DOI: https://doi.org/10.26089/NumMet.v27r102

Numerical solution of determining the initial condition problem in Cauchy problem for hyperbolic equation with small parameter

Authors

  • Daniil S. Andrianov

Keywords:

inverse problem
hyperbolic equation
singular perturbation
quasi-inversion methods
numerical methods

Abstract

The textconsiders the inverse problem of determining theinitialcondition fora singularly perturbed hyperbolic equation with a small parameter. This involves finding an unknown odd function using additional information about the derivative of the solution at a fixed spatial point. The main result of the text is to reduce the ill-posed initial problem to a Volterra integral equation of the second kind. An iterative method has been developed for numerically solving this equation. Computational experiments were conducted to test the effectiveness of this algorithm.


Downloads

Published

2026-01-28

Issue

Vol. 27 (2026): Issue 1.

Section

Methods and algorithms of computational mathematics and their applications

Author

Daniil S. Andrianov

Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

• PhD Student

References

  1. A. N. Tikhonov and V. Ia. Arsenin, Methods for Solving Incorrect Problems (Nauka, Moscow, 1986) [in Russian].
  2. A. M. Denisov, Introduction to the Theory of Inverse Problems (Mosk. Gos. Univ., Moscow, 1994) [in Russian].
  3. M. M. Lavrentév, V. G. Romanov, and S. P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis (Nauka, Moscow, 1980) [in Russian].
  4. V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, Moscow, 1978) [in Russian].
  5. V. G. Romanov, Inverse Problems of Mathematical Physics (Nauka, Moscow, 1984) [in Russian].
  6. S. I. Kabanikhin, Inverse and Ill-Posed Problems (Sib. Nauchn. Izd., Novosibirsk, 2009) [in Russian].
  7. R. Lattes and J.-L. Lions, The Method of Quasi-inversion and Its Applications (Mir, Moscow, 1970) [in Russian].
  8. V. K. Ivanov, “The Quasi-inversion Problem for the Heat Equation in the Uniform Metric,” Differ. Uravn. 8 (4), 652–658 (1972).
    https://www.mathnet.ru/de1532 Cited January 19, 2026.
  9. A. A. Samarskii and P. N. Vabishchevich, Numerical Methods for Solving Inverse Problems of Mathematical Physics (Editorial URSS, Moscow, 2004) [in Russian].
  10. E. V. Tabarintseva, L. D. Menikhes, and A. D. Drozin, “On solving an inverse boundary problem for a parabolic equation by the quasi-revesibility method,” Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Mekh. Fiz., No. 6, 8–13 (2012).
    https://www.mathnet.ru/rus/vyurm100 Cited January 19, 2026.
  11. A. I. Korotkii, I. A. Tsepelev, and A. T. Ismail-Zadeh, “Numerical Modeling of the Inverse Retrospective Problem of Thermal Convection with Applications to Geodynamic Problems,” Izv. Ural. Gos. Univ., No. 58, 78–87 (2008).
    http://elar.urfu.ru/handle/10995/24610 Cited January 20, 2026.
  12. A. M. Denisov, “Approximate Solution of an Inverse Problem for a Singularly Perturbed Integro-Differential Heat Equation,” Zh. Vychisl. Mat. Mat. Fiz. 63 (5), 795–802 (2023) [Comput. Math. Math. Phys. 63 (5), 837–844 (2023)].
    https://doi.org/10.1134/S0965542523050081 Cited January 20, 2026.
  13. N. Levashova, A. Gorbachev, R. Argun, and D. Lukyanenko, “The Problem of the Non-Uniqueness of the Solution to the Inverse Problem of Recovering the Symmetric States of a Bistable Medium with Data on the Position of an Autowave Front,” Symmetry 13 (5), Article Number 860 (2021).
    doi 10.3390/sym13050860
  14. D. V. Lukyanenko, A. A. Borzunov, and M. A. Shishlenin, “Solving Coefficient Inverse Problems for Nonlinear Singularly Perturbed Equations of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front,” Commun. Nonlinear Sci. Numer. Simul. 99, Article Number 105824 (2021).
    doi 10.1016/j.cnsns.2021.105824
  15. D. V. Lukyanenko, M. A. Shishlenin, and V. T. Volkov, “Asymptotic Analysis of Solving an Inverse Boundary Value Problem for a Nonlinear Singularly Perturbed Time-Periodic Reaction-Diffusion-Advection Equation,” J. Inverse Ill-Posed Probl. 27 (5), 745–758 (2019).
    doi 10.1515/jiip-2017-0074
  16. A. M. Denisov, “Approximate Solution of Inverse Problems for the Heat Equation with a Singular Perturbation,” Zh. Vychisl. Mat. Mat. Fiz. 61 (12), 2040–2049 (2021) [Comput. Math. Math. Phys. 61 (12), 2004–2014 (2021)].
    doi 10.1134/S0965542521120071
  17. A. M. Denisov and S. I. Solovéva, “Numerical Solution of Problems of Determining the Initial Condition in Cauchy Problems for a Hyperbolic Equation with a Small Parameter,” Prikl. Mat. Inform. 54, 5–12 (2017).
  18. A. M. Denisov, “Asymptotic Expansions of Solutions to Inverse Problems for a Hyperbolic Equation with a small Parameter Multiplying the Highest Derivative,” Zh. Vychisl. Mat. Mat. Fiz. 53 (5), 744–752 (2013) [Comput. Math. Math. Phys. 53 (5), 580–587 (2013)].
    https://doi.org/10.1134/S0965542513050047 Cited January 21, 2026.
  19. B. M. Budak, A. A. Samarskii, and A. N. Tikhonov, A Collection of Problems on Mathematical Physics (Gostekhizdat, Moscow, 1956) [in Russian].
  20. A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics , 4th ed. (Nauka, Moscow, 1972) [in Russian].
  21. H. Bateman, Higher Transcendental Functions , Vol. 2 (McGraw-Hill, New York, 1953).

License

Copyright (c) 2026 Д.С. Андрианов

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.