On a New Class of Fractional Calculus of Variations and Related Fractional Differential Equations

TL;DR

This paper introduces a novel fractional calculus of variations framework based on weak fractional derivatives, leading to new types of fractional differential equations and boundary conditions, with established well-posedness and regularity results.

Contribution

It develops a new fractional calculus of variations using weak fractional derivatives and Sobolev spaces, expanding the theory beyond classical fractional derivatives.

Findings

Established well-posedness of fractional variational problems.

Derived regularity results for fractional differential equations.

Introduced new boundary operators for fractional problems.

Abstract

This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. The primary objective of this paper is to establish the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.