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High-Order Non-uniform Grid Scheme for Numerical Analysis of Singularly Perturbed Fokker-Planck Equation

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Software Engineering Methods in Systems and Network Systems (CoMeSySo 2023)

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 934))

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Abstract

The problems of stochastic analysis have a significant place in various fields of science and technology at the present time. If we solve such problems, we must take into account fluctuation effects. The causes of these fluctuations are different in these problems. The problems, which may relate to the study such such as turbulence of gaseous and liquid substances, thermal noise in materials, noise immunity in telecommunication networks, but the methods of their theoretical research are very similar. There are mathematical methods using the theory of Brownian motion, the theory of diffusion-type processes and the theory of Markov random processes, which makes it possible to solve complex problems of these types at the present time. The aim of this work is numerical analysis of solutions of the singularly perturbed Fokker-Planck equation on a high-order non-uniform grid scheme of various problems with a small parameter. These problems can be solved on the basis of the generalized theory of Brownian motion. Numerical examples demonstrated that applied numerical scheme can be used to analyze processes in queuing theory for 5G/6G network modeling, statistical radiophysics, plasma physics, solid state theory, magnetohydrodynamics, etc.

This paper has been supported by the RUDN University Strategic Academic Leadership Program.

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References

  1. Almuslimani, I., Crouseilles, N.: Conservative stabilized Runge-Kutta methods for the Vlasov-Fokker-Planck equation. J. Comput. Phys. 488, 112241 (2023). https://doi.org/10.1016/j.jcp.2023.112241

    Article  MathSciNet  Google Scholar 

  2. Baddour, A., Malykh, M., Sevastianov, L.: On periodic approximate solutions of dynamical systems with quadratic right-hand side. J. Math. Sci. 261, 698–708 (2022). https://doi.org/10.1007/s10958-022-05781-4

    Article  MathSciNet  Google Scholar 

  3. Bouatta, M.A., Vasilyev, S.A., Vinitsky, S.I.: The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation, Discrete and Continuous Models and Applied Computational Science, vol. 29 (2), 126–145 (2021). https://doi.org/10.22363/2658-4670-2021-29-2-126-145

  4. Carrillo, J.A., Roux, P., Solem, S.: Noise-driven bifurcations in a nonlinear Fokker-Planck system describing stochastic neural fields. Phys. D. 449, 133736 (2023). https://doi.org/10.1016/j.physd.2023.133736

    Article  MathSciNet  Google Scholar 

  5. Danilyuk, E., Moiseeva, S., Nazarov, A.: Asymptotic diffusion analysis of an retrial queueing system M/M/1 with impatient calls. In: Vishnevskiy, V.M., Samouylov, K.E., Kozyrev, D.V. (eds) Distributed Computer and Communication Networks. DCCN 2021. Commun. Comput. Inf. Sci. Springer, Cham, vol. 1552 (2022).https://doi.org/10.1007/978-3-030-97110-6_18

  6. Geiser, J.: Operator splitting approaches of deposition of Brownian particles based on Fokker-Planck equation, AIP Conference Proceedings, vol. 2116 (1) (2019). https://doi.org/10.1063/1.5114469

  7. Hu, J., Liu, J.-G., Xie, Y., Zhou, Z.: A structure preserving numerical scheme for Fokker-Planck equations of neuron networks: numerical analysis and exploration. J. Comput. Phys. 433, 110195 (2021). https://doi.org/10.1016/j.jcp.2021.110195

    Article  MathSciNet  Google Scholar 

  8. Hu, R., Zhang, D., Gu, X.: Reliability analysis of a class of stochastically excited nonlinear markovian jump systems. Chaos, Solitons Fractals 155, 111737 (2022). https://doi.org/10.1016/j.chaos.2021.111737

    Article  MathSciNet  Google Scholar 

  9. Iwankiewicz, R.: Integro-differential Chapman-Kolmogorov equation for continuous-jump Markov processes and its use in problems of multi-component renewal impulse process excitations. Probab. Eng. Mech. 26(1), 16–25 (2011). https://doi.org/10.1016/j.probengmech.2010.06.002

    Article  Google Scholar 

  10. Kaushik, A., Choudhary, M.: A higher-order uniformly convergent defect correction method for singularly perturbed convection-diffusion problems on an adaptive mesh. Alex. Eng. J. 61(12), 9911–9920 (2022). https://doi.org/10.1016/j.aej.2022.03.005

    Article  Google Scholar 

  11. Langtangen, H.P.: A general numerical solution method for Fokker-Planck equations with applications to structural reliability. Probab. Eng. Mech. 6(1), 33–48 (1991). https://doi.org/10.1016/S0266-8920(05)80005-0

    Article  Google Scholar 

  12. Li, B., Xie, L.: Global dynamics and zero-diffusion limit of a parabolic-elliptic-parabolic system for ion transport networks. Nonlinear Anal. Real World Appl. 60, 103304 (2021). https://doi.org/10.1016/j.nonrwa.2021.103304

    Article  MathSciNet  Google Scholar 

  13. Li, Y., Meredith, C.: Artificial neural network solver for time-dependent Fokker-Planck equations. Appl. Math. Comput. 457, 128185 (2023). https://doi.org/10.1016/j.amc.2023.128185

    Article  MathSciNet  Google Scholar 

  14. Li, Y., Wang, Z., Li, J., Wang, J., Wang, K.: Mean-field modelling of precipitation kinetics with a Fokker-Planck equation in the Lifshitz-Slyozov-Wagner space. J. Cryst. Growth 618, 127312 (2023). https://doi.org/10.1016/j.jcrysgro.2023.127312

    Article  Google Scholar 

  15. Song, Y.: Global existence and decay rates of solutions for Vlasov-Navier-Stokes-Fokker-Planck equations with magnetic field. Math. Open 2, 2350001 (2023). https://doi.org/10.1142/S2811007223500013

    Article  Google Scholar 

  16. Tabandeh, A., Sharma, N., Iannacone, L., Gardoni, P.: Numerical solution of the Fokker-Planck equation using physics-based mixture models. Comput. Methods Appl. Mech. Eng. 399, 115424 (2022). https://doi.org/10.1016/j.cma.2022.115424

    Article  MathSciNet  Google Scholar 

  17. Tang, K., Wan, X., Liao, Q.: Adaptive deep density approximation for Fokker-Planck equations. J. Comput. Phys. 457, 111080 (2022). https://doi.org/10.1016/j.jcp.2022.111080

    Article  MathSciNet  Google Scholar 

  18. Vasilyev, S.A., Bouatta, M.A., Kanzitdinov, S.K., Tsareva, G.O.: Numerical analysis of shortest queue problem for time-scale queueing system with a small parameter. In: Dudin, A., Nazarov, A., Moiseev, A. (eds.) ITMM 2022. CCIS, vol. 1803, pp. 16–28. Springer, Cham (2023). https://doi.org/10.1007/978-3-031-32990-6_2

    Chapter  Google Scholar 

  19. Vasilyev, S.A., Bouatta, M.A., Kanzitdinov, S.K., Tsareva, G.O.: High-Order Non-uniform Grid Scheme for Numerical Analysis of Shortest Queue Control Problem with a Small Parameter. In: Silhavy, R., Silhavy, P. (eds) Networks and Systems in Cybernetics. CSOC 2023. Lecture Notes in Networks and Systems, vol. 723, Springer, Cham (2023). https://doi.org/10.1007/978-3-031-35317-8_58

  20. Wan, Y., Zheng, W., Wang, Y.: Identification of chloride diffusion coefficient in concrete using physics-informed neural networks, Construction and Building Materials, vol. 393, 132049 (2023). https://doi.org/10.1016/j.conbuildmat.2023.132049

  21. Zhang, J., Liu, X.: Uniform convergence of a weak Galerkin finite element method on Shishkin mesh for singularly perturbed convection-diffusion problems in 2D. Appl. Math. Comput. 432, 127346 (2022). https://doi.org/10.1016/j.amc.2022.127346

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia named after Patrice Lumumba (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

    Sergey A. Vasilyev, Mohamed A. Bouatta, Evgenii V. Mukaseev & Alexey A. Rukavishnikov

Authors
  1. Sergey A. Vasilyev
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  2. Mohamed A. Bouatta
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  3. Evgenii V. Mukaseev
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  4. Alexey A. Rukavishnikov
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Correspondence to Sergey A. Vasilyev .

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Editors and Affiliations

  1. Faculty of Applied Informatics, Tomas Bata University in Zlin, Zlin, Czech Republic

    Radek Silhavy

  2. Faculty of Applied Informatics, Tomas Bata University in Zlin, Zlin, Czech Republic

    Petr Silhavy

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Vasilyev, S.A., Bouatta, M.A., Mukaseev, E.V., Rukavishnikov, A.A. (2024). High-Order Non-uniform Grid Scheme for Numerical Analysis of Singularly Perturbed Fokker-Planck Equation. In: Silhavy, R., Silhavy, P. (eds) Software Engineering Methods in Systems and Network Systems. CoMeSySo 2023. Lecture Notes in Networks and Systems, vol 934. Springer, Cham. https://doi.org/10.1007/978-3-031-54813-0_23

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