Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator

Shivani Sharma,Arun Chaudhary,Mykola Gorbatenko,Igor V Malyk,Ravi Shanker Dubey

doi:10.3390/fractalfract5030064

Shivani Sharma, Arun Chaudhary + Show 3 more

Open Access

https://doi.org/10.3390/fractalfract5030064

Copy DOI

Abstract

In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.

Highlights

We will find the analytical solution of the following non-linear fractional diffusion equation in sense of YAC
The time-fractional nonlinear diffusion equation is taken into consideration in regards to the Yang–Abdel–Cattani fractional derivative operator
The analytical solutions obtained from the α-HATM, in the sense of the YAC derivative operator, are compared to the analytical solutions obtained via the Riemann–Liouville derivative operator and with the exact solution for all the three cases at distinct values of time t, and we observe that the analytical solutions obtained using the YAC derivative operator coincide with the exact solution more closely as compared to the Riemann–Liouville derivative operator when the value of fractional order is close to 1, i.e., γ = 0.99

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. In today’s era, fractional calculus [1,2,3,4,5,6] is emerging as an efficient and powerful tool in the field of science and technology It is the branch of mathematics pertaining to the derivatives and integrals of arbitrary order and it is fruitful in explaining the concepts of damping, wave propagation and diffusion, biology, genetic algorithms, control systems, economy and finance, signal processing, robotics, system identification, electromagnetism, heat transfer, and many more. Dt (w(η, t)) represents the Yang–Abdel–Cattani (YAC) fractional derivative of w(η, t), w is the density of the diffusing medium at point η and at time t This generalized fractional derivative was presented by Yang et al with the Rabotnov exponential function as the non-singular kernel. Graphical representations of the analytical solutions are given for a better understanding of the α-HATM

Definitions

Case 1

Case 2

Case 3

Conclusions