International Journal of


EISSN: 2313-3724, Print ISSN: 2313-626X

Frequency: 12

line decor
line decor

 Volume 7, Issue 6 (June 2020), Pages: 48-56


 Original Research Paper

 Title: Numerical treatment of fourth-order singular boundary value problems using new quintic B-spline approximation technique

 Author(s): Muhammad Kashif Iqbal 1, Muhammad Waseem Iftikhar 2, Muhammad Shahid Iqbal 3, Muhammad Abbas 4, *


 1Department of Mathematics, Government College University, Faisalabad, Pakistan
 2Department of Mathematics, National Textile University, Faisalabad, Pakistan
 3Department of Mathematics, University of Okara, Okara, Pakistan
 4Department of Mathematics, University of Sargodha, Sargodha, Pakistan

  Full Text - PDF          XML

 * Corresponding Author. 

  Corresponding author's ORCID profile:

 Digital Object Identifier:


Singular boundary value problems (SBVPs) are cropped up in mathematical modeling of many real-life phenomena such as chemical reactions, electro-hydrodynamics, aerodynamics, thermal explosions, fluid dynamics, and atomic nuclear reactions. In this work, a new quintic B-spline approximation technique has been presented for the numerical solution of fourth-order singular boundary value problems. The fifth-degree basis spline functions are brought into play together with a new approximation for fourth-order derivative. The proposed numerical technique is proved to be uniformly convergent in the entire domain. In order to corroborate this work, the proposed scheme has been implemented on some test problems. The comparison of computational outcomes advocates the superior performance of the presented algorithm over current methods on the topic. 

 © 2020 The Authors. Published by IASE.

 This is an open access article under the CC BY-NC-ND license (

 Keywords: Singular boundary value problems, Quintic B-spline functions, Quintic B-spline collocation method, Emden flower type equations

 Article History: Received 4 October 2019, Received in revised form 8 March 2020, Accepted 11 March 2020


No Acknowledgment.

 Compliance with ethical standards

 Conflict of interest: The authors declare that they have no conflict of interest.


 Iqbal MK, Iftikhar MW, and Iqbal MS et al. (2020). Numerical treatment of fourth-order singular boundary value problems using new quintic B-spline approximation technique. International Journal of Advanced and Applied Sciences, 7(6): 48-56

 Permanent Link to this page


 Fig. 1 Fig. 2 Fig. 3 Fig. 4 Fig. 5


 Table 1 Table 2 Table 3 Table 4 Table 5


 References (19) 

  1. Abbas M, Majid AA, Ismail AIM, and Rashid A (2014). Numerical method using cubic B-spline for a strongly coupled reaction-diffusion system. PloS One, 9(1): e83265.   [Google Scholar] PMid:24427270 PMCid:PMC3888394
  2. Abukhaled M, Khuri SA, and Sayfy A (2011). A numerical approach for solving a class of singular boundary value problems arising in physiology. International Journal of Numerical Analysis and Modeling, 8(2): 353-363.   [Google Scholar]
  3. Akram G (2011). Quartic spline solution of a third order singularly perturbed boundary value problem. Anziam Journal, 53: 44-58.   [Google Scholar]
  4. Akram G and Amin N (2012). Solution of a fourth order singularly perturbed boundary value problem using quintic spline. International Mathematical Forum, 7(44): 2179-2190.   [Google Scholar]
  5. Aruna K and Kanth AR (2013). A novel approach for a class of higher order nonlinear singular boundary value problems. International Journal of Pure and Applied Mathematics, 84(4): 321-329.   [Google Scholar]
  6. Caglar H, Caglar N, and Ozer M (2009). B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos, Solitons and Fractals, 39(3): 1232-1237.   [Google Scholar]
  7. Fyfe DJ (1969). The use of cubic splines in the solution of two-point boundary value problems. The Computer Journal, 12(2): 188-192.   [Google Scholar]
  8. Goh J, Majid AA, and Ismail AIM (2011). Extended cubic uniform B-spline for a class of singular boundary value problems. Science Asia, 37: 79-82.   [Google Scholar]
  9. Goh J, Majid AA, and Ismail AIM (2012). A quartic B-spline for second-order singular boundary value problems. Computers and Mathematics with Applications, 64(2): 115-120.   [Google Scholar]
  10. Iqbal MK, Abbas M, and Wasim I (2018). New cubic B-spline approximation for solving third order Emden–Flower type equations. Applied Mathematics and Computation, 331: 319-333.   [Google Scholar]
  11. Khuri SA (2001). An alternative solution algorithm for the nonlinear generalized Emden-Fowler equation. International Journal of Nonlinear Sciences and Numerical Simulation, 2(3): 299-302.   [Google Scholar]
  12. Khuri SA and Sayfy A (2014). Numerical solution for the nonlinear Emden–Fowler type equations by a fourth-order adaptive method. International Journal of Computational Methods, 11(01): 1350052.   [Google Scholar]
  13. Kim W and Chun C (2010). A modified Adomian decomposition method for solving higher-order singular boundary value problems. Zeitschrift für Naturforschung A, 65(12): 1093-1100.   [Google Scholar]
  14. Lodhi RK and Mishra HK (2016). Solution of a class of fourth order singular singularly perturbed boundary value problems by quintic B-spline method. Journal of the Nigerian Mathematical Society, 35(1): 257-265.   [Google Scholar]
  15. Parand K and Delkhosh M (2017). An effective numerical method for solving the nonlinear singular Lane-Emden type equations of various orders. Jurnal Teknologi, 79(1): 25-36.   [Google Scholar]
  16. Taiwo OA and Hassan MO (2015). Approximation of higher-order singular initial and boundary value problems by iterative decomposition and Bernstein polynomial methods. Journal of Advances in Mathematics and Computer Science, 9(6): 498-515.   [Google Scholar]
  17. Wazwaz AM (2015). The variational iteration method for solving new fourth-order Emden–Fowler type equations. Chemical Engineering Communications, 202(11): 1425-1437.   [Google Scholar]
  18. Wazwaz AM, Rach R, and Duan JS (2015). Solving new fourth–order emden–fowler-type equations by the adomian decomposition method. International Journal for Computational Methods in Engineering Science and Mechanics, 16(2): 121-131.   [Google Scholar]
  19. Xu XP and Lang FG (2014). Quintic B-spline method for function reconstruction from integral values of successive subintervals. Numerical Algorithms, 66(2): 223-240.   [Google Scholar]