Abstract:
We study the first boundary-value problem in a rectangle for an equation
of mixed type with a singular coefficient. We establish a criterion for
the uniqueness of solutions and construct the solution as the sum of a series
in the system of eigenfunctions of a one-dimensional eigenvalue problem.
Justifying the uniform convergence of the series encounters a problem of
small denominators. To deal with this we obtain bounds for the separation
of the small denominators from zero along with the corresponding asymptotic
results. These bounds enable us to justify the convergence of the series
in the class of regular solutions of the equation.
Keywords:
equation of mixed type, singular coefficient, Dirichlet problem, Keldysh problem,
survey, uniqueness, orthogonal series, small denominators, bounds, existence, stability.
This paper was written with the support of the Russian Foundation
for Basic Research (grant no.~16-31-50008~mol\_nr), Russian Foundation for
Basic Research and Republic Bashkortostan (grant no.~17-41-020516~r\_a).
Citation:
K. B. Sabitov, R. M. Safina, “The first boundary-value problem for an equation of mixed type with a singular coefficient”, Izv. Math., 82:2 (2018), 318–350
\Bibitem{SabSaf18}
\by K.~B.~Sabitov, R.~M.~Safina
\paper The first boundary-value problem for an equation of mixed type with a~singular coefficient
\jour Izv. Math.
\yr 2018
\vol 82
\issue 2
\pages 318--350
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Linking options:
https://www.mathnet.ru/eng/im8596
https://doi.org/10.1070/IM8596
https://www.mathnet.ru/eng/im/v82/i2/p79
This publication is cited in the following 18 articles:
K. B. Sabitov, “The Dirichlet problem for inhomogeneous mixed-type equation with Lavrent'ev–Bitsadze operator”, Izv. Math., 88:4 (2024), 655–677
B. Yu. Irgashev, “Boundary-value problem for a degenerate high-order equation with gluing conditions involving a fractional derivative”, Rend. Circ. Mat. Palermo, II. Ser, 2024
M. Kh. Ruziev, N. T. Yuldasheva, “Ob odnoi kraevoi zadache dlya uravneniya Gellerstedta s singulyarnym koeffitsientom”, Izv. vuzov. Matem., 2024, no. 12, 57–70
M. Kh. Ruziev, N. T. Yuldasheva, “On a Boundary Value Problem for the Gellerstedt Equation with a Singular Coefficient”, Russ Math., 68:12 (2024), 57
A. A. Abashkin, “O zadache Dirikhle v pryamougolnoi oblasti dlya uravneniya Lavrenteva–Bitsadze”, Izv. vuzov. Matem., 2023, no. 5, 3–10
A. A. Abashkin, “On the Dirichlet Problem in Rectangular Domain for Lavrentiev–Bitsadze Equation”, Russ Math., 67:5 (2023), 1
R. R. Ashurov, M. B. Murzambetova, “Kraevaya zadacha dlya uravneniya smeshannogo tipa s ellipticheskim operatorom vysokogo poryadka”, Vestnik KRAUNTs. Fiz.-mat. nauki, 39:2 (2022), 7–19
M. Ruziev, “A boundary value problem for mixed type equation with singular coefficient”, Topical issues Of Thermophysics, Energetics and Hydrogasdynamics in the Arctic Conditions, Dedicated to the 85th Birthday Anniversary of Professor E. A. Bondarev, AIP Conference Proceedings, 2528, 2022, 020013
K. B. Sabitov, “On the uniform convergence of the expansion of a function in Fourier–Bessel range”, Russian Math. (Iz. VUZ), 66:11 (2022), 79–85
M. Kh. Ruziev, “On a problem with shift on pieces of boundary characteristics for the Gellerstedt equation with singular coefficients”, Lobachevskii J Math, 43:2 (2022), 484
K. B. Sabitov, “The Dirichlet Problem for a mixed-type equation with fractional derivatives”, Russian Math. (Iz. VUZ), 66:9 (2022), 71–81
A. A. Abashkin, “On the Keldysh problem for mixed type equation with two singular lines”, Russian Math. (Iz. VUZ), 66:2 (2022), 1–14
M. Kh. Ruziev, “A boundary value problem for a mixed type equation with singular coefficients”, Russian Math. (Iz. VUZ), 66:7 (2022), 14–24
K. Sabitov, I. Burkhanova-Haji, “Reverse Problem to Find Right Parts Mixed Equations with the Chaplygin Operator”, Lobachevskii J Math, 42:15 (2021), 3726
R. M. Safina, “Dirichlet problem for mixed type equation with characteristic degeneration and singular coefficient”, Lobachevskii J. Math., 41:1, SI (2020), 80–88
V. N. Zaitseva, “First initial-boundary value problem for B -hyperbolic equation”, Lobachevskii J. Math., 40:2 (2019), 240–247
A. A. Abashkin, I. P. Egorova, “Dirichlet problem in parallelepiped for elliptic equation with three singular coefficients”, Russian Math. (Iz. VUZ), 63:10 (2019), 1–12
K. B. Sabitov, N. V. Zaitseva, “The second initial-boundary value problem for a B-hyperbolic equation”, Russian Math. (Iz. VUZ), 63:10 (2019), 66–76
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