A Linear Inverse Problem for a Multidimensional Mixed-Type Second-Order Equation of the First-Kind
In this paper, we study the well-posedness of a linear inverse problem for a mixed-type second-order multidimensional equation of the first kind. We prove the unique solvability of this problem in a certain function class with the help of “ε-regularization”, a priori estimates, and successive approximation methods.
Key wordsmultidimensional mixed-type second-order equation of the first kind linear inverse problem correctness of solution “ε-regularization ”methods a priori estimate successive approximation
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The authors are grateful to the academician of the Academy of Sciences of Republic of Uzbekistan Shavkat Arifzhanovich Alimov for the discussion of results obtained in this paper and for valuable advices, and to the reviewer for useful remarks.
This work was performed in accordance with the research plan OT-FA-88 and supported by the grant MRU-OT-1/2017.
- 1.Bitsadze, A.V. “Ill-Posedness of the Dirichlet Problem for Equations of Mixed Type”, Dokl. Akad, Nauk SSSR 122 (2), 167–170 (1953) [in Russian].Google Scholar
- 2.Frankl, F.I., “Subsonic Flow about a Profile with a Supersonic Zone Terminated by a Direct Shock Wave”, Prikl. Matem. i Mekhan. 20 (2), 196–202 (1956) [in Russian].Google Scholar
- 5.Tsybikov, B.N. Well-Posedness of a Periodic Problem for a Multidimensional Equation of Mixed Type (in: Nonclassical partial differential equations, Novosibirsk, 201–206 (1986)).Google Scholar
- 6.Anikonov, Yu.E. Some Methods of Investigations of Multidimensional Inverse Problems for Differential Equations (Nauka, Novosibirsk, 1978) [in Russian].Google Scholar
- 7.Kabanikhin, S.I. Inverse and Ill-Posed Problems (Sib. Nauch. Izd-vo, Novosibirsk, 2009) [in Russian].Google Scholar
- 8.Lavrent’ev, M.M., Romanov, V.G., Vasil’ev, V.G., Multidimensional Inverse Problems for Differential Equations (Nauka, Novosibirsk, 1969) [in Russian].Google Scholar
- 16.Vragov, V.N. Boundary Value Problems for Nonclassic Equations of Mathematical Physics (NGU, Novosibirsk, 1983) [in Russian].Google Scholar
- 17.Ladyzhenskaya, O.A. Boundary Value Problems of Mathematical Physics (Nauka, Moscow, 1973) [in Russian].Google Scholar