On the stable approximate solution of the ill-posed boundary value problem for the Laplace equation with homogeneous conditions of the second kind on the edges at inaccurate data on the approximated boundary

Cover Page

Cite item

Full Text

Abstract

In this paper, we consider the ill-posed continuation problem for harmonic functions from an ill-defined boundary in a cylindrical domain with homogeneous boundary conditions of the second type on the side faces. The value of the function and its normal derivative (Cauchy conditions) is known approximately on an approximated surface of arbitrary shape bounding the cylinder. In this case, the Cauchy problem for the Laplace equation has the property of instability with respect to the error in the Cauchy data, that is, it is ill-posed. On the basis of an idea about the source function of the original problem, the exact solution is represented as a sum of two functions, one of which depends explicitly on the Cauchy conditions, and the second one can be obtained as a solution of the Fredholm integral equation of the first kind in the form of Fourier series on the eigenfunctions of the second boundary value problem for the Laplace equation. To obtain an approximate stable solution of the integral equation, the Tikhonov regularization method is applied when the solution is obtained as an extremal of the Tikhonov functional. For an approximated surface, we consider the calculation of the normal to this surface and its convergence to the exact value depending on the error with which the original surface is given. The convergence of the obtained approximate solution to the exact solution is proved when the regularization parameter is compared with the errors in the data both on the inexactly specified boundary and on the value of the original function on this boundary. A numerical experiment is carried out to demonstrate the effectiveness of the proposed approach for a special case, for a flat boundary and a specific initial heat source (a set of sharpened sources).

About the authors

Evgeniy B. Laneev

RUDN University

Author for correspondence.
Email: elaneev@yandex.ru
ORCID iD: 0000-0002-4255-9393
Scopus Author ID: 24366681900
ResearcherId: G-7887-2016

Doctor of Physical and Mathematical Sciences, Professor of the Mathematical Institute named after S. M. Nikolsky

6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

Alexander V. Klimishin

RUDN University

Email: sa-sha-02@yandex.ru
Post-Graduate student of the Mathematical Institute named after S. M. Nikolsky 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

References

  1. Ivanitskii, G. R. Thermovision in medicine. Russian. Vestnik RAN 76, 44–53 (2006).‌
  2. Tikhonov, A. N. & Glasko, V. B. Use of the regularization method in non-linear problems. U.S.S.R. Comput. Math. Math. Phys. 5, 93–107. doi: 10.1016/0041-5553(65)90150-3 (1965).
  3. Ivanov, V. K., Vasin, V. V. & Tanana, V. P. The theory of linear ill-posed problems and its applications Russian (Nauka, Moscow, 1978).
  4. Vasin, V. V. The stable evaluation of a derivative in space C(-∞,+∞). U.S.S.R. Comput. Math. Math. Phys. 13, 16–24. doi: 10.1016/0041-5553(73)90002-5 (1973).
  5. Tihonov, A. N. & Arsenin, V. Y. Methods for solving ill-posed problems Russian (Nauka, Moscow, 1979).
  6. Tihonov, A. N., Glasko, V. B., Litvinenko, O. K. & Melihov, V. R. On the continuation of the potential towards disturbing masses based on the regularization method. Russian. Vestnik RAN 1, 30–48 (1968).
  7. Laneev, E. B., Chernikova, N. Y. & Baaj, O. Application of the minimum principle of a Tikhonov smoothing functional in the problem of processing thermographic data. Advances in Systems Science and Applications 1, 139–149. doi: 10.25728/assa.2021.21.1.1055 (2021).
  8. Baaj, O., Chernikova, N. Y. & Laneev, E. B. Correction of Thermographic Images Based on the MIinimization Method of Tikhonov Functional. Yugoslav Journal of Operations Research 32, 407–424. doi: 10.2298/YJOR211015026B (2022).
  9. Laneev, E. B. & Klimishin, A. V. 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. Russian. Russian Universities Reports. Mathematics 29, 164–175. doi: 10.20310/2686-9667-2024-29-146-164-175 (2024).
  10. Laneev, E. B. & Baaj, O. On a stable calculation of the normal to a surface given approximately. Discrete and Continuous Models and Applied Computational Science 3, 228–241. doi: 10.22363/2658- 4670-2023-31-3-228-241 (2023).
  11. Laneev, E. B. & Muratov, M. N. On the stable solution of a mixed boundary value problem for the Laplace equation with an approximately given boundary. Russian. Vestnik RUDN. Seriya Matematika 1, 102–111 (2002).
  12. Laneev, E. B. & Muratov, M. N. Ob odnoy obratnoy zadache k kraevoy zadache dlya uravneniya Laplasa s usloviem tret’ego roda na netochno zadannoy granitse. Russian. Vestnik RUDN. Seriya Matematika 1, 100–110 (2003).
  13. Chernikova, N. Y., Laneev, E. B., Muratov, M. N. & Ponomarenko, E. Y. On an Inverse Problem to a Mixed Problem for the Poisson Equation. Mathematical Analysis With Applications. CONCORD-90 2018 318, 141–146 (2020).
  14. Laneev, E. B. & Bhuvana, V. Ob ustojchivom reshenii odnoj smeshannoj zadachi dlya uravneniya Laplasa. Russian. Vestnik RUDN. Seriya Prikladnaya matematika i informatika 1, 128–133 (1999).
  15. Laneev, E. B., Lesik, P. A., Klimishin, A. V., Kotyukov, A. M. & Romanov A. A. and, K. A. G. On a stable approximate solution of an ill-posed boundary value problem for the metaharmonic equation. Russian. Russian Universities Reports. Mathematics 25, 156–164. doi: 10.20310/2686-9667- 2020-25-130-156-164 (2020).‌
  16. Laneev, E. B., Mouratov, M. N. & Zhidkov, E. P. Discretization and its proof for numerical solution of a Cauchy problem for Laplace equation with inaccurately given Cauchy conditions on an inaccurately defined arbitrary surface. Phys. Part. Nuclei Lett 5, 164–167. doi:10.1134/ S1547477108030059 (2008).
  17. Baaj, O. 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, 205–216. doi: 10.22363/2658-4670-2022-30-3-205-216 (2022).
  18. Hamming, R. W. Numerical methods for scientists and engineers (McGraw-Hill Book Company, New York, 1962).
  19. Laneev, E. B. & Baaj, O. 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, 342–356. doi: 10.22363/2658-4670- 2022-30-4-342-356 (2022).
  20. Morozov, V. A. On a stable method for computing the values of unbounded operators. Russian. Dokladi AN SSSR 185, 267–270 (1969).

Supplementary files

Supplementary Files
Action
1. JATS XML
Download