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.Keywords nonlocal diffusion · fractional diffusion · anomalous diffusion 1 IntroductionThe one-dimensional form of Fick's second law w t = cwxx, c > 0, (1.1) postulates the diffusion in time undergone by the scalar field w representing the particle density. An alternate model for diffusion is of interest when the underlying assumption of classical mass balance and Fick's first law, e.g.,respectively, are questionable. See, for instance, the papers [4,13] for discussions and citations to the literature. One alternative is the anomalous diffusion model given by the fractional diffusion equation v t = −c (−∆) α/2 v, 0 < α ≤ 2, (1.2)
Abstract. The purpose of this paper is to consider the exit-time problem for a finite-range Markov jump process, i.e, the distance the particle can jump is bounded independent of its location. Such jump diffusions are expedient models for anomalous transport exhibiting superdiffusion or nonstandard normal diffusion. We refer to the associated deterministic equation as a volume-constrained nonlocal diffusion equation. The volume constraint is the nonlocal analogue of a boundary condition necessary to demonstrate that the nonlocal diffusion equation is well-posed and is consistent with the jump process. A critical aspect of the analysis is a variational formulation and a recently developed nonlocal vector calculus. This calculus allows us to pose nonlocal backward and forward Kolmogorov equations, the former equation granting the various moments of the exit-time distribution.
This paper investigates the exit-time for a broad class of symmetric finite-range jump processes via the corresponding master equation, a nonlocal diffusion equation suitably constrained. In direct analogy to the classical diffusion equation with a homogeneous Dirichlet boundary condition, the nonlocal diffusion equation is augmented with a homogeneous volume-constraint. The volume-constrained master equation provides an efficient alternative over Monte Carlo simulation for computing an important statistic of the process. Several numerical examples are given.
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