Geodesic
On a problem for a three-dimensional elliptic equation with Bessel operators
Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 3, pp. 375-388.

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

The Keldysh problem for a three-dimensional elliptic equation with three singular coefficients in a rectangular parallelepiped is studied. Based on the property of completeness of systems of eigenfunctions of two one-dimensional spectral problems, a uniqueness theorem is proved. The solution to the problem posed is constructed as the sum of a double Fourier — Bessel series.
Keywords: Keldysh problem, elliptic equation, spectral method, singular coefficient, Bessel function. 
@article{CHFMJ_2024_9_3_a2,
     author = {K. T. Karimov},
     title = {On a problem for a three-dimensional elliptic equation with {Bessel} operators},
     journal = {\v{C}el\^abinskij fiziko-matemati\v{c}eskij \v{z}urnal},
     pages = {375--388},
     publisher = {mathdoc},
     volume = {9},
     number = {3},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHFMJ_2024_9_3_a2/}
}
TY  - JOUR
AU  - K. T. Karimov
TI  - On a problem for a three-dimensional elliptic equation with Bessel operators
JO  - Čelâbinskij fiziko-matematičeskij žurnal
PY  - 2024
SP  - 375
EP  - 388
VL  - 9
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHFMJ_2024_9_3_a2/
LA  - ru
ID  - CHFMJ_2024_9_3_a2
ER  - 
%0 Journal Article
%A K. T. Karimov
%T On a problem for a three-dimensional elliptic equation with Bessel operators
%J Čelâbinskij fiziko-matematičeskij žurnal
%D 2024
%P 375-388
%V 9
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHFMJ_2024_9_3_a2/
%G ru
%F CHFMJ_2024_9_3_a2
K. T. Karimov. On a problem for a three-dimensional elliptic equation with Bessel operators. Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 3, pp. 375-388. http://geodesic.mathdoc.fr/item/CHFMJ_2024_9_3_a2/

[1] Kipriyanov I.A., Singular elliptic boundary value problems, Nauka Publ., Moscow, 1997 (In Russ.) | MR

[2] Kipriyanov I.A., “Fourier-Bessel transforms and imbedding theorems for weight classes”, Proceedings of Steklov Mathematical Institute of Russian Academy of Sciences, 89 (1967), 130–213 (In Russ.) | Zbl

[3] Kipriyanov I.A., Kulikov A.A., “The Paley — Wiener — Schwartz theorem for the Fourier — Bessel transformation”, Doklady Mathematics, 37:1 (1988), 13–17 | MR | Zbl

[4] Katrakhov V.V., Sitnik S.M., “The transmutation method and boundary-value problems for singular elliptic equations”, Contemporary Mathematics. Fundamental Directions, 64:2 (2018), 211–426. (In Russ.) | MR

[5] Lyakhov L.N., B-Hypersingular integrals and their applications to the description of Kipriyanov functional classes and to integral equations with B-potential kernels, Lipetsk State Pedagogical University, Lipetsk, 2007 (In Russ.)

[6] Sitnik S.M., Shishkina E.L., Method of transformation operators for differential equations with Bessel operators, Fizmatlit, Moscow, 2019 (In Russ.)

[7] Muravnik A. B., “Fourier — Bessel transformation of compactly supported nonnegative functions and estimates of solutions of singular differential equations”, Functional Differential Equations, 8:3–4 (2001), 353–363 | MR | Zbl

[8] Keldysh M.V., “On some cases of degeneration of an equation of elliptic type on the domain boundary”, Reports of USSR Academy of Sciences, 77:2 (1951), 181–183 (In Russ.) | Zbl

[9] Vostrova L.E., Pul’kin S.P., “Singular problem with a normal derivative”, Volga mathematical collection, 1966, no. 5, 49–57 (In Russ.) | MR | Zbl

[10] Pul’kin S.P., “On uniqueness of a solution of a Gellerstedt singular problem”, News of universities. Mathematics, 1960, no. 6, 214–225 (In Russ.) | MR | Zbl

[11] Zaitseva N. V., “Keldysh type problem for B-hyperbolic equation with integral boundary value condition of the first kind”, Lobachevskii Journal of Mathematics, 38:1 (2017), 162–169 | DOI | MR | Zbl

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

[13] Urinov A.K., Karimov K.T., “The unique solvability of boundary value problems for a 3D elliptic equation with three singular coefficients”, Russian Mathematics, 63:2 (2019), 61–72 | DOI | MR | Zbl

[14] Karimov K. T., “Boundary value problems in a semi-infinite parallelepiped for an elliptic equation with three singular coefficients”, Lobachevskii Journal of Mathematics, 42:3 (2021), 560–571 | DOI | MR | Zbl

[15] Karimov K.T., “Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients in a semi-infinite parallelepiped”, Bulletin of Udmurt University. Mathematics. Mechanics. Computer sciences, 30:1 (2020), 31–48 (In Russ.) | MR | Zbl

[16] Karimov K.T., “Keldysh problem for a three-dimensional equation of mixed type with three singular coefficients”, Bulletin of KRAUNC. Physical and mathematical sciences, 34:1 (2021), 29–46 (In Russ.) | DOI | MR | Zbl

[17] Watson G.N., A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944 | MR | Zbl

[18] Kapilevich M.B., “On an equation of mixed elliptic-hyperbolic type”, Mathematical collection, 30:1 (1952), 11–38 (In Russ.) | Zbl

[19] Lebedev N.N., Special functions and their applications, Fizmatlit, Moscow, 1968 (In Russ.)

[20] Urinov A.K, Karimov Sh.T., The Erdelyi — Kober operators and its applications to partial differential equations, Fergana State University, Ferdgana, 2021 (In Russ.)

[21] Ilyin V.A., Poznyak E.G., Fundamentals of mathematical analysis, v. I, Fizmatlit, Moscow, 2005 (In Russ.) | MR