Weighted Generalized Fractional Integration by Parts and the Euler-Lagrange Equation

TL;DR

This paper introduces a new weighted generalized fractional derivative with an associated integration by parts formula, leading to an extended Euler-Lagrange equation applicable in dynamic optimization and quantum mechanics.

Contribution

It constructs a novel right-weighted generalized fractional derivative in the Riemann-Liouville sense and derives an extended Euler-Lagrange equation for variational problems.

Findings

Derived an integration by parts formula for the new fractional derivative.

Extended the Euler-Lagrange equation to include weighted generalized fractional derivatives.

Applied the framework to a quantum mechanics variational problem.

Abstract

Integration by parts plays a crucial role in mathematical analysis, e.g., during the proof of necessary optimality conditions in the calculus of variations and optimal control. Motivated by this fact, we construct a new, right-weighted generalized fractional derivative in the Riemann-Liouville sense with its associated integral for the recently introduced weighted generalized fractional derivative with Mittag-Leffler kernel. We rewrite these operators equivalently in effective series, proving some interesting properties relating to the left and the right fractional operators. These results permit us to obtain the corresponding integration by parts formula. With the new general formula, we obtain an appropriate weighted Euler-Lagrange equation for dynamic optimization, extending those existing in the literature. We end with the application of an optimization variational problem to the quantum mechanics framework.