On a Finite-Difference Scheme for an Hereditary Oscillatory Equation

Abstract

In this paper we suggest an explicit finite-difference scheme for numerical simulation of the Cauchy problem with an integro-differential nonlinear equation that describes an oscillatory process with friction and memory (hereditarity), and with the corresponding local initial conditions. The problems of approximation, stability, and convergence of the proposed finite-difference scheme are investigated. The results of computer experiments that implement the proposed numerical scheme, confirming the theoretical estimates obtained in theorems, are given.

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Correspondence to R. I. Parovik.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 154, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018.

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Parovik, R.I. On a Finite-Difference Scheme for an Hereditary Oscillatory Equation. J Math Sci 253, 547–557 (2021). https://doi.org/10.1007/s10958-021-05252-2

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Keywords and phrases

  • stability
  • convergence
  • explicit finite-difference scheme
  • hereditarity
  • integrodifferential equation
  • memory function
  • Runge rule
  • approximation

AMS Subject Classification

  • 37M05
  • 34A08