Highly Accurate Method for Solving Singular Boundary-Value Problems Via Padé Approximation and Two-Step Quartic B-Spline Collocation

  • H. AlloucheEmail author
  • A. Tazdayte
  • K. Tigma


We propose a highly accurate method to solve a class of singular two-point boundary-value problems having singular coefficients. These problems often arise when a partial differential equation is reduced to an ordinary differential equation by physical symmetry. The proposed method is based on Padé approximation and two-step collocation. The resolution domain is divided, Padé approximant is used on the subdomain in the vicinity of the singular point and provides a new boundary condition, and then, two-step quartic B-spline is used on the remaining subdomain where the problem is transformed to a regular boundary-value problem. The subdomains are chosen optimally using a suitable optimization procedure. Some examples are presented along with a comparison of the numerical results with other methods.


Singularity Boundary-value problems Quartic B-spline Two-step collocation method Padé approximation 

Mathematics Subject Classification

65N 34B16 41A15 41A21 65N15 



  1. 1.
    Albasiny, E.L., Hoskins, W.D.: Cubic spline solutions to two-point boundary value problems. Comput. J. 12(2), 151–153 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Allouche, H., Marhraoui, N.: Numerical solution of singular regular boundary value problems by pole detection with qd-algorithm. J. Comput. Appl. Math. 233(2), 420–436 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Allouche, H., Tazdayte, A.: Numerical solution of singular boundary value problems with logarithmic singularities by padé approximation and collocation methods. J. Comput. Appl. Math. 311, 324–341 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Baker, G.A., Graves-Morris, P.R.: Padé approximants, vol. 59. Cambridge University Press, Cambridge (1996)zbMATHCrossRefGoogle Scholar
  5. 5.
    Bickley, W.G.: Piecewise cubic interpolation and two-point boundary problems. Comput. J. 11(2), 206–208 (1968)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Christara, C., Liu, G.: Quartic spline collocation for second-order boundary value problems. In: Proceedings of the 9th HERCMA Conference on Computer Mathematics and Applications, Athens University of Economics and Business, pp. 1–8 (2009)Google Scholar
  7. 7.
    Cohen, A.M., Jones, D.E.: A note on the numerical solution of some singular second order differential equations. IMA J. Appl. Math. 13(3), 379–384 (1974)zbMATHCrossRefGoogle Scholar
  8. 8.
    Fyfe, D.J.: The use of cubic splines in the solution of two-point boundary value problems. Comput. J. 12(2), 188–192 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Geng, F., Cui, M.: Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space. Appl. Math. Comput. 192(2), 389–398 (2007)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Goh, J., Majid, A.A., Ismail, A.I.M.: Numerical method using cubic b-spline for the heat and wave equation. Comput. Math. Appl. 62(12), 4492–4498 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Kadalbajoo, M.K., Aggarwal, V.K.: Numerical solution of singular boundary value problems via chebyshev polynomial and b-spline. Appl. Math. Comput. 160(3), 851–863 (2005)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ravi Kanth, A.S.V., Reddy, Y.N.: Cubic spline for a class of singular two-point boundary value problems. Appl. Math. Comput. 170(2), 733–740 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Koch, O., Weinmüller, E.: The convergence of shooting methods for singular boundary value problems. Math. Comput. 72(241), 289–305 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Li, Z., Wang, Y., Tan, F., Wan, X., Nie, T.: The solution of a class of singularly perturbed two-point boundary value problems by the iterative reproducing kernel method. In: Abstract and Applied Analysis, volume 2012. Hindawi Publishing Corporation (2012)Google Scholar
  15. 15.
    Mohanty, R.K., Sachdev, P.L., Jha, N.: An o (h 4) accurate cubic spline tage method for nonlinear singular two point boundary value problems. Appl. Math. Comput. 158(3), 853–868 (2004)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mohsen, A., El-Gamel, M.: On the galerkin and collocation methods for two-point boundary value problems using sinc bases. Comput. Math. Appl. 56(4), 930–941 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Prenter, P.M.: Splines and variational methods. Wiley, New York (1975)zbMATHGoogle Scholar
  18. 18.
    Rashidinia, J., Mahmoodi, Z., Ghasemi, M.: Parametric spline method for a class of singular two-point boundary value problems. Appl. Math. Comput. 188(1), 58–63 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Robin, W.: Frobenius series solution of fuchs second-order ordinary differential equations via complex integration. In: International Mathematical Forum, vol. 9, pp. 953–965. Citeseer (2014)Google Scholar
  20. 20.
    Secer, A., Kurulay, M.: The sinc-galerkin method and its applications on singular dirichlet-type boundary value problems. Bound. Value Probl. 2012(1), 126 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Shampine, Lawrence F., Kierzenka, Jacek, Reichelt, Mark W.: Solving boundary value problems for ordinary differential equations in matlab with bvp4c (2000)Google Scholar
  22. 22.
    Wazwaz, A.-M.: The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models. Commun. Nonlinear Sci. Numer. Simul. 16(10), 3881–3886 (2011)zbMATHCrossRefGoogle Scholar
  23. 23.
    Zhang, J.: Bi-quartic Spline Collocation Methods for Fourth-order Boundary Value Problems with an Application to the Biharmonic Dirichlet Problem. PhD thesis, University of Toronto (2008)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.MAN-TA Research Team Department of Mathematics and Computer Sciences Faculty of SciencesMoulay Ismail UniversityMeknesMorocco
  2. 2.Higher Institute of Maritime Studies (ISEM)CasablancaMorocco

Personalised recommendations