# Fractional Herglotz variational problems of variable order

## TL;DR

This paper develops necessary optimality conditions for fractional variational problems of Herglotz type with variable order, extending classical calculus of variations to fractional derivatives of variable order in multiple variables.

## Contribution

It introduces a new framework for fractional Herglotz variational problems of variable order, deriving optimality conditions involving fractional differential equations.

## Key findings

- Derived fractional differential equations for optimality conditions.
- Extended results to multiple independent variables.
- Provided illustrative examples demonstrating the theory.

## Abstract

We study fractional variational problems of Herglotz type of variable order. Necessary optimality conditions, described by fractional differential equations depending on a combined Caputo fractional derivative of variable order, are proved. Two different cases are considered: the fundamental problem, with one independent variable, and the general case, with several independent variables. We end with some illustrative examples of the results of the paper.