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International Journal for Multiscale Computational Engineering

Impact factor: 0.760

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2011002402
pages 661-674


Nathanial Burch
Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523, USA
Richard Lehoucq
Applied Math and Applications, Sandia National Laboratories, Albuquerque, New Mexico 87185, USA


The purpose of this paper is to compare the solutions of one-dimensional boundary value problems corresponding to classical, fractional, and nonlocal diffusion on bounded domains. The latter two diffusions are viable alternatives for anomalous diffusion when Fick's first law is an inaccurate model. In the case of nonlocal diffusion, a generalization of Fick's first law in terms of a nonlocal flux is demonstrated to hold. A relationship between nonlocal and fractional diffusion is also reviewed, where the order of the fractional Laplacian can lie in the interval (0, 2]. The contribution of this paper is to present boundary value problems for nonlocal diffusion including a variational formulation that leads to a conforming finite-element method using piecewise discontinuous shape functions. The nonlocal Dirichlet and Neumann boundary conditions used represent generalizations of the classical boundary conditions. Several examples are given where the effect of nonlocality is studied. The relationship between nonlocal and fractional diffusion explains that the numerical solution of boundary value problems, where the order of the fractional Laplacian can lie in the interval (0, 2], is possible.


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