Numerical Solution to Anomalous Diffusion Equations for Levy Walks

Viacheslav V Saenko,Ruslan V Fedorov,Vladislav N Kovalnogov,Yuri E Chamchiyan

doi:10.3390/math9243219

Viacheslav V Saenko, Ruslan V Fedorov + Show 2 more

Open Access

https://doi.org/10.3390/math9243219

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Journal: Mathematics	Publication Date: Dec 13, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Ulyanovsk State Technical University

Abstract

The process of Levy random walks is considered in view of the constant velocity of a particle. A kinetic equation is obtained that describes the process of walks, and fractional differential equations are obtained that describe the asymptotic behavior of the process. It is shown that, in the case of finite and infinite mathematical expectation of paths, these equations have a completely different form. To solve the obtained equations, the method of local estimation of the Monte Carlo method is described. The solution algorithm is described and the advantages and disadvantages of the considered method are indicated.

Highlights

At present, the theory of anomalous diffusion is rarely used to describe the combustion processes of a substance, there are all the prerequisites for this
The diffusion packet width ∆(t) stops obeying the law ∆(t) ∝ tγ with an exponent γ = 1/2 and starts growing with time by the law with an exponent γ 6= 1/2, which testifies to the appearance of anomalous diffusion
Signs of the appearance of anomalous diffusion at thermal transport in a low-dimensional system are indicated in the papers [1,2]

Summary

Introduction

The theory of anomalous diffusion is rarely used to describe the combustion processes of a substance, there are all the prerequisites for this. The work [34] examines the influence of the final velocity on the spatial distribution of particles in Levy walks with exponential traps In this work, it Mathematics 2021, 9, 3219 was found that, in the case 1 < α 6 2, taking account of the constant velocity of particle motion is reduced to a decrease in the diffusion coefficient in the equation of anomalous diffusion (2). It was shown that, in the case of an infinite mathematical expectation of the distribution of paths, to take account of the finite velocity, it is necessary to replace the fractional Laplacian in the equation of anomalous diffusion by a material derivative of a fractional order.

Kinetic Equation of the Random Walk Process

Asymptotic Solution to a Kinetic Equation

Numerical Solution to a Kinetic Equation

Conclusions