Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method.

Rasool Shah,Hassan Khan,Muhammad Arif,Saima Mustafa,Poom Kumam

doi:10.3390/e21060557

Abstract

In the present article, fractional-order diffusion equations are solved using the Natural transform decomposition method. The series form solutions are obtained for fractional-order diffusion equations using the proposed method. Some numerical examples are presented to understand the procedure of the Natural transform decomposition method. The Natural transform decomposition method has shown the least volume of calculations and a high rate of convergence compared to other analytical techniques, the proposed method can also be easily applied to other non-linear problems. Therefore, the Natural transform decomposition method is considered to be one of the best analytical technique, to solve fractional-order linear and non-linear partial deferential equations, particularly fractional-order diffusion equation.

Highlights

The idea of fractional calculus and entropy are attractive and more prevalent for investigating the dynamics of complex systems
The results revealed the highest agreement with the exact solutions for the problems
It is investigated that the fractional-order solutions are convergent to the exact solution for the problems as fractional-order approaches integer-order

Summary

Introduction

The idea of fractional calculus and entropy are attractive and more prevalent for investigating the dynamics of complex systems. The idea of entropy for calculating the dynamics of multi-particle systems with integer- and fractional-order behavior was suggested in [3]. Entropy 2019, 21, 557 space-time fractional modified Korteweg–de Vries equations [7], nonlinear partial differential equations, third-order dispersion [8,9] in entropy and convexity, fractional derivative advection-diffusion in two-dimensional semi-conductor systems, and the dynamics of a national soccer league [10]. The exact solution to differential equations (DEs) of fractional-order with mixed partial derivatives [11] and space-fractional diffusion equation and Tsallis relative entropy [12]. We are applying the Natural transform decomposition method (NTDM) to solve the following types of diffusion equations.

Preliminaries

Idea of Fractional Natural Transform Decomposition Method

Example

Conclusions