Geodesic
On an ill-posed boundary value problem for a metaharmonic equation in a circular cylinder
Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 136, pp. 394-403
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this paper, we consider a mixed problem for a metaharmonic equation in a domain in a circular cylinder. The cylindrical area is bounded on one side by an arbitrary surface on which the Cauchy conditions are set, i. e. the function and its normal derivative are set. The other border of the cylindrical area is free. On the lateral surface of the cylindrical domain, homogeneous boundary conditions of the first kind are given. The problem is ill-posed and its approximate solution, stable to errors in the Cauchy data, is constructed using regularization methods. The problem is reduced to a first kind Fredholm integral equation. Based on the solution of the integral equation obtained in the form of a Fourier series by the eigenfunctions of the first boundary value problem for the Laplace equation in a circle, an explicit representation of the exact solution of the problem is constructed. A stable solution of the integral equation is obtained by the method of Tikhonov regularization. The extremal of the Tikhonov functional is considered as an approximate solution. Based on this solution, an approximate solution of the problem as a whole is constructed. A theorem on convergence of the approximate solution of the problem to the exact one as the error in the Cauchy data tends to zero and the regularization parameter is matched with the error in the data, is given. The results can be used for mathematical processing of thermal imaging data in early diagnostics in medicine.
Keywords: ill-posed problem, metaharmonic equation, Bessel function, integral equation of the first kind, method of Tikhonov. 
@article{VTAMU_2021_26_136_a5,
     author = {E. B. Laneev and V. A. Anisimov and P. A. Lesik and V. I. Remezova and A. A. Romanov and A. G. Khegai},
     title = {On an ill-posed boundary value problem for a metaharmonic equation in a circular cylinder},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {394--403},
     year = {2021},
     volume = {26},
     number = {136},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2021_26_136_a5/}
}
TY  - JOUR
AU  - E. B. Laneev
AU  - V. A. Anisimov
AU  - P. A. Lesik
AU  - V. I. Remezova
AU  - A. A. Romanov
AU  - A. G. Khegai
TI  - On an ill-posed boundary value problem for a metaharmonic equation in a circular cylinder
JO  - Vestnik rossijskih universitetov. Matematika
PY  - 2021
SP  - 394
EP  - 403
VL  - 26
IS  - 136
UR  - http://geodesic.mathdoc.fr/item/VTAMU_2021_26_136_a5/
LA  - ru
ID  - VTAMU_2021_26_136_a5
ER  - 
%0 Journal Article
%A E. B. Laneev
%A V. A. Anisimov
%A P. A. Lesik
%A V. I. Remezova
%A A. A. Romanov
%A A. G. Khegai
%T On an ill-posed boundary value problem for a metaharmonic equation in a circular cylinder
%J Vestnik rossijskih universitetov. Matematika
%D 2021
%P 394-403
%V 26
%N 136
%U http://geodesic.mathdoc.fr/item/VTAMU_2021_26_136_a5/
%G ru
%F VTAMU_2021_26_136_a5
E. B. Laneev; V. A. Anisimov; P. A. Lesik; V. I. Remezova; A. A. Romanov; A. G. Khegai. On an ill-posed boundary value problem for a metaharmonic equation in a circular cylinder. Vestnik rossijskih universitetov. Matematika, Tome 26 (2021) no. 136, pp. 394-403. http://geodesic.mathdoc.fr/item/VTAMU_2021_26_136_a5/

[1] G. R. Ivanitskii, “Thermovision in medicine”, Herald of the Russian Academy of Sciences, 76:1 (2006), 48–58 (In Russian) | MR

[2] J. P. Agnelli, A. A. Barrea, C. V. Turner, “Tumor location and parameter estimation by thermography”, Mathematical and Computer Modelling, 53:7-8 (2011), 1527–1534 | DOI | Zbl

[3] E. B. Laneev, B. Vasudevan, “On a stable solution of a mixed problem for the Laplace equation”, PFUR Reports. Series: Applied mathematics and computer science, 1999, no. 1, 128–133 (In Russian) | MR | Zbl

[4] E. B. Laneev, “Construction of a Carleman Function Based on the Tikhonov Regularization Method in an Ill-Posed Problem for the Laplace Equation”, Differential Equations, 54:4 (2018), 476–485 | DOI | MR | Zbl

[5] E. B. Laneev, D. Yu. Bykov, A. V. Zubarenko, O. N. Kulikova, D. A. Morozova, E. V. Shunin, “On an ill-posed boundary value problem for the Laplace equation in a circular cylinder”, Russian Universities Reports. Mathematics, 26:133 (2021), 35–43 (In Russian) | Zbl

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

[7] A. N. Tikhonov A.N., V. B. Glasko, O. K. Litvinenko, V. R. Melikhov, “On the continuation of the potential towards the perturbing masses based on the regularization method”, Izv. AN SSSR. Fizika Zemli, 1968, no. 1, 30–48 (In Russian)

[8] E. B. Laneev, M. N. Muratov,, “An inverse problem to a boundary value problem for the Laplace equation with a condition of the third kind on an inexactly specified boundary”, PFUR Reports. Series: Mathematics, 10:1 (2003), 100–110 (In Russian)