# Applications of the Mittag-Leffler law to linear kinetic models & diffusion equations

## TL;DR

This paper explores solutions to fractional derivative models using a modified method, comparing their convergence and behavior with classical methods.

## Contribution

A modified method combining homotopy perturbation and Sumudu transform is introduced for solving fractional diffusion equations.

## Key findings

- The proposed method generates convergent power series solutions for fractional diffusion equations.
- The classical Caputo method converges faster than the Atangana-Baleanu-Caputo method.
- The ABC method may be better suited for non-local behaviors due to its unique properties.

## Abstract

In this paper, we find the solutions to kinetic models and a one-dimensional diffusion equation applied to the Atangana-Baleanu-Caputo fractional derivative (ABCFD). The homotopy perturbation method is combined with the Sumudu transform of the Atangana-Baleanu fractional derivative in the Caputo sense to form a modified technique to solve two and three-dimensional diffusion equations. The technique yields a solution in the form of a power series that contains easily computable terms that converge to the exact solution. We observe that when we increase the number of computing terms, the series becomes closer to the exact solution so that the absolute error between the exact solution and the approximate solution becomes very small. Mathematica software is utilized to perform the graphical representations of the exact and approximate solutions. These solutions are analyzed by manipulating the variables to observe their effect on each other. One of our objectives is to show that if we let the fractional order be one, we always obtain the solution of the traditional derivative. The approximate solutions of the ABCFD models are compared with the approximate solutions of the classical Caputo fractional derivative (CFD). Our results indicate that the classical Caputo method converges to the exact solution more rapidly than the Atangana–Baleanu–Caputo method, suggesting that the classical Caputo method may be more suitable for applications requiring rapid convergence, while the ABC method may be suitable for non-local behaviors and other specific contexts where its unique properties offer advantages.