Abstract
We study a mixed-type equation of the second kind with a singular coefficient. With the help of the spectral analysis method we establish a uniqueness criterion for a solution of the problem with incomplete boundary data. The solution represents the sum of the Fourier–Bessel series. The substantiation of its uniform convergence is based on an estimate of the separation from zero of the small denominator with the corresponding asymptotic behavior. This allows us to prove the convergence of the series in the class of regular solutions.
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References
Sokhadze, R. S. “The First Boundary Value Problem for Mixed-Type Equations in a Rectangle”, Differ. Equations 19, 108–113 (1983).
Khachev, M. M. The First Boundary Value Problem for Linear Mixed-Type Equations (“El’brus”, Nal’chik, 1998) [in Russian].
Sabitov, K. B. and Suleimanova, A. Kh. “The Dirichlet Problem for a Mixed Type Equation of the Second Kind in a Rectangular Domain”, RussianMathematics 51, No. 4, 42–50 (2007).
Sabitov K.B. and Suleimanova A.Kh., “The Dirichlet Problem for aMixed-Type Equation with Characteristic Degeneration in a Rectangular Domain”, RussianMathematics 53, No. 11, 37–45 (2009).
Khairullin, R. S. “On the Dirichlet Problem for a Mixed-Type Equation of the Second Knd With Strong Degeneration”, Differ. Equ. 49, No. 4, 510–516 (2013).
Arnold, V. I. “Small Denominators and Problems of Stability of Motions in Classical and CelestialMechanics”, Russ. Math. Surv. 18, No. 6, 85–191 (1963).
Lomov, S. A. and Lomov, I. S. Fundamentals of theMathematical Theory of the Boundary Layer (Mosk. Univ.,Moscow, 2011) [in Russian].
Safina, R. M. “Keldysh Problem for Pulkin’s Equation in a Rectangular Domain”, Vestnik SamGU, Estestvenno-Nauchnaya Ser., No. 3, 53–64 (2015) [in Russian].
Safina, R. M. “Keldysh Problem for a Mixed-Type Equation of the Second Kind with the Bessel Operator”, Differ. Equ. 51, No. 10, 1347–1359 (2015).
Keldysh, M. V. “On Certain Cases of Degeneration of Equations of Elliptic Type on the Boundary of a Domain”, Sov. Phys.Dokl. 77 (2), 181–183 (1951) [in Russian].
Pul’kin, S. P. “Uniqueness of Solution of a Singular Problem of Gellerstedt”, Izv. Vyssh. Uchebn. Zaved., Mat., No. 6, 214–225 (1960) [in Russian].
Sabitov, K. B. On the Theory of Mixed-Type Equations (Fizmatlit, Moscow, 2014) [in Russian].
Sabitov, K. B. and Vagapova, E. V. “Dirichlet Problem for an Equation ofMixed Type with TwoDegeneration Lines in a Rectangular Domain”, Differ. Equ. 49, No. 1, 68–78 (2013).
Olver, F.W. J. Introduction to Asymptotics and Special Functions (Academic Press, New York–London, 1974; Mir,Moscow, 1986).
Bateman, H. and Erdelyi, A. Higher Transcendental Functions (Nauka, Moscow, 1966), Vol. II [Russ. transl.].
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Original Russian Text © R.M. Safina, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 8, pp. 53–61.
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Safina, R.M. Keldysh problem for mixed type equation with strong characteristic degeneration and singular coefficient. Russ Math. 61, 46–54 (2017). https://doi.org/10.3103/S1066369X17080059
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DOI: https://doi.org/10.3103/S1066369X17080059