Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

Gábor Maros,Ferenc Izsák

doi:10.3390/fractalfract5030075

Abstract

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment.

Highlights

The numerical solution of fractional-order diffusion problems has a number of applications to simulate real-world phenomena such as groundwater flows [1], the dynamics of some proteins [2], geophysical electromagnetics [3], or population dynamics [4]
The objective of this work is to set the theoretical basis of the boundary element method for fractional-order elliptic problems, which can successfully deal with inhomogeneous boundary conditions
Their numerical approximations, which were investigated here, offer efficient the solution is defined everywhere, this corresponds to the non-local nature of these methods

Summary

Introduction

The numerical solution of fractional-order diffusion problems has a number of applications to simulate real-world phenomena such as groundwater flows [1], the dynamics of some proteins [2], geophysical electromagnetics [3], or population dynamics [4]. In the space-fractional case, the core of any investigation is the study of the underlying fractional elliptic problem. This phenomenon ( called anomalous diffusion) can be recognized by the sublinear dependence of the mean-squared displacement of single particles as a function of time. The main difficulty is to deal with these problems on bounded domains. While many different definitions of the full-space fractional Laplacian are equivalent [7], for bounded domains, even in the case of homogeneous boundary conditions, substantially different definitions exist [8].

Objectives

Methods

Conclusion