Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 180-196. ISSN 2079-6641

Содержание выпуска/Contents of this issue

Research Article

MSC 35A08, 35J25, 35J70, 35J75 

Holmgren problem for elliptic equation with singular coefficients

T. G. Ergashev¹³, A. Hasanov¹²

¹V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, M. Ulugbek str. 81, 100125, Tashkent, Uzbekistan

²Department of Mathematics, Analysis, Logic and Discrete Mathematics of Ghent University, B 9000, Gent, Krijgslaan 281, Belgium
³Institute of Irrigation and Agricultural Mechanization Engineers, 100100, Tashkent, Kari-Niyazi st., 39, Uzbekistan

E-mail: ergashev.tukhtasin@gmail.com

In the present work, we investigate the Holmgren problem for an multidimensional elliptic equation with several singular coefficients. We use a fundamental solution of the equation, containing Lauricella’s hypergeometric function in many variables. Then using an «abc» method, the uniqueness for the solution of the Holmgren problem is proved. Applying a method of Green’s function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition and summation formulae, formulae of differentiation and some adjacent relations for Lauricella’s hypergeometric functions in many variables were used in order to find the explicit solution for the formulated problem.

Keywords: Holmgren problem, multidimensional elliptic equations with several singular coefficients, decomposition formula, summation formula, Lauricella hypergeometric function in many variables, Green’s function

DOI: 10.26117/2079-6641-2020-32-3-180-196

Original article submitted: 04.07.2020

Revision submitted: 07.10.2020

For citation. Ergashev T. G., Hasanov A. Holmgren problem for elliptic equation with singular coefficients. Vestnik KRAUNC. Fiz.-mat. nauki. 2020, 32: 3, 180-196. DOI: 10.26117/2079-6641-2020-32-3-180-196

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Ergashev T. G., Hasanov A., 2020

Научная статья

УДК 519.644

Задача Холмгрена для эллиптического уравнения с сингулярными коэффициентами

Т. Г. Эргашев¹³, А. Хасанов¹²

¹Институт Математики имени В. И. Романовского Академии наук Узбекистана, г. Ташкент, ул. Мирзо Улугбека 85, 100170, Республика Узбекистан
²Кафедра математики, анализа, логики и дискретной математики Гентского университета, г. Гент, Бельгия
³Ташкентский институт инженеров ирригации и механизации сельского хозяйства, 100100, г. Ташкент, ул. Кари-Ниязи, 39, Республика Узбекистан

E-mail: ergashev.tukhtasin@gmail.com

В данной работе мы исследуем задачу Холмгрена для многомерного эллиптического уравнения с несколькими сингулярными коэффициентами. Мы используем фундаментальное решение уравнения, содержащее гипергеометрическую функцию Лауричеллы от многих переменных. Затем методом «abc» доказывается единственность решения проблемы Холмгрена. Применяя метод функции Грина, мы можем найти решение задачи в явном виде. Более того, формулы разложения и суммирования, формулы дифференцирования и некоторые смежные соотношения для гипергеометрических функций Лауричеллы от многих переменных были использованы для нахождения явного решения поставленной задачи.

Ключевые слова: задача Холмгрена, многомерные эллиптические уравнения с несколькими сингулярными коэффициентами, формула разложения, формула суммирования, гипергеометрическая функция Лауричеллы от многих переменных, функция Грина.

DOI: 10.26117/2079-6641-2020-32-3-180-196

Поступила в редакцию: 04.07.2020

В окончательном варианте: 10.10.2020

Для цитирования. Ergashev T. G., Hasanov A. Holmgren problem for elliptic equation with singular coefficients // Вестник КРАУНЦ. Физ.-мат. науки. 2020. Т. 32. № 3. C. 180-196. DOI: 10.26117/2079-6641-2020-32-3-180-196

Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Ergashev T. G., Hasanov A., 2020

References

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  2. Serbina L. I., “A problem for the linearized Boussinesq equation with a nonlocal Samarskii condition”, Differential Equations, 38(8) (2002), 1187–1194.
  3. Ergashev T. G., “Third double-layer potential for a generalized bi-axially symmetric Helmholtz equation”, Ufa Mathematical Journal, 10(4) (2018), 111–122.
  4. 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.
  5. Weinstein A., “Generalized axially symmetric potentials theory”, Bull. Amer. Math. Soc., 59 (1959), 20–38.
  6. Holmgren E., “Sur un probleme aux limites pour l’equation ymuxx+uyy=0”, Arkiv for matematik, astronomi och Fysik, 19B(14) (1926), 1–3.
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  9. Srivastava H. M., Karlsson P. W., Multiple Gaussian Hypergeometric Series, Halsted Press, New York, Chichester, Brisbane and Toronto, 1985, 428 pp.
  10. Burchnall J. L., Chaundy T. W., “Expansions of Appell’s double hypergeometric functions”, The Quarterly Journal of Mathematics, 11 (1940), 249–270.
  11. Burchnall J. L., Chaundy T. W., “Expansions of Appell’s double hypergeometric functions. II”, The Quarterly Journal of Mathematics, 12 (1941), 112–128.
  12. Hasanov A., Srivastava H. M., “Some decomposition formulas associated with the Lauricella function F(r) A and other multiple hypergeometric functions”, Appl. Math. Lett., 19(2) (2006), 113–121.
  13. Hasanov A., Srivastava H. M., “Decomposition Formulas Associated with the Lauricella Multivariable Hypergeometric Functions”, Computers and Mathematics with Applications, 53(7) (2007), 1119–1128.
  14. 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.
  15. Mikhlin S. G., An Advanced Course of Mathematical Physics. North Holland Series in Applied Mathematics and Mechanics. V. 11 Amsterdam, London, North-Holland Publishing.
  16. Rassias M., Lecture Notes on Mixed Type Partial Differential Equations, World Scientific, 1990.

References (GOST)

  1. Bers L. Mathematical aspects of subsonic and transonic gas dynamics. New York. London, 1958.
  2. Serbina L. I. A problem for the linearized Boussinesq equation with a nonlocal Samarskii condition // Differential Equations. 2002. vol. 38(8). pp. 1187–1194.
  3. Ergashev T. G. Third double-layer potential for a generalized bi-axially symmetric Helmholtz equation // Ufa Mathematical Journal. 2018. vol. 10(4). pp. 111–122.
  4. Srivastava H. M., Hasanov A., Choi J. Double-Layer Potentials for a Generalized Bi-Axially Symmetric Helmholtz Equation // Sohag Journal of Mathematics. 2015. vol. 2(1). pp. 1–10.
  5. Weinstein A. Generalized axially symmetric potentials theory // Bull. Amer. Math. Soc. 1959. vol. 59. pp. 20–38.
  6. Holmgren E. Sur un probleme aux limites pour l’equation ymuxx+uyy=0 // Arkiv for matematik, astronomi och Fysik. 1926. 19B(14). pp. 1–3.
  7. Karimov E.T. On a boundary problem for 3-D elliptic equation with singular coefficients. Progress in Analysis and Its Applications. Proceeding of the 7th International ISAAC Congress, Imperial College London, UK, 13–18 July 2009. pp. 619–625.
  8. Erd´elyi A., Magnus W., Oberhettinger F., Tricomi F.G. Higher Transcendental Functions. vol. I. New York, Toronto and London: McGraw-Hill Book Company, 1953.
  9. Srivastava H. M., Karlsson P.W. Multiple Gaussian Hypergeometric Series. New York, Chichester, Brisbane and Toronto: Halsted Press, 1985. 428 p.
  10. Burchnall J. L., Chaundy T.W. Expansions of Appell’s double hypergeometric functions // The Quarterly Journal of Mathematics. 1940. Ser. 11. pp. 249–270.
  11. Burchnall J. L., Chaundy T.W. Expansions of Appell’s double hypergeometric functions. II // The Quarterly Journal of Mathematics. 1941. Ser. 12. pp. 112–128.
  12. Hasanov A., Srivastava H. M. Some decomposition formulas associated with the Lauricella function F(r) A and other multiple hypergeometric functions // Appl. Math. Lett. 2006. vol. 19(2). pp. 113–121.
  13. Hasanov A., Srivastava H. M. Decomposition Formulas Associated with the Lauricella Multivariable Hypergeometric Functions // Computers and Mathematics with Applications. 2007. vol. 53(7). pp. 1119–1128.
  14. Ergashev T. G. Fundamental solutions for a class of multidimensional elliptic equations with several singular coefficients // Journal of Siberian Federal University. Mathematics and Physics. 2020. vol. 13(1). pp. 48–57.
  15. Mikhlin S. G. An Advanced Course of Mathematical Physics. North Holland Series in Applied Mathematics and Mechanics, V. 11. Amsterdam, London: North-Holland Publishing, 1970.
  16. Rassias M. Lecture Notes on Mixed Type Partial Differential Equations. World Scientific, 1990.

Эргашев Тухтасин Гуламжанович –кандидат физико-математических наук, доцент, докторант, Институт математики имени В. И. Романовского, г. Ташкент, Республика Узбекистан.

Ergashev Tuhtasin Gulamjanovich – Ph. D.(Phys. & Math.), Doctoral Student, Institute of Mathematics named after V.I. Romanovsky, Tashkent, Republic of Uzbekistan.

Хасанов Анвар – доктор физико-математических наук, профессор, ведущий научный сотрудник, Институт математики имени В. И. Романовского, г. Ташкент, Республика Узбекистан.


Hasanov Anvar – Dr. Sci. (Math. & Phys.), Professor, Leading Researcher, Institute of Mathematics named after V. I. Romanovsky, Tashkent, Republic of Uzbekistan.

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