# Combined Fractional Variational Problems of Variable Order and Some Computational Aspects

## TL;DR

This paper extends fractional variational calculus to include higher-order derivatives and time delays, deriving new optimality conditions and demonstrating computational methods with Chebfun.

## Contribution

It introduces generalized fractional variational problems with variable order derivatives and delays, providing new theoretical optimality conditions and computational approaches.

## Key findings

- Derived higher-order integration by parts formula for variable order Caputo derivatives
- Established necessary optimality and transversality conditions for delayed functionals
- Illustrated results with examples and computational methods using Chebfun

## Abstract

We study two generalizations of fractional variational problems by considering higher-order derivatives and a state time delay. We prove a higher-order integration by parts formula involving a Caputo fractional derivative of variable order and we establish several necessary optimality conditions for functionals containing a combined Caputo derivative of variable fractional order. Because the endpoint is considered to be free, we also deduce associated transversality conditions. In the end, we consider functionals with a time delay and deduce corresponding optimality conditions. Some examples are given to illustrate the new results. Computational aspects are discussed using the open source software package Chebfun.