Abstract
Using the method of spectral analysis, for the mixed type equation u xx + (sgny)u yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.
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References
Desin, A. A. “On the Solvable Extensions of Partial Differential Operators”, Outlines of the Joint Soviet- American Symposium on Partial Diff. Equat., Novosibirsk, 65–66 (1963).
Desin, A. A. “Operators Involving a First Derivative with Respect to Time and Nonlocal Boundary Conditions”, Mathematics of the USSR-Izvestiya 1, No. 1, 57–80 (1967).
Nakhusheva, Z. A. “On a Nonlocal Problem of A. A. Dezin for the Lavrent’ev–Bitsadze Equation”, Differential Equations 45, 1223–1228 (2009).
Nakhusheva, Z. A. Nonlocal Boundary-Value Problems for Basic orMixed Type Differential Equations (Nal’chik, KBNTz RAN, 2011) [in Russian].
Sabitov, K. B. “Dirichlet Problem forMixed Type Equations in a Rectangular Domain”, DokladyMathematics 75, No. 2, 193–196 (2007).
Sabitov, K. B. and Sidorenko, O. G. “Problem with Periodicity Conditions for a Degenerating Equation of Mixed Type”, Differential Equations 46, 1, 108–116 (2010).
Frankl’, F. I. “Subsonic Flow About a Profile with a Supersonic Zone”, Prikl. Matem. i Mekhan. 20, No. 2, 196–202 (1956) [in Russian].
Zhegalov, V. I. “Boundary-Value Problem for a Mixed Type Equation with Boundary Condition on Both its Characteristics, Discontinuous on Transition Line”, Uchen. Zap. Kazan Univ. 122, No. 3, 3–16 (1962) [in Russian].
Nakhushev, A.M. “Certain Boundary-Value Problems for Hyperbolic and Mixed Type Equations”, Differential Equations 5, No. 1, 44–69 (1969) [in Russian].
Arnol’d, V. I. “Small Denominators and Problems of Stability of Motion in Classical and Celestial Mechanics”, RussianMathematical Surveys 18, No. 6, 85–191 (1963).
Sabitov, K. B. and Safin, E.M. “The Inverse Problem for an Equation of Mixed Parabolic-Hyperbolic Type”, Mathematical Notes 87, No. 6, 880–889 (2010).
Khinchin, A. Ya. Continued Fractions (The University of Chicago Press, Chicago and London, 1964; Nauka, Moscow, 1978).
Sabitov, K. B. “Boundary Value Problem for a Parabolic-Hyperbolic Equation with a Nonlocal Integral Condition”, Differ. Equ. 46, No. 10, 1472–1481 (2010).
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Original Russian Text © K.B. Sabitov, V.A. Novikova, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 6, pp. 61–72.
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Sabitov, K.B., Novikova, V.A. Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation. Russ Math. 60, 52–62 (2016). https://doi.org/10.3103/S1066369X16060074
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DOI: https://doi.org/10.3103/S1066369X16060074