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Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations

Author

Listed:
  • Gurami Tsitsiashvili

    (Institute for Applied Mathematics, Far Eastern Branch of Russian Academy of Sciences, IAM FEB RAS, Radio Str. 7, 690041 Vladivostok, Russia)

  • Alexey Gudimenko

    (Institute for Applied Mathematics, Far Eastern Branch of Russian Academy of Sciences, IAM FEB RAS, Radio Str. 7, 690041 Vladivostok, Russia)

  • Marina Osipova

    (Institute for Applied Mathematics, Far Eastern Branch of Russian Academy of Sciences, IAM FEB RAS, Radio Str. 7, 690041 Vladivostok, Russia
    Institute for Applied Mathematics, Far Eastern Federal University, 690922 Vladivostok, Russia)

Abstract

In this paper, the problems of estimating the parameters of partial differential equations from numerous observations in the vicinity of some reference points are considered. The paper is devoted to estimating the diffusion coefficient in the diffusion equation and the parameters of one-soliton solutions of nonlinear partial differential equations. When estimating the diffusion coefficient, it was necessary to construct an estimate of the second derivative based on inaccurate observations of the solution of the diffusion equation. This procedure required consideration of two reference points when determining the first and second partial derivatives of the solution of the diffusion equation. To analyse one-soliton solutions of partial differential equations, a series of techniques have been developed that allow one to estimate the parameters of the solution itself, but not its equation. These techniques are used to estimate the parameters of one-soliton solutions of the equations kdv, mkdv, Sine–Gordon, Burgers and nonlinear Schrodinger. All the considered estimates were tested during computational experiments.

Suggested Citation

  • Gurami Tsitsiashvili & Alexey Gudimenko & Marina Osipova, 2023. "Fast Method for Estimating the Parameters of Partial Differential Equations from Inaccurate Observations," Mathematics, MDPI, vol. 11(22), pages 1-13, November.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:22:p:4586-:d:1276787
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    References listed on IDEAS

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    1. J. O. Ramsay & G. Hooker & D. Campbell & J. Cao, 2007. "Parameter estimation for differential equations: a generalized smoothing approach," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(5), pages 741-796, November.
    2. Gurami Tsitsiashvili & Alexey Gudimenko & Marina Osipova, 2023. "Mathematical and Statistical Aspects of Estimating Small Oscillations Parameters in a Conservative Mechanical System Using Inaccurate Observations," Mathematics, MDPI, vol. 11(12), pages 1-11, June.
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