Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation

Rafał Brociek,Agata Chmielowska,Damian Słota

doi:10.3390/fractalfract4030039

Abstract

This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm’s accuracy.

Highlights

Models with fractional derivatives have found applications in many fields of science and engineering, such as control theory [1], mechanics [2], image processing [3] and heat conduction
The use of the fractional derivatives in modeling the heat conduction phenomena has been presented in the papers [4,5], wherein the models with the fractional derivative of Riemann–Liouville and Caputo type was considered
In case of the heat conduction problems the equations with the fractional derivatives are useful for modeling the phenomena that take place in the porous materials and composites due to the fact that the anomalous diffusion process occurs there

Summary

Introduction

Models with fractional derivatives have found applications in many fields of science and engineering, such as control theory [1], mechanics [2], image processing [3] and heat conduction. In case of the heat conduction problems the equations with the fractional derivatives are useful for modeling the phenomena that take place in the porous materials and composites due to the fact that the anomalous diffusion process occurs there. For such materials the models with the fractional derivatives give better results than the models based on the classic derivatives, as has been shown, for example, in papers [6,7]. More about the applications of the fractional derivatives can be found in [9,10,11,12]

Objectives

Methods

Findings

Conclusion