Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion
In this paper, we study the time-space fractional order (fractional for simplicity) nonlinear subdiffusion and superdiffusion equations, which can relate the matter flux vector to concentration gradient in the general sense, describing, for example, the phenomena of anomalous diffusion, fractional Brownian motion, and so on. The semi-discrete and fully discrete numerical approximations are both analyzed, where the Galerkin finite element method for the space Riemann-Liouville fractional derivative with order 1 + beta is an element of [1, 2] and the finite difference scheme for the time Caputo derivative with order alpha is an element of (0, 1) (for subdiffusion) and (1, 2) (for superdiffusion) are analyzed, respectively. Results on the existence and uniqueness of the weak solutions, the numerical stability, and the error estimates are presented. Numerical examples are included to confirm the theoretical analysis. During our simulations, an interesting diffusion phenomenon of particles is observed, that is, on average, the diffusion velocity for 0 < alpha < 1 is slower than that for alpha = 1, but the diffusion velocity for 1 < alpha < 2 is faster than that for alpha = 1. For the spatial diffusion, we have a similar observation. (C) 2011 Elsevier Ltd. All rights reserved.
subdiffusion superdiffusion finite element method caputo derivative time-space fractional diffusion equation riemann-liouville derivative difference method