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 Petrov-Galerkin Finite Element Method for Fractional Convection-Diffusion Equations

In this work, we develop variational formulations of Petrov-Galerkin type for one-dimensional fractional boundary value problems involving either a Riemann-Liouville or Caputo derivative of order $\alpha\in(3/2, 2)$ in the leading term and both convection and potential terms. They arise in the mathematical modeling of asymmetric super-diffusion processes in heterogeneous media. The well-posedness of the formulations and sharp regularity pickup of the variational solutions are established. A novel finite element method is developed, which employs continuous piecewise linear finite elements and "shifted" fractional powers for the trial and test space, respectively. The new approach has a number of distinct features: It allows deriving optimal error estimates in both $L^2(D)$ and $H^1(D)$ norms; and on a uniform mesh, the stiffness matrix of the leading term is diagonal and the resulting linear system is well conditioned. Further, in the Riemann-Liouville case, an enriched FEM is proposed to improve the convergence. Extensive numerical results are presented to verify the theoretical analysis and robustness of the numerical scheme.Comment: 23 p