A Stable Difference Scheme for a Third-Order Partial Differential Equation

Cover Page

Abstract


The nonlocal boundary-value problem for a third order partial differential equation in a Hilbert space H with a self-adjoint positive definite operator A is considered. A stable three-step difference scheme for the approximate solution of the problem is presented. The main theorem on stability of this difference scheme is established. In applications, the stability estimates for the solution of difference schemes of the approximate solution of three nonlocal boundary value problems for third order partial differential equations are obtained. Numerical results for oneand two-dimensional third order partial differential equations are provided.

About the authors

A Ashyralyev

Near East University; RUDN University

Email: allaberen.ashyralyev@neu.edu.tr

Kh Belakroum

Fre´res Mentouri University

Email: kheireddinebelakroum@gmail.com

References

  1. Амиров Ш., Кожанов А. И. Смешанная задача для одного класса сильно нелинейных уравнений соболевского типа высокого порядка// Докл. РАН. - 2013. - 451, № 5. - С. 492-494.
  2. Власов В. В., Раутиан Н. А. Спектральный анализ функционально-дифференциальных уравнений. - М.: МАКС Пресс, 2016.
  3. Габов Г. А., Свешников А. Г. Задачи динамики стратифицированных жидкостей. - М.: Наука, 1986.
  4. Кожанов А. И. Смешанная задача для некоторых классов нелинейных уравнений третьего порядка// Мат. сб. - 1982. - 118, № 4. - С. 504-522.
  5. Кожанов А. И. Смешанная задача для одного класса квазилинейных уравнений третьего порядка// В сб.: «Краевые задачи для нелинейных уравнений математической физики». - Новосибирск: Ин-т математики СО АН СССР, 1982. - С. 118-128.
  6. Крейн С. Г. Линейные дифференциальные уравнения в банаховом пространстве. - М.: Наука, 1967.
  7. Скубачевский А. Л. Краевые задачи для эллиптических функционально-дифференциальных уравнений и их приложения// Усп. мат. наук. - 2016. - 71, № 5. - С. 3-112.
  8. Соболевский П. Е. Разностные методы приближенного решения дифференциальных уравнений. - Воронеж: Воронежский гос. ун-т, 1975.
  9. Apakov Y. On the solution of a boundary-value problem for a third-order equation with multiple characteristics// Ukr. Math. J. - 2012. - 64, № 1. - С. 1-12.
  10. Apakov Y., Irgashev B. Boundary-value problem for a generate high-odd order equation// Ukr. Math. J. - 2015. - 66, № 10. - С. 1475-1490.
  11. Apakov Y., Rutkauskas S. On a boundary value problem to third order PDE with multiple characteristics// Nonlinear Anal. Model. Control. - 2011. - 16, № 3. - С. 255-269.
  12. Ashyralyev A. Fractional spaces generated by the positivite differential and difference operator in a Banach space// В сб.: «Mathematical methods in engineering. Selected papers of the International Symposium, MME06, Ankara, Turkey, April 27-29, 2006». - Dordrecht: Springer, 2007. - С. 13-22.
  13. Ashyralyev C., Akyuz G., Dedeturk M. Approximate solution for an inverse problem of multidimensional elliptic equation with multipoint nonlocal and Neumann boundary conditions// Electron. J. Differ. Equ. - 2017. - 2017, № 197. - С. 1-16.
  14. Ashyralyev A., Arjmand D., Koksal M. A note on the Taylor’s decomposition on four points for a thirdorder differential equation// Appl. Math. Comput. - 2007. - 188, № 2. - С. 1483-1490.
  15. Ashyralyev A., Arjmand D., Koksal M. Taylor’s decomposition on four points for solving third-order linear time-varying systems// J. Franklin Inst. - 2009. - 346, № 7. - С. 651-662.
  16. Ashyralyev A., Belakroum Kh., Guezane-Lakoud A. Stability of boundary-value problems for third order partial differential equations// Electron. J. Differ. Equ. - 2017. - 2017, № 53. - С. 1-11.
  17. Ashyralyev A., Simsek S. N. An operator method for a third-order partial differential equation// Numer. Funct. Anal. Optim. - 2017. - 38, № 9. - С. 1-19.
  18. Ashyralyev A., Sobolevskii P. E. A note on the difference schemes for hyperbolic equations// Abstr. Appl. Anal.- 2001.- 6, № 2. - С. 63-70.
  19. Ashyralyev A., Sobolevskii P. E. New difference schemes for partial differential equations. - Basel- Boston-Berlin: Birkha¨user, 2004.
  20. Belakroum Kh., Ashyralyev A., Guezane-Lakoud A. A note on the nonlocal boundary value problem for a third order partial differential equation// AIP Conf. Proc. - 2016. - 1759. - Article ID 020021.
  21. Denche M., Memou A. Boundary value problem with integral conditions for a linear third-order equation// J. Appl. Math. - 2003. - 11. - С. 533-567.
  22. Direk Z., Ashyraliyev M. FDM for the integral-differential equation of the hyperbolic type// Adv. Difference Equ. - 2014. - 2014, № 132. - С. 1-8.
  23. Fattorini H. O. Second order linear differential equations in Banach spaces. - Amsterdam: Elsevier, 1985.
  24. Kalmenov T. S., Suragan В. Initial-boundary value problems for the wave equation// Electron. J. Differ. Equ. - 2014. - 2014, № 48. - С. 1-6.
  25. Kudu M., Amirali I. Method of lines for third order partial differential equations// J. Appl. Math. Phys. - 2014. - 2, № 2. - С. 33-36.
  26. Latrous C., Memou A. A three-point boundary value problem with an integral condition for a third-order partial differential equation// Abstr. Appl. Anal. - 2005. - 2005, № 1. - С. 33-43.
  27. Lunardi A. Analytic semigroups and optimal regularity in parabolic problems. - Basel-Boston-Berlin: Birkha¨user, 1995.
  28. Niu J., Li P. Numerical algorithm for the third-order partial differential equation with three-point boundary value problem// Abstr. Appl. Anal. - 2014. - 2014. - Article ID 630671.
  29. Shakhmurov V., Musaev H. Maximal regular convolution-differential equations in weighted Besov spaces// Appl. Comput. Math. - 2017. - 16, № 2. - С. 190-200.
  30. Skubachevskii A. L. Elliptic functional-differential equations and applications. - Basel-Boston-Berlin: Birkha¨user, 1997.

Statistics

Views

Abstract - 154

PDF (Russian) - 124

Cited-By


PlumX

Dimensions

Refbacks

  • There are currently no refbacks.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies