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Numerical scheme for the Fokker-Planck equations describing anomalous diffusions with two internal states. (English) Zbl 1444.76070
Summary: Recently, the fractional Fokker-Planck equations (FFPEs) with multiple internal states are built for the particles undergoing anomalous diffusion with different waiting time distributions for different internal states, which describe the distribution of positions of the particles [P. Xu and W. Deng, Math. Model. Nat. Phenom. 13, No. 1, Paper No. 10, 22 p. (2018; Zbl 1405.35189)]. In this paper, we first develop the Sobolev regularity of the FFPEs with two internal states, including the homogeneous problem with smooth and nonsmooth initial values and the inhomogeneous problem with vanishing initial value, and then we design the numerical scheme for the system of fractional partial differential equations based on the finite element method for the space derivatives and convolution quadrature for the time fractional derivatives. The optimal error estimates of the scheme under the above three different conditions are provided for both space semidiscrete and fully discrete schemes. Finally, one- and two-dimensional numerical experiments are performed to confirm our theoretical analysis and the predicted convergence order.

76M10 Finite element methods applied to problems in fluid mechanics
76M99 Basic methods in fluid mechanics
76R50 Diffusion
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
26A33 Fractional derivatives and integrals
35R11 Fractional partial differential equations
Full Text: DOI
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