Iterative method for the Dirichlet problem and its modifiations

  • Vasil’ev Vasily I., vasvasil@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 58 Belinsky Street, Yakutsk, 677000 Russia
  • Kardashevsky Anatoly M., kardam123@gmail.com M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 58 Belinsky Street, Yakutsk, 677000 Russia
  • Popov Vasily V., imi.pm.pvvl@mail.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics and Informatics, 58 Belinsky Street, Yakutsk, 677000 Russia; Institute of Oil and Gas Problems SB RAS, 1 Oktyabrskaya Street, Yakutsk 677000, Russia
Keywords: hyperbolic equation, inverse problem, Dirichlet problem, finite difference method, iterative method, conjugate gradients method, random error

Abstract

A series of works of S. I. Kabanikhin’s scientific school are devoted to study of the existence, uniqueness, and numerical methods for the inverse Dirichlet problem for the second-order hyperbolic equations. We consider a numerical solution to the non-classical Dirichlet problem and its modifications for the two-dimensional hyperbolic second-order equations. The method of iterative refinement of the missing initial condition is applied by means of an additional condition specified at the final time. Moreover, the direct problem is numerically realized at each iteration. The efficiency of the proposed computational algorithm is confirmed by calculations for two-dimensional model problems, including additional conditions with random errors.

References

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How to Cite
Vasil’ev, V., Kardashevsky, A. and Popov, V. ( ) “Iterative method for the Dirichlet problem and its modifiations”, Mathematical notes of NEFU, 24(3), pp. 38-51. doi: https://doi.org/10.25587/SVFU.2018.3.10888.
Section
Mathematical Modeling