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On a stable approximate solution of an ill-posed boundary value problem for the metaharmonic equation

Annotation

In this paper, we consider a mixed problem for a metaharmonic equation in a region in a rectangular cylinder. On the side faces cylinder region is set to homogeneous conditions of the first kind. 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 boundary of the cylindrical region, which is flat, is free. This problem is illposed, and to construct its approximate solution in the case of Cauchy data known with some error, it is necessary to use regularizing algorithms. In this paper, the problem is reduced to the Fredholm integral equation of the first kind. Based on the solution of the integral equation, an explicit representation of the exact solution of the problem is obtained. 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. The convergence theorem of the approximate solution of the problem to the exact one is given when the error in the Cauchy data tends to zero and when the regularization parameter is agreed with the error in the data. The results can be used for mathematical processing of thermal imaging data in medical diagnostics.

Keywords

ill-posed problem; metaharmonic equation; integral equation of the first kind; method of Tikhonov regularization

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DOI

10.20310/2686-9667-2020-25-130-156-164

UDC

519.6

Pages

156-164

References

[1] 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. [2] 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, №1, 128-133 (In Russian). [3] 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. [4] A.N. Tikhonov, V.YA. Arsenin, Metody Resheniya Nekorrektnyh Zadach, Nauka, Moscow, 1979 (In Russian). [5] A.N. Tikhonov, V.B. Glasko, O.K. Litvinenko, V.R. Melikhov, "O prodolzheni potenciala v storonu vozmushchayushchih mass na osnove metoda regulyarizacii", Izv. AN SSSR. Fizika Zemli, 1968, №1, 30-48 (In Russian). [6] E.B. Laneev, M. Muratov, "Ob odnoy obratnoy zadache k kraevoy zadache dlya uravneniya Laplasa s usloviem tret'ego roda na netochno zadannoy granitse", PFUR Reports. Series: Mathematics, 10:1 (2003), 100-110 (In Russian). [7] G.R. Ivanitskii, "Thermovision in medicine", Herald of the Russian Academy of Sciences, 76:1 (2006), 48-58 (In Russian).

Received

2020-04-10

Section of issue

Scientific articles

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