546 research outputs found
Wavelet Galerkin method for fractional elliptic differential equations
Under the guidance of the general theory developed for classical partial
differential equations (PDEs), we investigate the Riesz bases of wavelets in
the spaces where fractional PDEs usually work, and their applications in
numerically solving fractional elliptic differential equations (FEDEs). The
technique issues are solved and the detailed algorithm descriptions are
provided. Compared with the ordinary Galerkin methods, the wavelet Galerkin
method we propose for FEDEs has the striking benefit of efficiency, since the
condition numbers of the corresponding stiffness matrixes are small and
uniformly bounded; and the Toeplitz structure of the matrix still can be used
to reduce cost. Numerical results and comparison with the ordinary Galerkin
methods are presented to demonstrate the advantages of the wavelet Galerkin
method we provide.Comment: 20 pages, 0 figure
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
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