# The Mellin Transform and Non-local Derivatives of Fractal Calculus

## TL;DR

This paper introduces fractal calculus tools, including Mellin transforms and non-local derivatives, to analyze fractal curves and solve related differential equations, expanding the mathematical framework for fractal analysis.

## Contribution

It defines fractal analogues of classical calculus operators and transforms, enabling the solution of fractal non-local differential equations.

## Key findings

- Defined fractal Mellin and non-local transforms.
- Established methods to solve fractal differential equations.
- Provided examples illustrating the application of these tools.

## Abstract

In this paper, the fractal calculus of fractal sets and fractal curves are compared. The analogues of the Riemann-Liouville and the Caputo integrals and derivatives are defined for the fractal curves which are non-local derivatives. The analogous for the fractional Laplace concepts are defined to solve fractal non-local differential equations on fractal curves. The fractal local Mellin and fractal non-local transforms are defined to solve fractal differential equations. We present tables and examples to illustrate the results.