Жазықтықтағы жалпыланған екi өске симметриялық эллиптикалық теңдеудiңпотенциалдық теориясы жайында
DOI:
https://doi.org/10.26577/JMMCS.2021.v109.i1.01Кілттік сөздер:
Appell hypergeometric function, generalized bi-axially symmetric elliptic equation, potential theory, Green’s function, Dirichlet problemАннотация
Fundamental solutions of the generalized biaxially symmetric elliptic equation are expressed in terms of the well-known Appel hypergeometric function in two variables, the properties of which are necessary for studying boundary value problems for the above equation. In this paper, using some properties of the Appel hypergeometric function, we prove limit theorems and derive integral equations for the double- and simple-layer potentials and apply the results of the constructed potential theory to the study of the Dirichlet problem for a two-dimensional elliptic equation with two singular coefficients in a domain bounded in the first quarter of the plane.Библиографиялық сілтемелер
1] Mikhlin S.G., An Advanced Course of Mathematical Physics, North Holland Series in Applied Mathematics and Mechanics,11 North-Holland Publishing, Amsterdam, London, 1970.
[2] G¨unter N. M., Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Frederick UngarPublishing Company, New York, 1967.
[3] Kondratev B. P. , Potential theory. New methods and problems with solutions, Mir, Moscow, 2007.
[4] Gellerstedt S., Sur un probleme aus limites pour l’equation y2szxx+ zyy= 0, Arkiv Mat. Ast och Fysik, 25A(10) (1935),1–12.
[5] Frankl F.I., Selected works on gas dynamics, Nauka, Moscow, 1973.
[6] Pulkin S. P. , Some boundary-value problems for the equation uxx± uyy+pxux= 0, Scientistic Notes Kuibyshev Pedag.Inst., 21 (1958), 3–54.
[7] Smirnov M. M., Degenerate Elliptic and Hyperbolic Equations, Nauka, Moscow, 1966.
[8] Mavlyaviev R.M. Solution of fundamental boundary value problems for a B-elliptic equation by the potential method,Russian Mathematics, 46(9), 61—63 (2002).
[9] Khismatullin A.Sh. Solution of boundary value problems for one degenerate B-elliptic equation of the 2nd kind by themethod of potentials, Russian Mathematics, 51(1), 58—70 (2007).
[10] Mukhlisov F. G., Nigmedzyanova A. M. Solution of boundary value problems for a degenerating elliptic equation of thesecond kind by the method of potentials, Russian Mathematics, 53(8), 46–57 (2009).
[11] Ergashev T.G. Potentials for three-dimensional singular elliptic equation and their application to the solving a mixedproblem, Lobachevskii Journal of Mathematics, 41(6), 1067–1077 (2020).
[12] Ergashev T. G. Double- and simple-layer potentials for a three-dimensional elliptic equation with a singular coefficientand their applications, Russian Mathematics, 65(1), 72–86 (2021).
[13] Ergashev T.G. , Potentials for the Singular Elliptic Equations and Their Application, Results in Applied Mathematics,7 (2020), 1–15. https://doi.org/10.1016/j.rinam.2020.100126
[14] Srivastava H.M. , Hasanov A. , Ergashev T.G. , A family of potentials for elliptic equations with one singular coefficientand their applications , Mathematical Methods in Applied Sciences, 43(10) (2020), 6181–6199.
[15] Srivastava H. M., Hasanov A., Choi J., Double-layer potentials for a generalized bi-axially symmetric Helmholtz equation,Sohag Journal of Mathematics, 2(1) (2015), 1–10.
[16] Berdyshev A. S. , Hasanov A., Ergashev T.G., Double-layer potentials for a generalized bi-axially symmetric Helmholtzequation.II , Complex Variables and Elliptic Equations, 65(2) (2020), 316–332.
[17] Ergashev T.G., Third double-layer potential for a generalized bi-axially symmetric Helmholtz equation, Ufa MathematicalJournal, 10(4) (2018) , 111–121.
[18] Ergashev T.G., The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation, Tomsk StateUniversity Journal of Mathematics and Mechanics, 50 (2017), 45–56.
[19] Erd´elyi A., Magnus W., Oberhettinger F. and Tricomi F.G., Higher Transcendental Functions, Vol. I, McGraw-Hill BookCompany, New York, Toronto and London, 1953; Russian edition, Izdat. Nauka, Moscow, 1973.
[20] Copson E.T., On Hadamard’s Elementary Solution, Proceedings of the Poyal Society of Edinburgh Section A:Mathematics, 69(1) (1970), 19–27.
[21] Ergashev T.G., Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients,Journal of Siberian Federal University. Mathematics and Physics, 13(1) (2020), 48–57.
[22] Gradshteyn I.S., Ryzhik I.M., Tables of Integrals, Series, and Products (Corrected and Enlarged edition prepared by A.Jeffrey and D. Zwillinger), Academic Press, New York, 1980; Eighth edition, 2014.
[2] G¨unter N. M., Potential Theory and Its Applications to Basic Problems of Mathematical Physics, Frederick UngarPublishing Company, New York, 1967.
[3] Kondratev B. P. , Potential theory. New methods and problems with solutions, Mir, Moscow, 2007.
[4] Gellerstedt S., Sur un probleme aus limites pour l’equation y2szxx+ zyy= 0, Arkiv Mat. Ast och Fysik, 25A(10) (1935),1–12.
[5] Frankl F.I., Selected works on gas dynamics, Nauka, Moscow, 1973.
[6] Pulkin S. P. , Some boundary-value problems for the equation uxx± uyy+pxux= 0, Scientistic Notes Kuibyshev Pedag.Inst., 21 (1958), 3–54.
[7] Smirnov M. M., Degenerate Elliptic and Hyperbolic Equations, Nauka, Moscow, 1966.
[8] Mavlyaviev R.M. Solution of fundamental boundary value problems for a B-elliptic equation by the potential method,Russian Mathematics, 46(9), 61—63 (2002).
[9] Khismatullin A.Sh. Solution of boundary value problems for one degenerate B-elliptic equation of the 2nd kind by themethod of potentials, Russian Mathematics, 51(1), 58—70 (2007).
[10] Mukhlisov F. G., Nigmedzyanova A. M. Solution of boundary value problems for a degenerating elliptic equation of thesecond kind by the method of potentials, Russian Mathematics, 53(8), 46–57 (2009).
[11] Ergashev T.G. Potentials for three-dimensional singular elliptic equation and their application to the solving a mixedproblem, Lobachevskii Journal of Mathematics, 41(6), 1067–1077 (2020).
[12] Ergashev T. G. Double- and simple-layer potentials for a three-dimensional elliptic equation with a singular coefficientand their applications, Russian Mathematics, 65(1), 72–86 (2021).
[13] Ergashev T.G. , Potentials for the Singular Elliptic Equations and Their Application, Results in Applied Mathematics,7 (2020), 1–15. https://doi.org/10.1016/j.rinam.2020.100126
[14] Srivastava H.M. , Hasanov A. , Ergashev T.G. , A family of potentials for elliptic equations with one singular coefficientand their applications , Mathematical Methods in Applied Sciences, 43(10) (2020), 6181–6199.
[15] Srivastava H. M., Hasanov A., Choi J., Double-layer potentials for a generalized bi-axially symmetric Helmholtz equation,Sohag Journal of Mathematics, 2(1) (2015), 1–10.
[16] Berdyshev A. S. , Hasanov A., Ergashev T.G., Double-layer potentials for a generalized bi-axially symmetric Helmholtzequation.II , Complex Variables and Elliptic Equations, 65(2) (2020), 316–332.
[17] Ergashev T.G., Third double-layer potential for a generalized bi-axially symmetric Helmholtz equation, Ufa MathematicalJournal, 10(4) (2018) , 111–121.
[18] Ergashev T.G., The fourth double-layer potential for a generalized bi-axially symmetric Helmholtz equation, Tomsk StateUniversity Journal of Mathematics and Mechanics, 50 (2017), 45–56.
[19] Erd´elyi A., Magnus W., Oberhettinger F. and Tricomi F.G., Higher Transcendental Functions, Vol. I, McGraw-Hill BookCompany, New York, Toronto and London, 1953; Russian edition, Izdat. Nauka, Moscow, 1973.
[20] Copson E.T., On Hadamard’s Elementary Solution, Proceedings of the Poyal Society of Edinburgh Section A:Mathematics, 69(1) (1970), 19–27.
[21] Ergashev T.G., Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients,Journal of Siberian Federal University. Mathematics and Physics, 13(1) (2020), 48–57.
[22] Gradshteyn I.S., Ryzhik I.M., Tables of Integrals, Series, and Products (Corrected and Enlarged edition prepared by A.Jeffrey and D. Zwillinger), Academic Press, New York, 1980; Eighth edition, 2014.
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Ergashev, T., & Hasanov, A. (2021). Жазықтықтағы жалпыланған екi өске симметриялық эллиптикалық теңдеудiңпотенциалдық теориясы жайында. Қазұу Хабаршысы. Математика, механика, информатика сериясы, 109(1), 3–24. https://doi.org/10.26577/JMMCS.2021.v109.i1.01
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