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Numerical Method for Solving of the Anomalous Diffusion Equation Based on a Local Estimate of the Monte Carlo Method

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  • Viacheslav V. Saenko

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

  • Ruslan V. Fedorov

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

  • Dmitry A. Generalov

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

  • Ekaterina V. Tsvetova

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

Abstract

This paper considers a method of stochastic solution to the anomalous diffusion equation with a fractional derivative with respect to both time and coordinates. To this end, the process of a random walk of a particle is considered, and a master equation describing the distribution of particles is obtained. It has been shown that in the asymptotics of large times, this process is described by the equation of anomalous diffusion, with a fractional derivative in both time and coordinates. The method has been proposed for local estimation of the solution to the anomalous diffusion equation based on the simulation of random walk trajectories of a particle. The advantage of the proposed method is the opportunity to estimate the solution directly at a given point. This excludes the systematic component of the error from the calculation results and allows constructing the solution as a smooth function of the coordinate.

Suggested Citation

  • Viacheslav V. Saenko & Vladislav N. Kovalnogov & Ruslan V. Fedorov & Dmitry A. Generalov & Ekaterina V. Tsvetova, 2022. "Numerical Method for Solving of the Anomalous Diffusion Equation Based on a Local Estimate of the Monte Carlo Method," Mathematics, MDPI, vol. 10(3), pages 1-19, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:3:p:511-:d:742731
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    References listed on IDEAS

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    1. Metzler, Ralf & Glöckle, Walter G. & Nonnenmacher, Theo F., 1994. "Fractional model equation for anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 211(1), pages 13-24.
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    4. Romanovsky, M.Yu., 1999. "Model space of economic events," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 265(1), pages 264-278.
    5. 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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