Fractional Laplacian in Bounded Domains

Details Fractional Laplacian in Bounded Domains Fractional Laplacian in Bounded Domains

2007-06-08

arxiv.org/abs/0706.1254v1

PHYSICAL REVIEW E 76, 021116 (2007)

Physics

Condensed Matter

Statistical Mechanics

11 pages, 11 figures

Scientific paper

10.1103/PhysRevE.76.021116

The fractional Laplacian operator, $-(-\triangle)^{\frac{\alpha}{2}}$, appears in a wide class of physical systems, including L\'evy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalues spectrum are also obtained.

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