The Neumann Problem for a Multidimensional Elliptic Equation with Several Singular Coefficients in an Infinite Domain
In this article the Neumann problem for a multidimensional elliptic equation with several singular coefficients in the infinite domain is studied. Using the method of the integral energy, the uniqueness of solution is proved. In proof of existence of the explicit solution of the Neumann problem a di...
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| Vydáno v: | Lobachevskii journal of mathematics Ročník 43; číslo 1; s. 199 - 206 |
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| Jazyk: | angličtina |
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Moscow
Pleiades Publishing
01.01.2022
Springer Nature B.V |
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| ISSN: | 1995-0802, 1818-9962 |
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| Abstract | In this article the Neumann problem for a multidimensional elliptic equation with several singular coefficients in the infinite domain is studied. Using the method of the integral energy, the uniqueness of solution is proved. In proof of existence of the explicit solution of the Neumann problem a differentiation formula, some adjacent and limiting relations for the Lauricella hypergeometric functions and the values of some multidimensional improper integrals are used. |
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| AbstractList | In this article the Neumann problem for a multidimensional elliptic equation with several singular coefficients in the infinite domain is studied. Using the method of the integral energy, the uniqueness of solution is proved. In proof of existence of the explicit solution of the Neumann problem a differentiation formula, some adjacent and limiting relations for the Lauricella hypergeometric functions and the values of some multidimensional improper integrals are used. |
| Author | Ergashev, T. G. Tulakova, Z. R. |
| Author_xml | – sequence: 1 givenname: T. G. surname: Ergashev fullname: Ergashev, T. G. email: ergashev.tukhtasin@gmail.com organization: Tashkent Institute of Irrigation and Agricultural Mechanization Engineers – sequence: 2 givenname: Z. R. surname: Tulakova fullname: Tulakova, Z. R. email: ziyodacoders@gmail.com organization: Ferghana branch of the Tashkent University of Information Technologies |
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| Keywords | Neumann problem Lauricella hypergeometric function of many variables adjacent and limiting relations multidimensional elliptic equations with several singular coefficients multidimensional improper integrals |
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| References | BerdyshevA. S.RyskanA. R.The Neumann and Dirichlet problems for one four-dimensional degenerate equationLobachevskii J. Math.20204110511066413593610.1134/S1995080220060062 GradshteynI. S.RyzhikI. M.Table of Integrals, Series and Products2007AmsterdamAcademic1208.65001 BersL.Mathematical Aspects of Subsonic and Transonic Gas Dynamics1958New YorkWiley0083.20501 ErgashevT. G.Potentials for three-dimensional singular elliptic equation and their application to the solving a mixed problemLobachevskii J. Math.20204110671077413593710.1134/S1995080220060086 ErgashevT. G.Fundamental solutions of the generalized Helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variablesLobachevskii J. Math.2020411526413105210.1134/S1995080220010047 MoiseevE. I.MogimiM.On the completeness of eigenfunctions of the Tricomi problem for a degenerate equation of mixed typeDiffer. Equat.20054114621466224146310.1007/s10625-005-0298-8 K. B. Sabitov and I. A. Khadzhi, ‘‘The boundary-value problem for the Lavrent’ev–Bitsadze equation with unknown right-hand side,’’ Russ. Math. (Iz. VUZ) 55 (5), 35–42 (2011). M. S. Salakhitdinov and B. I. Islomov, ‘‘A nonlocal boundary-value problem with conormal derivative for a mixed-type equation with two inner degeneration lines and various orders of degeneracy, ’’ Russ. Math. (Iz. VUZ) 55 (1), 42–49 (2011). SalakhitdinovM. S.MirsaburovM.A problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed typeMath. Notes200986704715264134810.1134/S0001434609110133 MoiseevE. I.MogimiM.On the completeness of eigenfunctions of the Neumann–Tricomi problem for a degenerate equation of mixed typeDiffer. Equat.20054117891791224402810.1007/s10625-006-0015-2 M. S. Salakhitdinov and N. B. Islamov, ‘‘Nonlocal boundary-value problem with Bitsadze–Samarskii condition for equation of parabolic-hyperbolic type of the second kind,’’ Russ. Math. (Iz. VUZ) 59 (6), 34–42 (2015). SrivastavaH. M.KarlssonP. W.Multiple Gaussian Hypergeometric Series1985ChichesterHalsted, Ellis Horwood0552.33001 FranklF. I.Selected Works on the Gas Dynamics1973MoscowNauka ErgashevT. G.Generalized Holmgren problem for an elliptic equation with several singular coefficientsDiffer. Equat.202056842856413688610.1134/S0012266120070046 SalakhitdinovM. S.HasanovA.To the theory of multidimensional Gellerstedt equationUzb. Matem. Zh.200739510925691481189.35185 YuldashevT. K.IslomovB. I.AbdullaevA. A.On solvability of a Poincare–Tricomi type problem for an elliptic-hyperbolic equation of the second kindLobachevskii J. Math.202142663675426024210.1134/S1995080221030239 T. G. Ergashev and Z. R. Tulakova, ‘‘Dirichlet problem for an elliptic equation with several singular coefficients in an infinite domain,’’ Russ. Math. (Iz. VUZ) 65 (7), 71–80 (2021). ErgashevT. G.Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficientsJ. Sib. Fed. Univ. Math. Phys.2020134857407425810.17516/1997-1397-2020-13-1-48-57 KarimovK. T.Nonlocal problem for an elliptic equation with singular coefficients in semi-infinite parallelepipedLobachevskii J. Math.2020414657413105610.1134/S1995080220010084 HasanovA.ErgashevT. G.New decomposition formulas associated with the Lauricella multivariable hypergeometric functionsMont. Taur. J. Pur. Appl. Math.20213317326 K. B. Sabitov and V. A. Novikova, ‘‘Nonlocal A. A. Dezin’s problem for Lavrent’ev–Bitsadze equation,’’ Russ. Math. (Iz. VUZ) 60 (6), 52–62 (2016). MarichevO. I.Singular boundary value problem for a generalized biaxially symmetric Helmholtz equationDokl. Akad. Nauk SSSR1976230523526422914 MoiseevE. I.MoiseevT. E.KholomeevaA. A.Non-uniqueness of the solution of the internal Neumann–Hellerstedt problem for the Lavrentiev–Bitsadze equationSib. Zh. Chist. Prikl. Mat.20171752571438.35291 ErdelyiA.MagnusW.OberhettingerF.TricomiF. G.Higher Transcendental Functions1953New YorkMcGraw-Hill0051.30303 SalakhitdinovM. S.UrinovA. K.Eigenvalue problems for a mixed-type equation with two singular coefficientsSib. Math. J.20074870771710.1007/s11202-007-0072-7 AppellP.Kampé de FérietJ.Fonctions Hypergéometriques et Hypersphériques: Polynômes d’Hermite1926ParisGauthier-Villars52.0361.13 K. B. Sabitov, ‘‘On the theory of the Frankl problem for equations of mixed type,’’ Russ. Math. (Iz. VUZ) 81 (1), 99–136 (2017). SalakhitdinovM. S.IslomovB.Boundary value problems for an equation of mixed type with two inner lines of degeneracyDokl. Math.19914323523811133260782.35045 KarimovK. T.Boundary value problems in a semi-infinite parallelepiped for an elliptic equation with three singular coefficientsLobachevskii J. Math.202142560571426023110.1134/S1995080221030124 6678_CR23 K. T. Karimov (6678_CR21) 2021; 42 T. K. Yuldashev (6678_CR14) 2021; 42 P. Appell (6678_CR16) 1926 E. I. Moiseev (6678_CR4) 2005; 41 E. I. Moiseev (6678_CR5) 2017; 17 T. G. Ergashev (6678_CR15) 2020; 13 H. M. Srivastava (6678_CR26) 1985 O. I. Marichev (6678_CR24) 1976; 230 T. G. Ergashev (6678_CR17) 2020; 56 A. Erdelyi (6678_CR25) 1953 T. G. Ergashev (6678_CR29) 2020; 41 A. S. Berdyshev (6678_CR22) 2020; 41 F. I. Frankl (6678_CR2) 1973 T. G. Ergashev (6678_CR18) 2020; 41 M. S. Salakhitdinov (6678_CR9) 1991; 43 6678_CR8 6678_CR7 6678_CR6 6678_CR11 6678_CR10 E. I. Moiseev (6678_CR3) 2005; 41 M. S. Salakhitdinov (6678_CR13) 2007; 48 L. Bers (6678_CR1) 1958 A. Hasanov (6678_CR27) 2021; 3 K. T. Karimov (6678_CR20) 2020; 41 I. S. Gradshteyn (6678_CR28) 2007 M. S. Salakhitdinov (6678_CR19) 2007; 3 M. S. Salakhitdinov (6678_CR12) 2009; 86 |
| References_xml | – reference: SalakhitdinovM. S.MirsaburovM.A problem with a nonlocal boundary condition on the characteristic for a class of equations of mixed typeMath. Notes200986704715264134810.1134/S0001434609110133 – reference: AppellP.Kampé de FérietJ.Fonctions Hypergéometriques et Hypersphériques: Polynômes d’Hermite1926ParisGauthier-Villars52.0361.13 – reference: T. G. Ergashev and Z. R. Tulakova, ‘‘Dirichlet problem for an elliptic equation with several singular coefficients in an infinite domain,’’ Russ. Math. (Iz. VUZ) 65 (7), 71–80 (2021). – reference: HasanovA.ErgashevT. G.New decomposition formulas associated with the Lauricella multivariable hypergeometric functionsMont. Taur. J. Pur. Appl. Math.20213317326 – reference: MarichevO. I.Singular boundary value problem for a generalized biaxially symmetric Helmholtz equationDokl. Akad. Nauk SSSR1976230523526422914 – reference: ErgashevT. G.Fundamental solutions of the generalized Helmholtz equation with several singular coefficients and confluent hypergeometric functions of many variablesLobachevskii J. Math.2020411526413105210.1134/S1995080220010047 – reference: MoiseevE. I.MogimiM.On the completeness of eigenfunctions of the Tricomi problem for a degenerate equation of mixed typeDiffer. Equat.20054114621466224146310.1007/s10625-005-0298-8 – reference: MoiseevE. I.MoiseevT. E.KholomeevaA. A.Non-uniqueness of the solution of the internal Neumann–Hellerstedt problem for the Lavrentiev–Bitsadze equationSib. Zh. Chist. Prikl. Mat.20171752571438.35291 – reference: K. B. Sabitov, ‘‘On the theory of the Frankl problem for equations of mixed type,’’ Russ. Math. (Iz. VUZ) 81 (1), 99–136 (2017). – reference: MoiseevE. I.MogimiM.On the completeness of eigenfunctions of the Neumann–Tricomi problem for a degenerate equation of mixed typeDiffer. Equat.20054117891791224402810.1007/s10625-006-0015-2 – reference: M. S. Salakhitdinov and B. I. Islomov, ‘‘A nonlocal boundary-value problem with conormal derivative for a mixed-type equation with two inner degeneration lines and various orders of degeneracy, ’’ Russ. Math. (Iz. VUZ) 55 (1), 42–49 (2011). – reference: ErdelyiA.MagnusW.OberhettingerF.TricomiF. G.Higher Transcendental Functions1953New YorkMcGraw-Hill0051.30303 – reference: KarimovK. T.Nonlocal problem for an elliptic equation with singular coefficients in semi-infinite parallelepipedLobachevskii J. Math.2020414657413105610.1134/S1995080220010084 – reference: GradshteynI. S.RyzhikI. M.Table of Integrals, Series and Products2007AmsterdamAcademic1208.65001 – reference: SalakhitdinovM. S.HasanovA.To the theory of multidimensional Gellerstedt equationUzb. Matem. Zh.200739510925691481189.35185 – reference: BerdyshevA. S.RyskanA. R.The Neumann and Dirichlet problems for one four-dimensional degenerate equationLobachevskii J. Math.20204110511066413593610.1134/S1995080220060062 – reference: BersL.Mathematical Aspects of Subsonic and Transonic Gas Dynamics1958New YorkWiley0083.20501 – reference: YuldashevT. K.IslomovB. I.AbdullaevA. A.On solvability of a Poincare–Tricomi type problem for an elliptic-hyperbolic equation of the second kindLobachevskii J. Math.202142663675426024210.1134/S1995080221030239 – reference: SalakhitdinovM. S.UrinovA. K.Eigenvalue problems for a mixed-type equation with two singular coefficientsSib. Math. J.20074870771710.1007/s11202-007-0072-7 – reference: SalakhitdinovM. S.IslomovB.Boundary value problems for an equation of mixed type with two inner lines of degeneracyDokl. Math.19914323523811133260782.35045 – reference: FranklF. I.Selected Works on the Gas Dynamics1973MoscowNauka – reference: SrivastavaH. M.KarlssonP. W.Multiple Gaussian Hypergeometric Series1985ChichesterHalsted, Ellis Horwood0552.33001 – reference: K. B. Sabitov and V. A. Novikova, ‘‘Nonlocal A. A. Dezin’s problem for Lavrent’ev–Bitsadze equation,’’ Russ. Math. (Iz. VUZ) 60 (6), 52–62 (2016). – reference: M. S. Salakhitdinov and N. B. Islamov, ‘‘Nonlocal boundary-value problem with Bitsadze–Samarskii condition for equation of parabolic-hyperbolic type of the second kind,’’ Russ. Math. (Iz. VUZ) 59 (6), 34–42 (2015). – reference: KarimovK. T.Boundary value problems in a semi-infinite parallelepiped for an elliptic equation with three singular coefficientsLobachevskii J. Math.202142560571426023110.1134/S1995080221030124 – reference: ErgashevT. G.Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficientsJ. Sib. Fed. Univ. Math. Phys.2020134857407425810.17516/1997-1397-2020-13-1-48-57 – reference: K. B. Sabitov and I. A. Khadzhi, ‘‘The boundary-value problem for the Lavrent’ev–Bitsadze equation with unknown right-hand side,’’ Russ. Math. (Iz. VUZ) 55 (5), 35–42 (2011). – reference: ErgashevT. G.Generalized Holmgren problem for an elliptic equation with several singular coefficientsDiffer. Equat.202056842856413688610.1134/S0012266120070046 – reference: ErgashevT. G.Potentials for three-dimensional singular elliptic equation and their application to the solving a mixed problemLobachevskii J. Math.20204110671077413593710.1134/S1995080220060086 – ident: 6678_CR11 doi: 10.3103/S1066369X15060067 – volume: 3 start-page: 317 year: 2021 ident: 6678_CR27 publication-title: Mont. Taur. J. Pur. Appl. Math. – volume: 41 start-page: 15 year: 2020 ident: 6678_CR29 publication-title: Lobachevskii J. Math. doi: 10.1134/S1995080220010047 – volume: 41 start-page: 46 year: 2020 ident: 6678_CR20 publication-title: Lobachevskii J. Math. doi: 10.1134/S1995080220010084 – volume: 41 start-page: 1051 year: 2020 ident: 6678_CR22 publication-title: Lobachevskii J. Math. doi: 10.1134/S1995080220060062 – volume-title: Table of Integrals, Series and Products year: 2007 ident: 6678_CR28 – volume: 56 start-page: 842 year: 2020 ident: 6678_CR17 publication-title: Differ. Equat. doi: 10.1134/S0012266120070046 – volume: 48 start-page: 707 year: 2007 ident: 6678_CR13 publication-title: Sib. Math. J. doi: 10.1007/s11202-007-0072-7 – volume: 13 start-page: 48 year: 2020 ident: 6678_CR15 publication-title: J. Sib. Fed. Univ. Math. Phys. doi: 10.17516/1997-1397-2020-13-1-48-57 – volume-title: Selected Works on the Gas Dynamics year: 1973 ident: 6678_CR2 – volume: 17 start-page: 52 year: 2017 ident: 6678_CR5 publication-title: Sib. Zh. Chist. Prikl. Mat. – ident: 6678_CR7 doi: 10.3103/S1066369X11050069 – volume: 42 start-page: 560 year: 2021 ident: 6678_CR21 publication-title: Lobachevskii J. Math. doi: 10.1134/S1995080221030124 – volume: 41 start-page: 1789 year: 2005 ident: 6678_CR4 publication-title: Differ. Equat. doi: 10.1007/s10625-006-0015-2 – volume: 3 start-page: 95 year: 2007 ident: 6678_CR19 publication-title: Uzb. Matem. Zh. – volume: 42 start-page: 663 year: 2021 ident: 6678_CR14 publication-title: Lobachevskii J. Math. doi: 10.1134/S1995080221030239 – ident: 6678_CR10 doi: 10.3103/S1066369X11010051 – volume-title: Fonctions Hypergéometriques et Hypersphériques: Polynômes d’Hermite year: 1926 ident: 6678_CR16 – volume-title: Multiple Gaussian Hypergeometric Series year: 1985 ident: 6678_CR26 – volume: 43 start-page: 235 year: 1991 ident: 6678_CR9 publication-title: Dokl. Math. – volume-title: Mathematical Aspects of Subsonic and Transonic Gas Dynamics year: 1958 ident: 6678_CR1 – volume: 41 start-page: 1462 year: 2005 ident: 6678_CR3 publication-title: Differ. Equat. doi: 10.1007/s10625-005-0298-8 – volume: 41 start-page: 1067 year: 2020 ident: 6678_CR18 publication-title: Lobachevskii J. Math. doi: 10.1134/S1995080220060086 – ident: 6678_CR6 doi: 10.1070/IM8401 – volume: 86 start-page: 704 year: 2009 ident: 6678_CR12 publication-title: Math. Notes doi: 10.1134/S0001434609110133 – volume: 230 start-page: 523 year: 1976 ident: 6678_CR24 publication-title: Dokl. Akad. Nauk SSSR – ident: 6678_CR23 doi: 10.3103/S1066369X21070082 – ident: 6678_CR8 doi: 10.3103/S1066369X16060074 – volume-title: Higher Transcendental Functions year: 1953 ident: 6678_CR25 |
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| Title | The Neumann Problem for a Multidimensional Elliptic Equation with Several Singular Coefficients in an Infinite Domain |
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