Radial basis functions method for solving the fractional diffusion equations

Fractional order diffusion equations are generalizations of classical diffusion equations, treating super-diffusive flow processes. The paper presents a meshless method based on spatial trial spaces spanned by the radial basis functions (RBFs) for the numerical solution of a class of initial-boundary value fractional diffusion equations with variable coefficients on a finite domain. The space fractional derivatives are defined by using Riemann-Liouville fractional derivative. We first provide Riemann-Liouville fractional derivatives for the five kinds of RBFs, including the Powers, Gaussian, Multiquadric, Matérn and Thin-plate splines, in one dimension. The time-dependent fractional diffusion equation is discretized in space with the RBF collocation method and the remaining system of ordinary differential equations (ODEs) is advanced in time with an ODE method using a method of lines approach. Some numerical results are given in order to demonstrate the efficiency and accuracy of the method. Additionally, some physical properties of this fractional diffusion system are simulated, which further confirm the effectiveness of our method. The stability of the linear systems arising from discretizing Riemann-Liouville fractional differential operator with RBFs is also analysed.