Abstract
We consider an equation of mixed elliptic-hyperbolic type, whose right-hand side represents a product of two one-dimensional functions. We establish a criterion for the unique solvability of this equation and construct its solution as a sum of series on the set of its eigenfunctions. Under certain constraints imposed on the ratio of the rectangle sides, on boundary functions, and on known multipliers in the right-hand side of the equation, we obtain estimates separating small denominators that appear in coefficients of constructed series from zero.
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References
Frankl, F. I., Selected Works on Gas Dynamics (Nauka, Moscow, 1973) [in Russian].
Kraiko, A. N., Makarov, V. E., and 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, No. 11, 1814–1819 (1999)
Krajko, 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, No. 12, 1816–1829 (2000).
Alexandrov, V. G., Krajko, A. N., Reent, K. S. “Integral and Local Characteristics of Supersonic Pulse Detonation Ramjet Engines” Mat. Model. 15, No. 6, 17–26 (2003) [in Russian].
Babich, V. M. Linear Equations ofMathematical Physics (Nauka, Moscow, 1964) [in Russian].
Sabitov, K. B. and Khadzhi, I. A., “The Boundary-Value Problem for the Lavrent’ev–Bitsadze Equation with Unknown Right-Hand Side”, RussianMathematics 55, No. 5, 35–42 (2011).
Sabitov, K. B. and Martem’yanova, N. V. “A Nonlocal Inverse Problem for aMixed-Type Equation”, Russian Mathematics 55, No. 5, 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”, Sib.Math. J. 53, No. 3, 507–519 (2012).
Arsenin, V. Ya. and Tikhonov, A. N. Methods for Solving Ill-Posed Problems (Nauka, Moscow, 1974) [in Russian].
Ivanov, V. K., Vasin, V. V., and Tanana, V. P. The Theory of Linear Ill-Posed Problems and its Applications (Nauka, Moscow, 1978) [in Russian].
Lavrent’ev, M. M., Romanov, V. G., and Shishatskii, S. T., Ill-Posed Problems of Mathematical Physics and Analysis (Nauka, Moscow, 1980) [in Russian].
Lavrent’ev, M. M., Reznitskaya, K. G., and Yakhno, V. G. One-Dimensional Ill-Posed Problems in Mathematical Physics (Nauka SO, Novosibirsk, 1982) [in Russian].
Romanov, V. G., Inverse Problems ofMathematical Physics (Nauka, Moscow, 1984) [in Russian].
Denisov, A.M., Introduction to the Theory of Inverse Problems (MGU,Moscow, 1994) [in Russian].
Kabanikhin, S. I. Inverse and Ill-Posed Problems (Novosibirsk, Sib. Nauchn. Izd-vo, 2009) [in Russian].
Arnol’d, V. I. “Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics” Russ. Math. Surv. 18, No. 6, 85–191 (1963).
Sabitov, K. B. “Dirichlet Problem for Mixed-Type Equations in a Rectangular Domain”, Dokl.Math. 75, No. 2, 193–196 (2007).
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Original Russian Text ©K.B. Sabitov, N.V. Martem’yanova, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 2, pp. 44–57.
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Sabitov, K.B., Martem’yanova, N.V. The inverse problem for the Lavrent’ev–Bitsadze equation connected with the search of elements in the right-hand side. Russ Math. 61, 36–48 (2017). https://doi.org/10.3103/S1066369X17020050
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DOI: https://doi.org/10.3103/S1066369X17020050