Нажмите "Enter", чтобы перейти к контенту

Доклады АМАН. Т. 19, №1. С. 22-30. ISSN 1726-9946

Доклады АМАН. Т. 19, №1. С. 22-30. ISSN 1726-9946

Содержание выпуска/Contents of this issue

МАТЕМАТИКА

УДК 517.956.32 Научная статья

On the boundary conditions of the wave potential in a domain with a curviline borders

Kal’menov T.Sh.1 – academician of AIAS, academician of NAS RK,

Derbissaly B.O.2

1Институт математики и математического моделирования, Алматы
2Казахский национальный университет имени аль-Фараби, Алматы
E-mail: 1kalmenov.t@mail.ru, 2derbissaly@math.kz

    В работе мы рассматриваем одномерный объемный волновой потенциал в области с криволинейными границами. В качестве ядра волнового потенциала выбрано фундаментальное решение задачи Коши. Хорошо известно, что в этом случае объемный волновой потенциал удовлетворяет однородным начальным условиям Коши. Мы построили краевые условия, которым удовлетворяет волновой потенциал на боковых границах области. Показано, что сформулированная начально-краевая задача имеет единственное классическое решение.

Ключевые слова: волновое уравнение, начально-краевая задача, уравнение гиперболического типа, граничное условие, волновой потенциал.

© Т.Ш. Кальменов,
Б.О. Дербисалы, 2019

 

MATHEMATICS

Research Article

On the boundary conditions of the wave potential in a domain with a curviline borders

Kal’menov T.Sh.1 – academician of AIAS, academician of NAS RK,

Derbissaly B.O.2

1Institute of Mathematics and Mathematical Modeling, Almaty
2Al-Farabi Kazakh National University, Almaty
E-mail: 1kalmenov.t@mail.ru, 2derbissaly@math.kz

   We study a one-dimensional volume wave potential in a domain with curvilinear boundaries. As a kernel of the wave potential we have chosen the fundamental solution of the Cauchy problem. It is well-known that in this case the volume wave potential satisfies one-dimensional initial conditions of Cauchy. We have constructed boundary conditions to which the wave potential satisfies at lateral boundaries of the domain. It is shown that the formulated initial-boundary value problem has the unique classical solution.

Keywords: wave equation, initial-boundary value problem, equation hyperbolic type, boundary condition, wave potential.

© T.Sh. Kal’menov,
B.O. Derbissaly, 2019

Список литературы (ГОСТ)

  1. Kal’menov T.Sh., Suragan D. To spectral problems for the volume potential // Doklady Mathematics. 2009. V. 80, № 2. Pp. 646-649.
  2. Kal’menov T.Sh., Suragan D. A boundary сondition and Speсtral Problems for the Newton Potentials // Operator Theory: Advanсes and Appliсations. 2011. V. 216. Pp. 187-210.
  3. Kal’menov T.Sh., Suragan D. Boundary conditions for the volume potential for the polyharmonic equation // Differential Equations. 2012. V. 48, № 4. Pp. 604-608.
  4. Kal’menov T.Sh., Tokmagambetov N.E. On a nonlocal boundary value problem for the multidimensional heat equation in a noncylindrical domain // Siberian Mathematical Journal. 2013. V. 54, № 6. Pp. 1023-1028.
  5. Suragan D., Tokmagambetov N. On transparent boundary conditions for the high-order heat equation // Siberian Electronic Mathematical Reports. 2013. V. 10. Pp. 141-149.
  6. Kal’menov T.Sh., Suragan D. Initial-boundary value problems for the wave equation // Electronic Journal of Differential Equations. 2014. V. 2014. Pp. 1-7.
  7. Kal’menov T.Sh., Suragan D. On permeable potential boundary conditions for the Laplace-Beltrami operator // Siberian Mathematical Journal. 2015. V. 55, № 6. Pp. 1060-1064.
  8. Ruzhansky M., Suragan D. On Kac’s principle of not feeling the boundary for the Kohn Laplacian on the Heisenberg group // Proceedings of the American Mathematical Society. 2016. V. 144. Pp. 709-721.
  9. Sadybekov M.A., Oralsyn G. Nonlocal initial boundary value problem for the time-fractional diffusion equation // Electronic Journal of Differential Equations. 2017. V. 2017, № 201. Pp. 1-7.
  10. Engquist B. and Majda A. Radiation boundary conditions for acoustic and elastic wave calculations // Comm. Pure Appl. Math. 1979. V. 32. Pp. 313-357.
  11. Givoli D. Recent advances in the DtN finite element method for unbounded domains // Arch. Comput. Methods Eng. 1999. V. 6. Pp. 71-116.
  12. Givoli D. Numerical methods for problems in infinite domains. Elsevier, 2013. V. 33. 299 p.
  13. Givoli D. Non-reffecting boundary conditions: a review // J. Comput. Phys. 1991. V. 94. Pp. 1-29.
  14. Li J.R., Greengard L. On the numerical solution of the heat equation I: Fast solvers in free space // J. Comput. Phys. 2007. V. 226. Pp. 1891-1901.
  15. Hagstrom T. Radiation boundary conditions for the numerical simulation of waves // Acta Numer. 1999. V. 8. Pp. 47-106.
  16. Tsynkov S.V. Numerical solution of problems on unbounded domains // Appl. Numer. Math. 1998. V. 27. Pp. 465-532.
  17. Saito N. Data analysis and representation on a general domain using eigenfunctions of Laplacian // Appl. Comput. Harmon. Anal. 2008. V. 25. Pp. 68-97.
  18. Wu X. and Zhang J. High-order local absorbing boundary conditions for heat equation in unbounded domains // Journal of Computational Mathematics. 2011. V. 1(29). P. 74-90.
  19. Riley K.F., Hobson M.P. and Bence S.J. Mathematical methods for physics and engineering. Cambridge University Press. 2010. 1333 p

Для цитирования. Kal’menov T.Sh., Derbissaly B.O. On the boundary conditions of the wave potential in a domain with a curviline borders // Докл.  Адыгской (Черкесской) международной академии наук. 2019. Т. 19, № 1. C. 22-30.
For citation. Kal’menov T.Sh., Derbissaly B.O. On the boundary conditions of the wave potential in a domain with a curviline borders. Reports Adyghe (Circassian) International Academy of Sciences. 2019, vol. 19, no. 1, pp. 22-30.

Читать статью/Read article

©​ | 2020 | Адыгская (Черкесская) Международная академия наук