Abstract
We consider an equation of mixed elliptic-hyperbolic type. The right-hand part of the equation is represented as a product of two functions, each which only depends on one of variables. We study an inverse problem for this equation; it consists in finding the unknown multipliers. We establish a uniqueness criterion for this problem. The solution is constructed as a series in eigenfunctions for the corresponding one-dimensional spectral problem. Under some conditions on the boundary of domain, boundary values, and the unknown multipliers, we establish separation from zero of small denominators of ratios containing in the coefficients of the series; the existence of solution and its stability are proved as well.
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References
Frankl, F.I. Selected Papers on Gas Dynamics (Nauka, Moscow, 1973) [in Russian].
Kraiko, A.N., Makarov, V.E., Pudovikov, D.E. “Construction of the bow wave through upstream computation of a supersonic flow by the method of characteristics”, Comput. Math. Math. Phys. 39 (11), 1814–1819 (1999).
Kraiko, A.N., P’yankov, K.S. “Construction of airfoils and engine nacelles that are supercritical in a transonic perfect-gas flow Comput”. Math. Math. Phys. 40 (12), 1816–1829 (2000)
Alexandrov, V.G., Kraiko, A.N., Reent, K.S. “Integral and local characteristics of supersonic pulse detonation ramjet engine”, Matem. Mod. 15 (6), 17–26 (2003).
Sabitov, K.B., Khadzhi, I.A. “The boundary-value problem for the Lavrent’ev-Bitsadze equation with unknown right-hand side”, Russian Math. (Iz. VUZ) 55 (5), 35–42 (2011).
Sabitov, K.B., Martem’yanova, N.V. “A nonlocal inverse problem for a mixed-type equation”, Russian Math. (Iz. VUZ) 55 (2), 61–74 (2011).
Sabitov, K.B., Martem’yanova, N.V. “An inverse problem for an equation of elliptic-hyperbolic type with a nonlocal boundary condition”, Siberian Math. J. 53 (3), 507–519 (2012).
Sabitov, K.B., Martemyanova, N.V. “Nonlocal boundary problem for the third order equation of mixed type”, Contemp. Anal. Appl. Math. 3 (2), 153–169 (2015).
Tikhonov, A.N. “On stability of inverse problems”, Dokl. AN SSSR 39 (5), 195–198 (1943) [in Russian].
Arsenin, V.Ya., Tikhonov, A.N., Method of solving ill-posed problems (Nauka, Moscow, 1974) [in Russian].
Lavrentiev, M.M., Romanov, V.G., Shishatskii, S.P., Ill-Posed Problems of Mathematical Physics and Analysis (American Mathematical Society, Providence, 1986).
LavrentTev, M.M., Reznitskay, K.G., Yakhno, V.G. One-Dimensional Inverse Problems of Mathematical Physics (American Mathematical Society Translations, Series 2, Volume 130, 1986).
Ivanov, V.K., Vasin, V.V., Tanana, V.P. The Theory of Linear Ill-Posed Problems and Its Applications (Nauka, Moscow, 1978) [in Russian].
Denisov, A.M. Introduction to the Theory of Inverse Problems (Moscow, Moscow State University Press, 1994) [in Russian].
Prilepko A.I. Methods for Solving Inverse Problems in Mathematical Physics (New York, Basel, Marcel Dekker Inc., 1999).
Kabanikhin, S.I., Inverse and Ill-Posed Problems. Theory and Applications (De Gruyter, Germany, 2011).
Denisov, A.M. “Problems of determining the unknown source in parabolic and hyperbolic equations”, Comput. Math. Math. Phys. 55(5), 829–833, (2015).
Kozhanov, A.I. “Inverse problems of recovering the right-hand side of a special type of parabolic equations”, Mathematical Notes of NEFU 23 (4), 31–45 (2016) [in Russian].
Kamynin, V.L. “Inverse problem of determining the right-hand side in a degenerating parabolic equation with unbounded coefficients”, Comput. Math. Math. Phys. 57 (5), 833–842 (2017).
Kostin, A.B. “Criteria of the uniqueness of solutions and well-posedness of inverse source problems”, J. Math. Sci. (N. Y.) 230 (6), 907–949 (2018).
Prilepko, A.I., Kostin, A.B., Solov’ev, V.V. “Inverse source and inverse coefficients problems for elliptic and parabolic equations in Hoelder and Sobolev spaces”, Sib. J. Pure and Appl. Math. 17 (3), 67–85 (2017) [in Russian].
Arnol’d, V.I. “Small denominators and problems of stability of motion in classical and celestial mechanics”, Russian Math. Surveys 18 (6), 85–191 (1963).
Sabitov, K.B. “Dirichlet problem for mixed-type equations in a rectangular domain”, Doklady Mathematics 75 (2), 193–196 (2007).
Sabitov, K.B., Safin, E.M. “The inverse problem for an equation of mixed parabolic-hyperbolic type”, Math. Notes 87 (6), 880–889 (2010).
Bukhshtab, A.A. Number Theory (Prosveschenie, Moscow, 1978) [in Russian].
Khinchin, A.Ya. Continued Fraction (Dover Publ. Inc., Mineola, New York, 1997).
Funding
The paper is fulfilled under financial support of the Russian Foundation for Basic Research, projects nos. 5-31-50018, 16-31-00421.
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Russian Text © The Author(s), 2020, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2020, 2020, No. 1, pp. 46–63.
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Martemyanova, N.V. Nonlocal Inverse Problem of Finding Unknown Multipliers in the Right-Hand Part of Lavrentiev-Bitsadze Equation. Russ Math. 64, 40–57 (2020). https://doi.org/10.3103/S1066369X20010053
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DOI: https://doi.org/10.3103/S1066369X20010053