

A Modified Numerical Method Based on Bernstein Wavelets for Numerical Assessment of Fractional Variational and Optimal Control Problems

Numerical approximation of fractional variational problems with several dependent variables using Jacobi poly-fractonomials

We discuss a new numerical scheme using Jacobi poly-fractonomials for the fractional variational problem (FVP) with several dependent variables. The FVP is defined using the Caputo fractional derivative. Using Jacobi poly-fractonomials in the discussed method, the considered FVP is reduced to a system of algebraic equations. By solving this system of algebraic equations, an approximate solution of FVP is accomplished. We also proved the convergence of the presented scheme and the fractional variational error. At last, we perform some figurative examples to exhibit the legitimacy and pertinence of the current method.

An optimum method for fractal–fractional optimal control and variational problems

An optimum method for fractal–fractional optimal control and variational problems

Duality Results for Fractional Variational Problems and Its Application

Duality Results for Fractional Variational Problems and Its Application

Sufficient conditions for extremum of fractional variational problems

Sufficient conditions for extremum of fractional variational problems are formulated with the help of Caputo fractional derivatives. The Euler–Lagrange equation is defined in the Caputo sense and Jacobi conditions are derived using this. Again, Wierstrass integral for the considered functional is obtained from the Jacobi conditions and the transversality conditions. Further, using the Taylor’s series expansion with Caputo fractional derivatives in the Wierstrass integral, the Legendre’s sufficient condition for extremum of the fractional variational problem is established. Finally, a suitable counterexample is presented to justify the efficacy of the fresh findings.

Necessary Conditions to A Fractional Variational Problem

The fractional variational calculus is a recent fifield, where classical variational problems are considered, but in the presence of fractional derivatives. Since there are several defifinitions of fractional derivatives, it is logical to think of different types of optimality conditions. For this reason, in order to solve fractional variational problems, two theorems of necessary conditions are well known: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville fractional derivatives, and other Euler-Lagrange equation that involves only Caputo derivatives. However, it is undecided which of these two methods is convenient to work with. In this article, we make a comparison solving a particular fractional variational problem with both methods to obtain some conclusions about which one gives the optimal solution.

Numerical solution of fractional variational and optimal control problems via fractional-order Chelyshkov functions

<abstract><p>In this paper, we present a new numerical method based on the fractional-order Chelyshkov functions (FCHFs) for solving fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). The fractional derivatives are considered in the Caputo sense. The operational matrix of fractional integral for FCHFs, together with the Lagrange multiplier method, are used to reduce the fractional optimization problem into a system of algebraic equations. Some results concerning the approximation errors are discussed and the convergence of the presented method is also demonstrated. The performance of the introduced method is tested through several examples. Some comparisons with recent numerical methods are introduced to show the accuracy and effectiveness of the presented method.</p></abstract>