On an approximate solution to an ill-posed mixed boundary value problem for the Laplace equation in a cylindrical domain with homogeneous conditions of the second kind on the lateral surface of the cylinder

E. B. Laneev ; A. V. Klimishin

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 146, pp. 164-175

Résumé

We consider a problem mixed in boundary conditions for the Laplace equation in a domain that is a part of a cylinder of a rectangular cross-section with homogeneous boundary conditions of the second kind on the side surface of the cylinder. The cylindrical region is limited on one side by surface of a general kind on which the Cauchy conditions are specified, i. e. a function and its normal derivative are given, and the other boundary of the cylindrical region is free. In this case, the problem has the property of instability of the Cauchy problem for the Laplace equation with respect to the error in the Cauchy data, i. e. is ill-posed, and its approximate solution, robust to errors in Cauchy data, requires the use of regularization methods. The problem under consideration is reduced to the Fredholm integral equation of the first kind. Based on the solution of the integral equation obtained in the form of a Fourier series on the eigenfunctions of the second boundary value problem for the Laplace equation in a rectangle, an explicit representation of the exact solution of the problem was constructed. A stable approximate solution to the integral equation was constructed using the Tikhonov regularization method. The extremal of the Tikhonov functional is considered as an approximate solution to the integral equation. Based on the approximate solution of the integral equation, an approximate solution of the boundary value problem as a whole is constructed. A theorem is proved for the convergence of an approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is consistent with the error in the data.

Détail

Keywords: ill-posed problem, Cauchy problem for the Laplace equation, integral equation of the first kind, Tikhonov regularization method

Bibliographie

[1] M. Joachimiak, “Choice of the regularization parameter for the Cauchy problem for the Laplace equation”, International Journal of Numerical Methods for Heat Fluid Flow, 30:10 (2020), 4475–4492

[2] S. B. Sorokin, “Ekonomichnyi pryamoi metod chislennogo resheniya zadachi Koshi dlya uravneniya Laplasa”, Sib. zhurn. vychisl. matem., 22:1 (2019), 99–117 | MR

[3] M. M. Lavrent'ev, On Some Ill-Posed Problems of Mathematical Physics, Academy of Sciences Publ., Novosibirsk, 1962 (In Russian) | MR

[4] G. M. Goluzin, V. I. Krylov, “Generalized Carleman formula and its application to the analytic continuation of functions”, Mat. Sb., 40:2 (1933), 144–149 (In Russian) | Zbl

[5] E. B. Laneev, “O postroenii funktsii Karlemana na osnove metoda regulyarizatsii Tikhonova v nekorrektno postavlennoi zadache dlya uravneniya Laplasa”, Differentsialnye uravneniya, 54:4 (2018), 483–491 | DOI | MR | Zbl

[6] A. N. Tikhonov, V. {Ya}. Arsenin, Methods for Solving Ill-Posed Problems, Nauka Publ., Moscow, 1979 (In Russian)

[7] E. M. Landis, “Nekotorye voprosy kachestvennoi teorii ellipticheskikh uravnenii vtorogo poryadka (sluchai mnogikh nezavisimykh peremennykh)”, Uspekhi matematicheskikh nauk, 18:1(109) (1963), 3–62 | MR | Zbl

[8] V. B. Glasko, Inverse Problems of Mathematical Physics, MSU Publ., Moscow, 1984 (In Russian) | MR

[9] Sh. Yarmukhamedov, “Funktsiya Karlemana i zadacha Koshi dlya uravneniya Laplasa”, Sibirskii matematicheskii zhurnal, 45:3 (2004), 702–719 | MR | Zbl

[10] E. N. Sattorov, Z. E. Ermamatova, “Carleman’s formula of a solutions of the poisson equation in bounded domain”, Ural Mathematical Journal, 7:2 (2021), 110–120 | MR | Zbl

[11] O. Baaj, “On the application of the Fourier method to solve the problem of correction of thermographic images”, Discrete and Continuous Models and Applied Computational Science, 30:3 (2022), 205–216 | MR

[12] E. B. Laneev, O. Baaj, “On a modification of the Hamming method for summing discrete Fourier series and its application to solve the problem of correction of thermographic images”, Discrete and Continuous Models and Applied Computational Science, 30:4 (2022), 342–356

[13] E. B. Laneev, N. Yu. Chernikova, O. Baaj, “Application of the minimum principle of a Tikhonov smoothing functional in the problem of processing thermographic data”, Advances in Systems Science and Applications, 2021, no. 1, 139–149

[14] E. B. Laneev, E. Yu. Ponomarenko, “On a linear inverse potential problem with approximate data on the potential field on an approximately given surface”, Eurasian Mathematical Journal, 14:1 (2023), 57–70 | MR