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Numerical approximation of nonlinear fractional differential equations with subdiffusion and superdiffusion. (English) Zbl 1228.65190
Summary: 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 $1+\beta \in \left[1,2\right]$ and the finite difference scheme for the time Caputo derivative with order $\alpha \in \left(0,1\right)$ (for subdiffusion) and $\left(1,2\right)$ (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.
MSC:
 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals (real functions) 45K05 Integro-partial differential equations 65M12 Stability and convergence of numerical methods (IVP of PDE)