New Derivatives on Fractal Subset of Real-line

TL;DR

This paper introduces new fractional derivatives and special functions on fractal sets, enabling better modeling of processes with memory effects on fractal subsets of the real line.

Contribution

It develops generalized fractional derivatives and functions on fractals, and applies non-local Laplace transforms to solve fractal differential equations, advancing mathematical tools for fractal analysis.

Findings

Defined fractional derivatives on fractals.

Applied Laplace transform to fractal equations.

Enhanced modeling of memory processes on fractals.

Abstract

In this manuscript we introduced the generalized fractional Riemann-Liouville and Caputo like derivative for functions defined on fractal sets. The Gamma, Mittag-Leffler and Beta functions were defined on the fractal sets. The non-local Laplace transformation is given and applied for solving linear and non-linear fractal equations. The advantage of using these new nonlocal derivatives on fractals subset of real-line lies in the fact that they are used for better modelling of processes with memory effect.