Caputo-Hadamard fractional derivatives of variable order

TL;DR

This paper introduces three types of Caputo-Hadamard derivatives of variable fractional order, providing their relations, approximation formulas, error estimates, and numerical comparisons to exact derivatives.

Contribution

It presents new definitions of variable-order Caputo-Hadamard derivatives, along with approximation methods and error analysis, advancing fractional calculus theory.

Findings

Derived three types of variable-order Caputo-Hadamard derivatives.

Provided approximation formulas using integer derivatives.

Compared numerical approximations with exact derivatives.

Abstract

In this paper we present three types of Caputo-Hadamard derivatives of variable fractional order, and study the relations between them. An approximation formula for each fractional operator, using integer-order derivatives only, is obtained, and an estimation for the error is given. At the end we compare the exact fractional derivative of a concrete example with some numerical approximations.