Computing Hadamard type operators of variable fractional order

TL;DR

This paper introduces a novel numerical method for approximating Hadamard type fractional operators of variable order using series expansions of integer derivatives, with applications demonstrated in differential equations and variational problems.

Contribution

It develops a series representation for Hadamard fractional derivatives and integrals, including a new Hadamard-Marchaud operator, enabling finite sum approximations with error bounds.

Findings

Effective numerical approximation of Hadamard fractional operators.

Application to fractional differential equations and variational problems.

Error bounds for the series approximation.

Abstract

We consider Hadamard fractional derivatives and integrals of variable fractional order. A new type of fractional operator, which we call the Hadamard-Marchaud fractional derivative, is also considered. The objective is to represent these operators as series of terms involving integer-order derivatives only, and then approximate the fractional operators by a finite sum. An upper bound formula for the error is provided. We exemplify our method by applying the proposed numerical procedure to the solution of a fractional differential equation and a fractional variational problem with dependence on the Hadamard-Marchaud fractional derivative.