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

A New Parameter In Accretion Disk Model

Taking optically thin accretion flows as an example, we investigate the dynamics and the emergent spectra of accretion flows with different outer boundary conditions (OBCs) and find that OBC plays an important role in accretion disk model. This is because the accretion equations describing the behavior of accretion flows are a set of {\em differential} equations, therefore, accretion is intrinsically an initial-value problem. We argue that optically thick accretion flow should also show OBC-dependent behavior. The result means that we should seriously consider the initial physical state of the accretion flow such as its angular momentum and its temperature. An application example to Sgr A$^*$ is presented.Comment: 6 pages, 4 figures, to appear in the Proceeding of "Pacific Rim Conference on Stellar Astrophysics", Aug. 1999, HongKong, Chin