# 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

 [1] Kabanikhin S. I., Inverse and ill-posed problems. Theory and applications, De Gruyter, Berlin, 2011 [2] Lavrent'ev M. M., Romanov V. G., Shishatskii S. P., Ill-posed problems of mathematical physics and analysis, Amer. Math. Soc., Providence, RI, 1986 [3] Samarskii A. A., Vabishchevich P. N., Numerical methods for solving inverse problems of mathematical physics, De Gruyter, Berlin, 2007 [4] Kabanikhin S. I., Bektemesov M. A., Nurseitov D. B., Krivorotko O. I., Alimova A. N., “An optimization method in the Dirichlet problem for the wave equation”, J. Inverse Ill-posed Probl., 20:2 (2012), 193–211 [5] Kabanikhin S. I. and Krivorotko O. I., “A numerical method for solving the Dirichlet problem for the wave equation”, J. Appl. Ind. Math., 7:2 (2013), 187–198 [6] Samarskii A. A., Vabishchevich P. N., and Vasil'ev V. I., “Iterative solution of a retrospective inverse problem of heat conduction”, Mat. Model., 9:5 (1997), 119–127 [7] Vasil'ev V. I., Popov V. V., Eremeeva M. S., Kardashevsky A. M., “Iterative solution of a non classical problem for the equation of string vibrations”, Vestn. Mosk. Gos. Tekh. Univ. Im. N. Eh. Baumana, Ser. Estestv. Nauki, 2015, no. 3, 77–87 [8] Vabishchevich P. N. and Vasil'ev V. I., “terative solution of the Dirichlet problem for hyperbolic equations”, Setochnye Metody dlya Kraevykh Zadach i Prilozheniya, Mat. X Mezhdunar. Konf., Izd-vo Kazansk. Univ., Kazan, 2014, 162–166 [9] Vasil'ev V. I. and Kardashevsky A. M., “Iterative solutions to some inverse problems for second-order hyperbolic equations”, Aktualnye Problemy Vychislitelnoi i Prikladnoi Matematiki, Tr. Mezhdunar. Konf., Abvei, Novosibirsk, 2015, 150–156 [10] Samarskii A. A., The theory of difference schemes, Marcel Dekker, New York; Basel, 2001 [11] Saad Yu., Iterative methods for sparse linear systems. 2nd ed., SIAM, Philadelphia, PA, 2003
How to Cite
Vasil’ev, V., Kardashevsky, A. and Popov, V. (&nbsp;) “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.
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Mathematical Modeling