27,458 research outputs found
Differential quadrature method for space-fractional diffusion equations on 2D irregular domains
In mathematical physics, the space-fractional diffusion equations are of
particular interest in the studies of physical phenomena modelled by L\'{e}vy
processes, which are sometimes called super-diffusion equations. In this
article, we develop the differential quadrature (DQ) methods for solving the 2D
space-fractional diffusion equations on irregular domains. The methods in
presence reduce the original equation into a set of ordinary differential
equations (ODEs) by introducing valid DQ formulations to fractional directional
derivatives based on the functional values at scattered nodal points on problem
domain. The required weighted coefficients are calculated by using radial basis
functions (RBFs) as trial functions, and the resultant ODEs are discretized by
the Crank-Nicolson scheme. The main advantages of our methods lie in their
flexibility and applicability to arbitrary domains. A series of illustrated
examples are finally provided to support these points.Comment: 25 pages, 25 figures, 7 table
Boundary value problems for the diffusion equation of the variable order in differential and difference settings
Solutions of boundary value problems for a diffusion equation of fractional
and variable order in differential and difference settings are studied. It is
shown that the method of energy inequalities is applicable to obtaining a
priori estimates for these problems exactly as in the classical case. The
credibility of the obtained results is verified by performing numerical
calculations for a test problem.Comment: 19 pages. Presented at the 4-th IFAC Workshop on Fractional
Differentiation and Its Applications, Badajoz, Spain, October 18-20, 201
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