V.P. Tanana, A.I. Sidikova. Approximate solution of an inverse boundary value problem for a system of differential equations of parabolic type and estimation of the error of this solution ... P. 247-264

We study the problem of finding a boundary condition in the heat equation for a hollow ball made of a composite material consisting of two homogeneous components. The Dirichlet condition is considered as boundary conditions inside the ball at $r= r_0$. In the inverse problem, the temperature inside the ball is assumed to be unknown on an infinite time interval. For finding it, the temperature of the heat flux at the media interface for $r = r_1$ is measured. We analyze the direct problem, which allows us to give a strict formulation of the inverse problem and determine the functional spaces in which it is natural to solve the inverse problem. Estimating the error of the approximate solution presents a major difficulty, which is dealt with in this paper by the method of projection regularization. Using this method, we find order-exact estimates.

Keywords: error estimation, modulus of continuity, Fourier transform, ill-posed problem

Received June 28, 2019

Revised August 22, 2019

Accepted August 26, 2019

Vitalii Pavlovich Tanana, Dr. Phys.-Math. Sci., Prof., South Ural State University, Chelyabinsk, 454080 Russia, e-mail: tananavp@susu.ru

Anna Ivanovna Sidikova, Сand. Phys.-Math. Sci, South Ural State University, Chelyabinsk, 454080 Russia, e-mail: sidikovaai@susu.ru

REFERENCES

1.   Wildemann V.E., Sokolkin Yu.V., Tashkinov A.A. Mekhanika neuprugogo deformirovaniya i razrusheniya kompozitsionnykh materialov [Mechanics of inelastic deformation and fracture of composite materials]. Moscow: Nauka Publ., 1997, 288 p. ISBN: 5-02-015078-9 .

2.   Alifanov O.M., Artyukhin E.A., Rumyantsev S.V. Ekstremal’nye metody resheniya nekorrektnykh zadach [Extremal methods for the solution of ill-posed problems]. Moscow: Nauka Publ., 1988, 288 p. ISBN: 5-02-013774-X .

3.   Tanana V.P., Danilin A.R. The optimality of regularizing algorithms in the solution of ill-posed problems. Differ. Uravn., 1976, vol. 12, no. 7, pp. 1323–1326 (in Russian).

4.   Tanana V.P., Ershova A.A. On the solution of an inverse boundary value problem for composite materials. Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2018, vol. 28, no. 4, pp. 474–488 (in Russian).

5.   Kabanikhin S.I. Inverse and Ill-posed problems. Berlin: de Gruyter, 2012, 459 p. ISBN: 978-3-11-022400-9 . Original Russian text published in Kabanikhin S. I. Obratnye i nekorrektnye zadachi. Novosibirsk: Sib. Nauch. Izd-vo, 2009, 457 p.

6.   Ivanov V.K., Vasin V.V., Tanana V.P. Theory of linear ill-posed problems and its applications. Utrecht: VSP, 2002, 281 p. ISBN: 9789067643672 . Original Russian text published in Ivanov V.K., Vasin V.V., Tanana V.P. Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya. Moscow: Nauka Publ., 1978, 206 p.

7.   Tanana V.P. On the order-optimality of the projection regularization method in solving inverse problems. Sib. Zh. Ind. Mat., 2004, vol. 7, no. 2, pp. 117–132 (in Russian).

8.   Tikhonov V.N., Glasko V.B. Methods of determining the surface temperature of a body. U.S.S.R. Comput. Math. Math. Phys., 1967, vol. 7, no. 4, pp. 267–273. DOI: 10.1016/0041-5553(67)90161-9 

9.   Lavrent’ev M.M., Romanov V.G., Shishatskii S.P. Ill-posed problems of mathematical physics and analysis. Translations of Mathematical Monographs, 64. Providence, R.I.: American Mathematical Society, 1986, 290 p. ISBN: 9780821845172 . Original Russian text published in Lavrent’ev M.M., Romanov V.G., Shishatskii S.P. Nekotorye zadachi matematicheskoi fiziki i analiza. Moscow: Nauka Publ., 1980, 285 p.

10.   Budak B.M., Samarskii A.A., Tikhonov A. N. Sbornik zadach po matematicheskoi fizike [Collection of problems in mathematical physics]. Moscow: Tekhniko-Teoret. Literatura Publ., 1956, 683 p.

11.   Lykov A.V. Teoriya teploprovodnosti [Theory of heat conduction]. Moscow: Vysshaya Shkola Publ., 1967, 599 p.

12.   Tanana V., Sidikova A. Optimal methods for ill-posed problems: with applications to heat conduction. Berlin: De Gruyter, 2018. 130 p. ISBN: 978-3-11-057721-1 .

13.   Fikhtengol’ts G.M. Osnovy matematicheskogo analiza [The Fundamentals of Mathematical Analysis]. Vol. 2. Moscow: Fiz. Mat. Lit. Publ., 2006, 864 p.

14.   Privalov I.I. Vvedenie v teoriyu funktsii kompleksnogo peremennogo [Introduction to the theory of functions of a complex variable]. Moscow: Nauka Publ., 1984, 432 p. ISBN(14th ed.): 5-06-003612-X .

Cite this article as: V.P. Tanana, A.I. Sidikova. Approximate solution of an inverse boundary value problem for a system of differential equations of parabolic type and estimation of the error of this solution, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 3, pp. 247–264.