Abstract
For the equation K(t)u xx + u tt − b 2 K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t|m, m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u x (0, t) = u x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.
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References
Lerner, M.E. and Repin, O.A., On Frankl Type Problems for Some Elliptic Equations with Degeneration of Various Types, Differ. Uravn., 1999, vol. 35, no. 8, pp. 1087–1093.
Moiseev, E.I., On the Solution of a Nonlocal Boundary Value Problem by the Spectral Method, Differ. Uravn., 1999, vol. 35, no. 8, pp. 1094–1100.
Moiseev, E.I., On the Solvability of a Nonlocal Boundary Value Problem, Differ. Uravn., 2001, vol. 37, no. 11, pp. 1565–1567.
Sabitov, K.B. and Sidorenko, O.G., On the Unique Solvability of a Nonlocal Problem for a Degenerate Elliptic Equation by the Spectral Method, in Tr. mezhdunar. konf. “Spektral’naya teoriya differentsial’nykh operatorov i rodstvennye problemy,” posv. yubileyu akad. V.A. Il’ina (Proc. Int. Conf. “Spectral Theory of Differential Operators and Related Problems,” Devoted to the Jubilee of Acad. V. A. Il’in), Sterlitamak-Ufa, 2003, vol. 1, pp. 213–219.
Sabitov, K.B. and Sidorenko, O.G., Essentially Nonlocal Problems for Degenerate Parabolic Equation, in Mater. mezhdunar. rossiisko-kazakhskogo simpoziuma “Nelokal’nye kraevye zadachi i problemy sovremennogo analiza i informatiki” (Proc. Int. Russian-Kazakhstan Symp. “Nonlocal Boundary Value Problems and Problems of Modern Analysis and Computer Science”), Nalchik, 2004, pp. 156–160.
Il’in, V.A., Uniqueness and Membership in W 12 of the Classical Solution of a Mixed Problem for a Self-Adjoint Hyperbolic Equation, Mat. Zametki, 1975, vol. 17, no. 1, pp. 91–101.
Il’in, V.A., A Theorem on the Uniqueness and Membership in the Class W 12 of the Classical Solution of a Mixed Problem for a Nonself-Adjoint Hyperbolic Equation in an Arbitrary Cylinder, Differ. Uravn., 1975, vol. 11, no. 1, pp. 60–65.
Il’in, V.A., Necessary and Sufficient Conditions for the Basis Property in L p and Equiconvergence with a Trigonometric Series of Spectral Expansions and Expansions in Systems of Exponentials, Dokl. Akad. Nauk SSSR, 1983, vol. 273, no. 5, pp. 789–793.
Sabitov, K.B., The Dirichlet Problem for Equations of Mixed Type in a Rectangular Domain, Dokl. Akad. Nauk, 2007, vol. 413, no. 1, pp. 23–26.
Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., Higher Transcendental Functions (Bateman Manuscript Project), New York: McGraw-Hill, 1953. Translated under the title Vysshie transtsendentnye funktsii, Moscow, 1955, vol. 2.
Tricomi, F., Differential Equations, London, 1961. Translated under the title Differentsial’nye uravneniya, Moscow: Inostrannaya Literatura, 1962.
Sabitov, K.B., The Dirichlet Problem for Equations of Mixed Type in a Half-Strip, Differ. Uravn., 2007, vol. 43, no. 10, pp. 1417–1422.
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Original Russian Text © K.B. Sabitov, O.G. Sidorenko, 2010, published in Differentsial’nye Uravneniya, 2010, Vol. 46, No. 1, pp. 105–113.
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Sabitov, K.B., Sidorenko, O.G. Problem with periodicity conditions for a degenerating equation of mixed type. Diff Equat 46, 108–116 (2010). https://doi.org/10.1134/S0012266110010118
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DOI: https://doi.org/10.1134/S0012266110010118