Fractional differential equations with dependence on the Caputo-Katugampola derivative

TL;DR

This paper introduces the Caputo-Katugampola derivative, a new fractional operator, and establishes foundational theorems and numerical methods for solving related fractional differential equations.

Contribution

The paper defines a novel fractional derivative, proves existence and uniqueness theorems, and develops a numerical solution approach for equations involving this operator.

Findings

Caputo-Katugampola derivative generalizes existing fractional derivatives.

Existence and uniqueness of solutions are established.

A simple numerical method for solving equations with this derivative is proposed.

Abstract

In this paper we present a new type of fractional operator, the Caputo-Katugampola derivative. The Caputo and the Caputo-Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy type problem, with dependence on the Caputo--Katugampola derivative, is proven. A decomposition formula for the Caputo-Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation.