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Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

Author

Listed:
  • Dmitry Lukyanenko

    (Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia
    Moscow Center of Fundamental and Applied Mathematics, 119234 Moscow, Russia)

  • Tatyana Yeleskina

    (Faculty of Physics, Lomonosov Moscow State University, Baku Branch, Baku 1143, Azerbaijan)

  • Igor Prigorniy

    (Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia)

  • Temur Isaev

    (Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia)

  • Andrey Borzunov

    (Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, Russia)

  • Maxim Shishlenin

    (Institute of Computational Mathematics and Mathematical Geophysics of SB RAS, 630090 Novosibirsk, Russia
    Department of Mathematics and Mechanics, Novosibirsk State University, 630090 Novosibirsk, Russia
    Mathematical Center in Akademgorodok, 630090 Novosibirsk, Russia)

Abstract

In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is the use of additional information about the value of the solution of the equation at the known position of a reaction front, measured experimentally with a delay relative to the initial moment of time. In this case, for the numerical solution of the inverse problem, the gradient method of minimizing the cost functional is applied. In the case when only the position of the reaction front is known, the method of deep machine learning is applied. Numerical experiments demonstrated the possibility of solving such kinds of considered inverse problems.

Suggested Citation

  • Dmitry Lukyanenko & Tatyana Yeleskina & Igor Prigorniy & Temur Isaev & Andrey Borzunov & Maxim Shishlenin, 2021. "Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay," Mathematics, MDPI, vol. 9(4), pages 1-12, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:342-:d:496395
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    References listed on IDEAS

    as
    1. Egger, H. & Fellner, K. & Pietschmann, J.-F. & Tang, B.Q., 2018. "Analysis and numerical solution of coupled volume-surface reaction-diffusion systems with application to cell biology," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 351-367.
    2. Natalia Levashova & Alla Sidorova & Anna Semina & Mingkang Ni, 2019. "A Spatio-Temporal Autowave Model of Shanghai Territory Development," Sustainability, MDPI, vol. 11(13), pages 1-13, July.
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    Citations

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    Cited by:

    1. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
    2. Raul Argun & Natalia Levashova & Dmitry Lukyanenko & Alla Sidorova & Maxim Shishlenin, 2023. "Modeling the Dynamics of Negative Mutations for a Mouse Population and the Inverse Problem of Determining Phenotypic Differences in the First Generation," Mathematics, MDPI, vol. 11(14), pages 1-17, July.

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