WITHDRAWN: New fractional operators and application to fractional variational problem

Abstract

Abstract Motivated by Hilfer fractional derivative, a class of generalized fractional integral and derivative operators is discussed. The generalized fractional integrals are the counterparts of generalized fractional derivatives. We study their semigroup property and fractional integration by parts formulae, which play an important role in deducing the Euler–Lagrange equation of extremal problem involving generalized fractional operators. As applications, we discuss a minimizing functional problem and an isoperimetric problem containing these new operators. Numerical simulations based on polynomial approximation and Galerkin approach are presented. Our study shows that the polynomial approximation and Galerkin approach can be easily applied to fractional variational problem depending on both left and right generalized fractional integrals and derivatives.