On potential theory for the generalized bi-axially symmetric elliptic equation in the plane
DOI:
https://doi.org/10.26577/JMMCS.2021.v109.i1.01Keywords:
Аппелдiң екi айнымалы гипергеометрияқ функциясы, жәй қабатты және қосқабатты потенциалдар, Грин функциясы, iргелi шешiм, Дирихле есебiAbstract
Жалпыланған екi өске симметриялық эллиптикалық теңдеудiң iргелi шешiмдерi екi айныма- лысы бар Аппелдiң гипергеометриялық функциясы арқылы өрнектеледi, олардың қасиеттерi жоғарыда келтiрiлген теңдеу үшiн шектi есептердi зерттеу үшiн қажет. Бұл жұмыста Аппел- дiң гипергеометриялық функциясының кейбiр қасиеттерiн қолдана отырып, бiз қос қабатты және жай қабатты потенциалдардың тығыздығы үшiн шектi теоремаларды дәлелдеймiз және интегралдық теңдеулер аламыз. Құрылған потенциалдар теориясының нәтижелерiн жазы- қтықтың бiрiншi ширегiнде шектелген облыста екi сингулярлы коэффициентi бар екi өлшемдi эллиптикалық теңдеу үшiн Дирихле есебiн зерттеуге қолданамыз.References
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[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.
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[6] Pulkin S. P. , Some boundary-value problems for the equation uxx± uyy+pxux= 0, Scientistic Notes Kuibyshev Pedag.Inst., 21 (1958), 3–54.
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[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). On potential theory for the generalized bi-axially symmetric elliptic equation in the plane. Journal of Mathematics, Mechanics and Computer Science, 109(1), 3–24. https://doi.org/10.26577/JMMCS.2021.v109.i1.01
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