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Numerical Scheme for the Fokker–Planck Equations Describing Anomalous Diffusions with Two Internal States

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Abstract

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 (Xu and Deng in Math Model Nat Phenom 13:10, 2018). 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.

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Acknowledgements

Funding was provided by National Natural Science Foundation of China (Grant No. 11671182).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Daxin Nie, Jing Sun & Weihua Deng

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  1. Daxin Nie
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  2. Jing Sun
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  3. Weihua Deng
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Correspondence to Weihua Deng.

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Nie, D., Sun, J. & Deng, W. Numerical Scheme for the Fokker–Planck Equations Describing Anomalous Diffusions with Two Internal States. J Sci Comput 83, 33 (2020). https://doi.org/10.1007/s10915-020-01218-9

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  • DOI: https://doi.org/10.1007/s10915-020-01218-9

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