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Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

Author

Listed:
  • Raul Argun

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

  • Alexandr Gorbachev

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

  • Natalia Levashova

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

  • Dmitry Lukyanenko

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

Abstract

The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.

Suggested Citation

  • Raul Argun & Alexandr Gorbachev & Natalia Levashova & Dmitry Lukyanenko, 2021. "Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduc," Mathematics, MDPI, vol. 9(18), pages 1-14, September.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:18:p:2342-:d:639736
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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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    Cited by:

    1. Raul Argun & Alexandr Gorbachev & Dmitry Lukyanenko & Maxim Shishlenin, 2021. "On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front," Mathematics, MDPI, vol. 9(22), pages 1-18, November.

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