Numerical algorithm for the space-time fractional Fokker–Planck system with two internal states

The fractional Fokker–Planck system with multiple internal states is derived in [Xu and Deng, Math. Model. Nat. Phenom., 13 , 10 (2018)], where the space derivative is Laplace operator. If the jump length distribution of the particles is power law instead of Gaussian, the space derivative should be...

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Bibliographic Details
Published in:Numerische Mathematik 2020-11, Vol.146 (3), p.481-511
Main Authors: Nie, Daxin, Sun, Jing, Deng, Weihua
Format: Article
Language:English
Subjects:
Algorithms
Approximation
Finite element method
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Operators (mathematics)
Regularity
Simulation
Theoretical
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Summary:The fractional Fokker–Planck system with multiple internal states is derived in [Xu and Deng, Math. Model. Nat. Phenom., 13 , 10 (2018)], where the space derivative is Laplace operator. If the jump length distribution of the particles is power law instead of Gaussian, the space derivative should be replaced with fractional Laplacian. This paper focuses on solving the two-state Fokker–Planck system with fractional Laplacian. We first provide a priori estimate for this system under different regularity assumptions on the initial data. Then we use L 1 scheme to discretize the time fractional derivatives and finite element method to approximate the fractional Laplacian operators. Furthermore, we give the error estimates for the space semidiscrete and fully discrete schemes without any assumption on regularity of solutions. Finally, the effectiveness of the designed scheme is verified by one- and two-dimensional numerical experiments.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-020-01148-6