Abstract
We consider a boundary-value problem for a mixed-type equation with two perpendicular singularity lines given in a domain whose elliptic part is a rectangle, while the hyperbolic one is a vertical half-strip. This problem differs from the Dirichlet one by the fact that at the left boundary of the rectangle and the half-strip we specify the vanishing order of the desired function rather than its value. We find a solution to the problem by a spectral method with the use of the Fourier–Bessel series and prove the uniqueness of the solution. We substantiate the uniform convergence of the corresponding series under certain requirements to the problem statement.
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References
Pulkin, S. P. Uniqueness of Solution of a Singular Problem of Gellerstedt, Izv. Vyssh. Uchebn. Zaved. Mat., No. 6, 214–225 (1960).
Volkodavov, V. F. and Nosov, V. A. Solution of T α Problem for a Mixed-Type Equation in an Unbounded Domain of a Special Type, Volzhsk.Matem. Sb.,Vyp. 9, 32–38 (1971).
Repin, O. A. A Nonlocal Problem for a Mixed-Type Equation with a Singular Coefficient, Vestn. Samarsk. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, No. 34, 5–9 (2005).
Marichev, O. I. Kilbas, A. A. and Repin, O. A. Boundary-Value Problems for Partial Differential Equations with Discontinuous Coefficients (SGEU, Samara, 2008) [in Russian].
Sabitov, K. B. The Dirichlet Problem for Mixed-Type Equations in a Rectangular Domain, Dokl.Math. 75, No. 2, 193–196 (2007).
Sabitov, K. B. and Suleimanova, A. Kh. The Dirichlet Problem for a Mixed Type Equation of the Second Kind in a Rectangular Domain, RussianMathematics (Iz. VUZ) 51, No. 4, 42–50 (2007).
Sabitova, Yu. K. A Nonlocal Problem for the Lavrent’ev–Bitsadze Equation in a Rectangular Domain, Trudy Sterlitamaksk. Filiala AN RB, 94–102 (2009).
Sabitova, Yu. K. and Bakhristova, A. A. The Dirichlet Problem for a Mixed-Type Equation with the Lavrent’ev–Bitsadze OPerator, Trudy Sterlitamaksk. Filiala AN RB, 103–110 (2009).
Sabitov, K. B. and Sidorenko, O. F. Problem with Periodicity Conditions for a Degenerating Equation of Mixed Type, Differ. Equ. 46, No. 1, 108–116 (2010).
Sabitov, K. B. Boundary Value Problem for a Parabolic-Hyperbolic Equation with a Nonlocal Integral Condition, Differ. Equ. 46, No. 10, 1472–1481 (2010).
Moiseev, E. I. Solvability of a Nonlocal Boundary Value Problem, Differ. Equ. 37, No. 11, 1643–1646 (2001).
Abashkin, A. A. One-Valued Solvability of a Nonlocal Problem for the Axisymmetric Helmholtz Equation, Vestn. Samarsk. Gos. Tekhn. Univ. Ser. Fiz.-Mat. Nauki, No. 3, 26–34 (2011).
Lebedev, N. N. Special Functions and Their Applications (Lan’, SPb., 2010) [in Russion].
Watson, G. N. A Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, 1922; In. Lit., Moscow, 1949).
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Original Russian Text © A.A. Abashkin, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 2, pp. 3–9.
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Abashkin, A.A. A spectral solution method for a boundary-value problem for a mixed-type equation with two singularity lines. Russ Math. 60, 1–6 (2016). https://doi.org/10.3103/S1066369X16020018
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DOI: https://doi.org/10.3103/S1066369X16020018