Geodesic
The Dirichlet problem for a mixed-type equation
Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 1, pp. 80-93.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider the first boundary value problem in a rectangular area for a mixed-type equation of the second kind with a singular coefficient. The criterion of the uniqueness of the problem solution is determined. The uniqueness of the problem solution is proved on the basis of completeness of the system of eigenfunctions of the corresponding one-dimensional spectral problem. The solution of the problem is built explicitly as a sum of Fourier–Bessel. There is the problem of the small denominators that appears when justifying the uniform convergence of the constructed series. In this regard, an evaluation of separateness from zero with a corresponding small denominator asymptotic behavior is found. This estimate has allowed to prove the convergence of the series and its derivatives up to the second order, and the existence theorem for the class of regular solutions of this equation.
Keywords: mixed-type equation, Dirichlet problem, spectral method, uniqueness, Fourier–Bessel series, small denominators
Mots-clés : singular coefficient, existence. 
@article{VSGTU_2017_21_1_a3,
     author = {R. M. Safina},
     title = {The {Dirichlet} problem for a mixed-type equation},
     journal = {Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences},
     pages = {80--93},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSGTU_2017_21_1_a3/}
}
TY  - JOUR
AU  - R. M. Safina
TI  - The Dirichlet problem for a mixed-type equation
JO  - Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
PY  - 2017
SP  - 80
EP  - 93
VL  - 21
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VSGTU_2017_21_1_a3/
LA  - ru
ID  - VSGTU_2017_21_1_a3
ER  - 
%0 Journal Article
%A R. M. Safina
%T The Dirichlet problem for a mixed-type equation
%J Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences
%D 2017
%P 80-93
%V 21
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VSGTU_2017_21_1_a3/
%G ru
%F VSGTU_2017_21_1_a3
R. M. Safina. The Dirichlet problem for a mixed-type equation. Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 21 (2017) no. 1, pp. 80-93. http://geodesic.mathdoc.fr/item/VSGTU_2017_21_1_a3/

[1] Frankl F., “To the theory of the Laval nozzle”, Izv. Akad. Nauk SSSR Ser. Mat., 9:2 (1945), 121–143 (In Russian) | MR | Zbl

[2] Bitsadze A. V., “The incorrectness of the Dirichlet problem for equations of mixed type”, Dokl. Akad. Nauk SSSR, 122:2 (1953), 167–170 (In Russian) | MR

[3] Shabat B. V., “Examples of solving the Dirichlet problem for equations of mixed type”, Dokl. Akad. Nauk SSSR, 112:3 (1957), 386–389 (In Russian) | Zbl

[4] Vakhaniya N. N., “On a particular problem for the mixed type”, Trudy Akad. Nauk Gruz. SSR, 3 (1963), 69–80 (In Russian)

[5] Cannon J. R., “A Dirichlet problem for an equation of mixed type with a discontinuous coefficient”, Annali di Matematica, 61:1 (1963), 371–377 | DOI | MR | Zbl

[6] Nakhushev A. M., “A uniqueness condition of the Dirichlet problem for an equation of mixed type in a cylindrical domain”, Differ. Uravn., 6:1 (1970), 190–191 (In Russian) | MR | Zbl

[7] Khachev M. M., “The Dirichlet problem for the Tricomi equation in a rectangle”, Differ. Uravn., 11:1 (1975), 151–160 (In Russian) | MR | Zbl

[8] Khachev M. M., “The Dirichlet problem for a certain equation of mixed type”, Differ. Uravn., 12:1 (1976), 137–143 (In Russian) | MR | Zbl

[9] Soldatov A. P., “Dirichlet-type problems for the Lavrent'ev–Bitsadze equation. I. Uniqueness theorems”, Dokl. Math., 48:2 (1994), 410–414 | MR | Zbl

[10] Soldatov A. N., “Dirichlet-type problems for the Lavrent'ev–Bitsadze equation. II. Existence theorems”, Dokl. Math., 48:3 (1994), 433–437 | MR | Zbl

[11] Khachev M. M., Pervaia kraevaia zadacha dlia lineinykh uravnenii smeshannogo tipa [The First Boundary Value Problem for Linear Equations of Mixed Type], Elbrus, Nalchik, 1998, 168 pp. (In Russian)

[12] Sokhadze R. I., “The first boundary value problem for an equation of mixed type with weighted glueing conditions along a line of parabolic degeneration”, Differ. Uravn., 17:1 (1981), 150–156 (In Russian) | MR | Zbl

[13] Sokhadze R. I., “The first boundary value problem for an equation of mixed type in a rectangle”, Differ. Uravn., 19:1 (1983), 127–134 (In Russian) | MR | Zbl

[14] Sabitov K. B., “The Dirichlet problem for equations of mixed type in a rectangular domain”, Dokl. Math., 75:2 (2007), 193–196 | MR | Zbl

[15] Sabitov K. B., Suleimanova A. Kh. The Dirichlet problem for a mixed-type equation of the second kind in a rectangular domain, Russian Math. (Iz. VUZ), 51:4 (2007), 42–50 | DOI | MR | Zbl

[16] Sabitov K. B., Suleimanova A. Kh., “The Dirichlet problem for a mixed-type equation with characteristic degeneration in a rectangular domain”, Russian Math. (Iz. VUZ), 53:11 (2009), 37–45 | DOI | MR | Zbl

[17] Sabitov K. B., Vagapova E. V., “Dirichlet problem for an equation of mixed type with two degeneration lines in a rectangular domain”, Differ. Equ., 49:1 (2013), 68–78 | DOI | MR | Zbl

[18] Khairullin R. S., “On the Dirichlet problem for a mixed-type equation of the second kind with strong degeneration”, Differ. Equ., 49:4 (2013), 510–516 | DOI | DOI | MR | Zbl

[19] Safina R. M., “Dirichlet problem for Pulkin's equation in a rectangular domain”, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2014, no. 10(121), 91–101 (In Russian) | Zbl

[20] Safina R. M., “Keldysh problem for a mixed-type equation of the second kind with the Bessel operator”, Differ. Equ., 51:10 (2015), 1347–1359 | DOI | DOI | MR | Zbl

[21] Watson G. N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944, viii+804 pp. | MR | Zbl

[22] Olver F. W. J., “Differential equations with irregular singularities; Bessel and confluent hypergeometric functions (Chapter 7)”, Asymptotics and Special Functions, Academic Press, Inc., Boston, 1974, 229–278 | DOI

[23] Erdélyi A., Magnus W., Oberhettinger F., Tricomi F. G., Higher transcendental functions, v. II, Bateman Manuscript Project, McGraw-Hill Book Co., Inc., New York, Toronto, London, 1953, xvii+396 pp. | Zbl

[24] Safina R. M., “Keldysh problem for Pulkin's equation in a rectangular domain”, Vestnik SamGU. Estestvenno-Nauchnaya Ser., 2015, no. 3(125), 53–63 (In Russian)