On nth Level Fractional Derivatives: An Equivalent Representation and Applications to Inverse Problem

TL;DR

This paper introduces an equivalent representation for nth level fractional derivatives, generalizes the concept, and applies it to solve an inverse diffusion problem involving 2nd level fractional derivatives.

Contribution

It provides a generalized representation of nth level fractional derivatives and demonstrates its application in solving inverse problems in diffusion equations.

Findings

Equivalent representation of 2nd level fractional derivative in terms of Riemann-Liouville derivative

Generalization to nth level fractional derivatives

Successful solution of an inverse diffusion problem using the new representation

Abstract

This work contributes to the theory of nth level fractional derivative, where $n$ is a positive integer. An equivalent representation of 2nd level fractional derivative in terms of Riemann-Liouville fractional derivative is presented. We generalized our result and provide representation of nth level fractional derivative. As an application, we solve an inverse problem defined for a diffusion equation involving 2nd level fractional derivative.