Numerical Scheme for a Quadratic Type Generalized Isoperimetric Constraint Variational Problems With A Operator

Abstract

This paper presents a numerical scheme for a class of isoperimetric constraint variational problems (ICVPs) defined in terms of an Aoperator introduced recently. In this scheme, Bernstein's polynomials are used to approximate the desired function and to reduce the problem from a functional space to an eigenvalue problem in a finite dimensional space. Properties of the eigenvalues and eigenvectors of this problem are used to obtain approximate solutions to the problem. Results for two examples are presented to demonstrate the effectiveness of the proposed scheme. In special cases, the Aoperator reduces to Riemann–Liouville, Caputo, Riesz–Riemann–Liouville, and Riesz–Caputo, and several other fractional derivatives defined in the literature. Thus, the approach presented here provides a general scheme for ICVPs defined using different types of fractional derivatives. Although, only Bernstein's polynomials are used here to approximate the solutions, many other approximation schemes are possible. Effectiveness of these approximation schemes will be presented in the future. While the presented numerical scheme is applied to a quadratic type generalized ICVPs, it can also be applied to other types of problems.