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Approximate Methods for Solving Problems of Mathematical Physics on Neural Hopfield Networks

Author

Listed:
  • Ilya Boykov

    (Department of Higher and Applied Mathematics, Penza State University, 40 Krasnaya Str., 440026 Penza, Russia)

  • Vladimir Roudnev

    (Department of Computational Physics, Saint Petersburg State University, 1 Ulyanovskaya Str., 198504 Saint Petersburg, Russia)

  • Alla Boykova

    (Department of Higher and Applied Mathematics, Penza State University, 40 Krasnaya Str., 440026 Penza, Russia)

Abstract

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations.

Suggested Citation

  • Ilya Boykov & Vladimir Roudnev & Alla Boykova, 2022. "Approximate Methods for Solving Problems of Mathematical Physics on Neural Hopfield Networks," Mathematics, MDPI, vol. 10(13), pages 1-22, June.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:13:p:2207-:d:847014
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    References listed on IDEAS

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    1. Ilya Boykov & Vladimir Roudnev & Alla Boykova, 2022. "Stability of Solutions to Systems of Nonlinear Differential Equations with Discontinuous Right-Hand Sides: Applications to Hopfield Artificial Neural Networks," Mathematics, MDPI, vol. 10(9), pages 1-16, May.
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