Hilfer-Katugampola fractional derivative

TL;DR

This paper introduces the Hilfer-Katugampola fractional derivative, a new mathematical tool that interpolates various existing derivatives, and demonstrates its application in solving nonlinear fractional differential equations with proven existence and uniqueness of solutions.

Contribution

The paper presents a novel fractional derivative that unifies multiple existing derivatives and applies it to nonlinear differential equations, establishing solution existence and uniqueness.

Findings

The Hilfer-Katugampola derivative interpolates several known fractional derivatives.

The nonlinear fractional differential equation is equivalent to a Volterra integral equation.

Existence and uniqueness of solutions are proven for the initial value problem.

Abstract

We propose a new fractional derivative, the Hilfer-Katugampola fractional derivative. Motivated by the Hilfer derivative this formulation interpolates the well-known fractional derivatives of Hilfer, Hilfer-Hadamard, Riemann-Liouville, Hadamard, Caputo, Caputo-Hadamard, Liouville, Weyl, generalized and Caputo-type. As an application, we consider a nonlinear fractional differential equation with an initial condition using this new formulation. We show that this equation is equivalent to a Volterra integral equation and demonstrate the existence and uniqueness of solution to the nonlinear initial value problem.