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## Archive of Issues

Russia Chelyabinsk
Year
2020
Volume
30
Issue
1
Pages
59-63
 Section Mathematics Title Completeness of the system of eigenfunctions of the Sturm-Liouville problem with the singularity Author(-s) Tanana V.P.a Affiliations South Ural State Universitya Abstract Mathematical modeling of composite materials plays an important role in modern technology, and the solution and study of inverse boundary value problems of heat transfer is impossible without the use of systems of eigenfunctions of the Sturm-Liouville problem for the differential equation with discontinuous coefficients. One of the most important properties of such systems is their completeness in the corresponding spaces. This property of systems allows to prove theorems of existence and uniqueness of both direct problems and inverse boundary value problems of thermal conductivity, and also to prove numerical methods of solving such problems. In this paper, we prove the completeness of the Sturm-Liouville problem in the space $L_2[r_0,r_2]$ for a second-order differential operator with a discontinuous coefficient. This problem arises when investigating and solving the inverse boundary problem of thermal conductivity for a hollow ball consisting of two balls with different temperature conductivity coefficients. Self-conjugacy, injectivity, and positive definiteness of this operator are proved. Keywords system of eigenfunctions, Sturm-Liouville problem, composite material, inverse boundary value problems UDC 517.983.54 MSC 34L10, 35P10 DOI 10.35634/vm200105 Received 19 January 2020 Language Russian Citation Tanana V.P. Completeness of the system of eigenfunctions of the Sturm-Liouville problem with the singularity, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 1, pp. 59-63. References Mukhtarov O.Sh., Kadakal M. Some spectral properties of one Sturm-Liouville type problem with discontinuous weight, Siberian Mathematical Journal, 2005, vol. 46, issue 4, pp. 681-694 (in Russian). https://doi.org/10.1007/s11202-005-0069-z Erdogan S., Mukhtarov O.Sh. Spectral properties of discontinuous Sturm-Liouville problems with a finite number of transmission conditions, Mediterranean Journal of Mathematics, 2016, vol. 13, issue 1, pp. 153-170. https://doi.org/10.1007/s00009-014-0487-x Tanana V.P., Sidikova A.I. 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, issue 3, pp. 247-264 (in Russian). https://doi.org/10.21538/0134-4889-2019-25-3-247-264 Lyusternik L.A., Sobolev V.I. Elementy funktsional'nogo analiza (Elements of functional analysis), Moscow: Nauka, 1965. Osipov Yu.S., Vasil'ev F.P., Potapov M.M. Osnovy metoda dinamicheskoi regulyarizatsii (Bases of the dynamical regularization method), Moscow: Moscow State University, 1999. Full text