On Third-Order Boundary Value Problems with Multiple Characteristics

  • N. Bendjazia
  • A. Guezane-LakoudEmail author
  • R. Khaldi
Original Research


In this paper, we apply the reproducing kernel Hilbert space method (RKHSM) for solving third order differential equations with multiple characteristics in a rectangular domain. The exact solution is expressed in a series form. The numerical examples are given to demonstrate the good performance of the presented method. The results obtained indicate that the method is simple and effective.


Reproducing kernel Hilbert space Third order differential equation Boundary value problems Existence of Solution Approximate solution 

Mathematics Subject Classification

35A35 35G20 



  1. 1.
    Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Caglar, H.N., Caglar, S.H., Twizell, E.H.: The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions. Int. J. Comput. Math. 71(3), 373–381 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Cui, M., Du, H.: Representation of exact solution for the nonlinear Volterra–Fredholm integral equations. Appl. Math. Comput. 182, 1795–1802 (2006)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Diesperov, V.N.: On Green’s function of the linearized viscous transonic equation. U.S.S.R. Comput. Math. Math. Phys. 12(5), 1265–1279 (1972)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Du, J., Cui, M.: Solving the forced Duffing equations with integral boundary conditions in the reproducing kernel space. Int. J. Comput. Math. 87, 2088–2100 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Geng, F., Cui, M.: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 327, 1167–1181 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Geng, F., Cui, M.: Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space. Appl. Math. Comput. 192, 389–398 (2007)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Geng, F., Cui, M.: New method based on the HPM and RKHSM for solving forced Duffing equations with integral boundary conditions. J. Comput. Appl. Math. 233, 165–172 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Geng, F.: Iterative reproducing kernel method for a beam equation with third-order nonlinear boundary conditions. Math. Sci. 6, 1 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Ghazala, A., Tehseen, M., Siddiqi, S.S.: Solution of a linear third order multi-point boundary value problem using RKM. Br. J. Math. Comput. Sci. 3(2), 180–194 (2013)CrossRefGoogle Scholar
  11. 11.
    Guezane-Lakoud, A., Bendjazia, N., Khaldi, R.: An approximation method for solving Volterra integro-differential equation with weighted integral condition. J. Funct. Spaces 2015, Article ID 758410MathSciNetCrossRefGoogle Scholar
  12. 12.
    Irgashev, Y., Apakov, Y.P.: First boundary value problem for a third-order equation of pseudo seudoelliptic type. Uzb. Mat. Zh. 2, 44–51 (2006). (in Russian)Google Scholar
  13. 13.
    Khan, A., Aziz, T.: The numerical solution of third-order boundary-value problems using quintic splines. Appl. Math. Comput. 137(2–3), 253–260 (2003)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Moradi, E., Yusefi, A., Abdollahzadeh, A., Tila, E.: New implementation of reproducing kernel Hilbert space method for solving a class of third-order differential equations. J. Math. Comput. Sci. 12, 253–262 (2014)CrossRefGoogle Scholar
  15. 15.
    Li, Z., Wang, Y., Tan, F.: The solution of a class of third-order boundary value problems by the reproducing kernel method. J. Funct. Spaces 2012, Article ID 195310Google Scholar
  16. 16.
    Rashidinia, J., Ghasemi, M.: B-spline collocation for solution of two-point boundary value problems. J. Comput. Appl. Math. 235(8), 2325–2342 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ryzhov, O.S.: Asymptotics map of the transonic flow of viscous heat-conducting gas around revolution bodies. Prikl. Mat. Mekh. 2(6), 1004–1014 (1952) (in Russian) Google Scholar
  18. 18.
    Tatari, M., Dehghan, M.: The use of the adomian decomposition method for solving multipoint boundary value problems. Phys. Scr. 73(6), 672–676 (2006)CrossRefGoogle Scholar
  19. 19.
    Tirmizi, I.A., Twizell, E.H., Siraj-Ul-Islam: A numerical method for third-order non-linear boundary-value problems in engineering. Int. J. Comput. Math. 82(1), 103–109 (2005)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wang, W., Cui, M., Han, B.: A new method for solving a class of singular two-point boundary value problems. Appl. Math. Comput. 206, 721–727 (2008)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Wu, B., Li, X.: Application of reproducing kernel method to third order three-point boundary value problems. Appl. Math. Comput. 217, 3425–3428 (2010)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Yao, H., Cui, M.: A new algorithm for a class of singular boundary value problems. Appl. Math. Comput. 186, 1183–1191 (2007)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Yusufjon, P.A., Rutkauskasb, S.: On a boundary value problem to third order PDE with multiple characteristics. Nonlinear Anal. Model. Control 16(3), 255–269 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2019

Authors and Affiliations

  1. 1.Laboratory of Advanced Materials, Faculty of SciencesBadji Mokhtar-Annaba UniversityAnnabaAlgeria
  2. 2.Department of MathematicsUniversity 08 Mai 45GuelmaAlgeria

Personalised recommendations