Numerical Algorithms

, Volume 73, Issue 2, pp 391–406 | Cite as

A posteriori analysis: error estimation for the eighth order boundary value problems in reproducing Kernel space

  • Taher Lotfi
  • Mehdi Rashidi
  • Katauoun Mahdiani
Original Paper


In the previous work (Akram and Rehman Numer. Algor. 62 527–540 2013), Akram and Rehman presented the reproducing kernel method (RKM) for solving various eighth order boundary value problems. However, an effective error estimation for this method has not yet been discussed. This work is devoted to deal with this problem. Some other aspects of the RKM will be considered such as convergence analysis and numerical implementations.


Error estimation Exact solution Approximate solution Gram-Schmidt orthogonal process Reproducing Kernel Searching least value (SLV) method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akram, G., Rehman, H.U.: Solution of first order singularly perturbed initial value problem in reproducing kernel hilbert space. Eur. J. Sci. Res. 53(4), 516–523 (2011)Google Scholar
  2. 2.
    Akram, G., Siddiqi, S.S.: Nonic spline solutions of eighth order boundary value problems. Appl. Math. Comput. 182, 829–845 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Akram, G., Rehman, H.U.: Numerical solution of eighth order boundary value problems in reproducing Kernel space. Numer. Algor. 62, 527–540 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bishop, R.E.D., Cannon, S.M., Miao, S.: On coupled bending and torsional vibration of uniform beams. J. Sound Vib. 131, 457–464 (1989)CrossRefMATHGoogle Scholar
  5. 5.
    Boutayeb, A., Twizell, E.H.: Finite-difference methods for the solution of eighth-order boundary-value problems. Int. J. Comput. Math. 48, 63–75 (1993)CrossRefMATHGoogle Scholar
  6. 6.
    Chandrasekhar, S.: Hydrodynamic and hydromagnetic stability. Clarendon Press, Oxford (1961)MATHGoogle Scholar
  7. 7.
    Cui, M.G., Geng, F.Z.: A computational method for solving one-dimensional variablecoefficient burgers equation. Appl. Math. Comput. 188, 1389–1401 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Geng, F.Z., Cui, M.G.: Solving singular two-point boundary value problem in reproducing Kernel space. J. Comput. Appl. Math. 205, 6–15 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Golbabai, A., Javidi, M.: Application of homotopy perturbation method for solving eighthorder boundary value problems. Appl. Math. Comput. 191, 334–346 (2007)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    He, J.-H.: The variational iteration method for eighth-order initial-boundary value problems. Phys. Scr. 76, 680–682 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Inc, M., Evans, D.J.: An efficient approach to approximate solutions of eighth-order boundaryvalue problems. Int. J. Comput. Math. 81(6), 685–692 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Li, C., Cui, M.: The exact solution for solving a class of nonlinear operator equations in the reproducing kernel space. Appl. Math. Comput. 143(23), 393–399 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Li, X.Y., Wu, B.Y.: Error estimation for the reproducing kernel method to solve linear boundary value problems. Appl. Math. Comput. 243, 10–15 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Liu, G.R., Wu, T.Y.: Differential quadrature solutions of eighth-order boundary-value differential equations. J. Comput. Appl. Math. 145, 223–235 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Mestrovic, M.: The modified decomposition method for eighth-order boundary value problems. Appl. Math. Comput. 188, 1437–1444 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Noor, M.A., Mohyud-Din, S.T.: Variational iteration decomposition method for solving eighth-order boundary value problems. Differ. Equat. Nonlinear Mech. (2007)Google Scholar
  17. 17.
    Porshokouhi, M.G., Ghanbari, B., Gholami, M., Rashidi, M.: Numerical solution of eighth order boundary value problems with variational iteration method. Gen. Math. Notes 2(1), 128–133 (2011)MATHGoogle Scholar
  18. 18.
    Siddiqi, S.S., Akram, G.: Solution of eighth-order boundary value problems using the nonpolynomial spline technique. Int. J. Comput. Math. 84(3), 347–368 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Siddiqi, S.S., Twizell, E.H.: Spline solution of linear eighth-order boundary value problems. Comput. Methods Appl. Mech. Eng. 131, 309–325 (1996)CrossRefMATHGoogle Scholar
  20. 20.
    Wazwaz, A.M.: The numerical solutions of special eighth-order boundary value problems by the modified decomposition method. Neural Parallel Sci. Comput. 8(2), 133–146 (2000)MathSciNetMATHGoogle Scholar
  21. 21.
    A Dezhbord, A., Lotfi, T., Mahdiani, K.: A new efficient method for cases of the singular integral equation of the first kind. J. Comput. Appl. Math. 296, 156–169 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics, Hamedan BranchIslamic Azad UniversityHamedanIran

Personalised recommendations