A Caputo fractional derivative of a function with respect to another function

TL;DR

This paper introduces a Caputo fractional derivative with respect to another function, explores its properties, proposes a numerical approximation method, and demonstrates its application in a population growth model for improved accuracy.

Contribution

It presents a new Caputo-type fractional derivative with respect to another function, along with properties, a numerical method, and an application example.

Findings

The numerical method effectively approximates the fractional derivative.

The derivative's properties include semigroup law and relationships with fractional integrals.

Application to population models shows improved accuracy.

Abstract

In this paper we consider a Caputo type fractional derivative with respect to another function. Some properties, like the semigroup law, a relationship between the fractional derivative and the fractional integral, Taylor's Theorem, Fermat's Theorem, etc, are studied. Also, a numerical method to deal with such operators, consisting in approximating the fractional derivative by a sum that depends on the first-order derivative, is presented. Relying on examples, we show the efficiency and applicability of the method. Finally, an application of the fractional derivative, by considering a Population Growth Model, and showing that we can model more accurately the process using different kernels for the fractional operator is provided.