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Numerical Solution to Anomalous Diffusion Equations for Levy Walks

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  • Viacheslav V. Saenko

    (Laboratory for Interdisciplinary Problems of Energy Reproduction, Ulyanovsk State Technical University, 32, Severny Venets St., 432027 Ulyanovsk, Russia
    S.P. Kapitsa Scientific Research Institute of Technology, Ulyanovsk State University, 42, L. Tolstoy St., 432017 Ulyanovsk, Russia)

  • Vladislav N. Kovalnogov

    (Laboratory for Interdisciplinary Problems of Energy Reproduction, Ulyanovsk State Technical University, 32, Severny Venets St., 432027 Ulyanovsk, Russia)

  • Ruslan V. Fedorov

    (Laboratory for Interdisciplinary Problems of Energy Reproduction, Ulyanovsk State Technical University, 32, Severny Venets St., 432027 Ulyanovsk, Russia)

  • Yuri E. Chamchiyan

    (Laboratory for Interdisciplinary Problems of Energy Reproduction, Ulyanovsk State Technical University, 32, Severny Venets St., 432027 Ulyanovsk, Russia)

Abstract

The process of Levy random walks is considered in view of the constant velocity of a particle. A kinetic equation is obtained that describes the process of walks, and fractional differential equations are obtained that describe the asymptotic behavior of the process. It is shown that, in the case of finite and infinite mathematical expectation of paths, these equations have a completely different form. To solve the obtained equations, the method of local estimation of the Monte Carlo method is described. The solution algorithm is described and the advantages and disadvantages of the considered method are indicated.

Suggested Citation

  • Viacheslav V. Saenko & Vladislav N. Kovalnogov & Ruslan V. Fedorov & Yuri E. Chamchiyan, 2021. "Numerical Solution to Anomalous Diffusion Equations for Levy Walks," Mathematics, MDPI, vol. 9(24), pages 1-17, December.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:24:p:3219-:d:701213
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    References listed on IDEAS

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    1. Uchaikin, Vladimir V., 1998. "Anomalous transport equations and their application to fractal walking," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 255(1), pages 65-92.
    2. Souza, J.W.G. & Santos, A.A.B. & Guarieiro, L.L.N. & Moret, M.A., 2015. "Fractal aspects in O2 enriched combustion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 434(C), pages 268-272.
    3. Saenko, Viacheslav V., 2016. "The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 444(C), pages 765-782.
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    Cited by:

    1. Camelia Petrescu & Valeriu David, 2022. "Preface to the Special Issue on “Modelling and Simulation in Engineering”," Mathematics, MDPI, vol. 10(14), pages 1-3, July.

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