A survey on fractional variational calculus

TL;DR

This survey reviews the main results and techniques in fractional calculus of variations, focusing on Caputo derivatives, optimality conditions, and approximation methods for solving related variational problems.

Contribution

It provides a comprehensive overview of the field, including new necessary optimality conditions and approximation techniques for fractional variational problems.

Findings

Derived Euler-Lagrange type conditions for fractional variational problems

Developed approximation methods using truncated Grünwald–Letnikov derivatives

Unified treatment of fundamental, higher-order, and isoperimetric problems

Abstract

Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary optimality conditions of Euler-Lagrange type for the fundamental, higher-order, and isoperimetric problems, and compute approximated solutions based on truncated Gr\"{u}nwald--Letnikov approximations of Caputo derivatives.