Numerical methods for nonlocal and fractional models

Author:

D’Elia Marta,Du Qiang,Glusa Christian,Gunzburger Max,Tian Xiaochuan,Zhou Zhi

Abstract

Partial differential equations (PDEs) are used with huge success to model phenomena across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDEs fail to adequately model observed phenomena, or are not the best available model for that purpose. On the other hand, in many situations,nonlocal modelsthat account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis and of specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modelling and algorithmic extensions, which serve to show the wide applicability of nonlocal modelling.

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics,Numerical Analysis

Reference276 articles.

1. Moving least-square reproducing kernel method Part II: Fourier analysis

2. Optimal distributed control of nonlocal steady diffusion problems;D’Elia;SIAM J. Control Optim.,2014

3. Numerical Methods for the Fractional Laplacian: A Finite Difference-Quadrature Approach

4. On the distribution of first hits for the symmetric stable processes;Blumenthal;Trans. Amer. Math. Soc.,1961

5. A second-order accurate numerical approximation for the fractional diffusion equation

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