CALCULUS OF VARIATIONS WITH COMBINATION OF CLASSICAL AND FRACTIONAL OPERATORS

Abstract

In this paper, we present the problem of generalized fractional calculus of variations. Proposed generalization differs in terms of describing the objective function, which involves a combination of classical and fractional (differential and integral) operators. We obtain the necessary conditions in order to find an extremizer of the problem. Provided examples illustrate fractional Euler-Lagrange equations with noticeable consequences. Additionally, generalized fractional isoperimetric problem is discussed. This paper conjointly presents a formulation of the solution scheme for fractional calculus of variations. Construction of this scheme is in terms of approximating the composition of fractional derivatives. This method shows that the solution of Euler-Lagrange equations can also be obtained by the approximation of the composition of left and right fractional derivatives occurring in fractional Euler-Lagrange equations. Moreover, examples demonstrating the formulation are given with sufficient numerical information.