A new extended B-spline approximation technique for second order singular boundary value problems arising in physiology

Volume 19, Issue 4, pp 258--267 http://dx.doi.org/10.22436/jmcs.019.04.06

Publication Date: July 10, 2019 Submission Date: April 23, 2019 Revision Date: June 12, 2019 Accteptance Date: June 15, 2019

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Authors

Imtiaz Wasim - Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan. Muhammad Abbas - Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan. Muhammad Kashif Iqbal - Department of Mathematics, , Government College University, Faisalabad, 38000, Pakistan.

Abstract

In this study, we have explored the approximate solution of \(2^{nd}\) order singular boundary value problems (SBVP's) using extended cubic B-spline (ECBS) collocation approach. The accuracy of the numerical algorithm has been enhanced by means of a novel ECBS approximation for $2^{nd}$ order derivative. To endorse our claim, few test examples have been considered and the experimental results are compared with the already existing methods. It is observed that the proposed technique is more accurate and efficient in comparison to the existing techniques on the topic.

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Keywords

• Singular boundary value problems
• extended B-spline functions
• quasi-linearization technique
• extended B-spline collocation method

MSC

•  34B15
•  34B16
•  74H15
•  65L10
•  65L11

References

• [1] M. Abbas, A. A. Majid, A. I. M. Ismail, A. Rashid, The application of cubic trigonometric B-spline to the numerical solution of the hyperbolic problems, Appl. Math. Comput., 239 (2014), 74--88
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] M. Abukhaled, S. A. Khuri, A. Sayfy, A numerical approach for solving a class of singular boundary value problems arising in physiology, Int. J. Numer. Anal. Model., 8 (2011), 353--363
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] E. Babolian, A. Iftikhari, A. Saadarmandi, A Sinc-Galerkin technique for the numerical solution of a class of singular boundary value problems, Comput. Appl. Math., 34 (2015), 45--63
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] N. Caglar, H. Caglar, B-spline solution of singular boundary value problems, Appl. Math. Comput., 182 (2006), 1509--1513
  • View Article
  • MathSciNet
  • Google Scholar


• [5] Z. D. Cen, Numerical study for a class of singular two-point boundary value problems using Green’s functions, Appl. Math. Comput., 183 (2006), 10--16
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] M. M. Chawala, R. Subramanian, H. L. Sathi, A fourth order method for a singular two-point boundary value problem, BIT Numer. Math., 28 (1988), 88--97
  • View Article
  • Google Scholar
  • MATH


• [7] U. Flesch, The distribution of heat sources in the human head: a theoretical consideration, J. Theor. Biol., 54 (1975), 285--287
  • View Article
  • Google Scholar


• [8] D. J. Fyfe, The use of cubic splines in the solution of two-point boundary value problems, Comput. J., 12 (1969/70), 188--192
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] J. B. Garner, R. Shivaji, Diffusion problems with a mixed nonlinear boundary condition, J. Math. Anal. Appl., 148 (1990), 422--430
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] J. Goh, A. A. Majid, A. I. M. Ismail, Extended cubic uniform B-spline for a class of singular boundary value problems, Sci. Asia, 37 (2011), 79--82
  • View Article
  • Google Scholar


• [11] J. Goh, A. A. Majid, A. I. M. Ismail, A quartic B-spline for second-order singular boundary value problems, Comput. Math. Appl., 64 (2012), 115--120
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] M. K. Iqbal, M. Abbas, N. Khalid, New Cubic B-spline Approximation for Solving Non-linear Singular Boundary Value Problems Arising in Physiology, Comm. Math. Appl., 9 (2018), 377--392
  • Google Scholar


• [13] M. K. Iqbal, M. Abbas, I. Wasim, New cubic B-spline approximation for solving third order Emden–Flower type equations, Appl. Math. Comput., 331 (2018), 319--333
  • View Article
  • MathSciNet
  • Google Scholar


• [14] M. K. Iqbal, M. Abbas, B. Zafar, New Quartic B-Spline Approximation for Numerical Solution of Third Order Singular Boundary Value Problems, Punjab. Univ. J. Math., 51 (2019), 43--59
  • View Article


• [15] S. A. Khuri, A. Sayfy, A novel approach for the solution of a class of singular boundary value problems arising in physiology, Math. Comp. Modelling, 52 (2010), 626--636
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] M. Kumar, Y. Gupta, Methods for solving singular boundary value problems using splines: a review, J. Appl. Math. Comput., 32 (2010), 265--278
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] F. G. Lang, X. P. Xu, A new cubic B-spline method for approximating the solution of a class of nonlinear second-order boundary value problem with two dependent variables, Sci. Asia, 40 (2014), 444--450
  • View Article
  • Google Scholar


• [18] P. M. Lima, M. P. Carpentier, Iterative methods for a singular boundary-value problem, J. Comput. Appl. Math., 111 (1999), 173--186
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] S. H. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics, J. Theor. Biol., 60 (1976), 449--457
  • View Article
  • Google Scholar
  • MATH


• [20] D. L. S. McElwain, A re-examination of oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics, J. Theor. Biol., 71 (1978), 255--263
  • View Article
  • Google Scholar


• [21] M. Mohsenyzadeh, K. Maleknejad, R. Ezzati, A numerical approach for the solution of a class of singular boundary value problems arising in physiology, Adv. Difference Equ., 2015 (2015), 10 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] R. K. Pandey, A. K. Singh, On the convergence of a finite difference method for a class of singular boundary value problems arising in physiology, J. Comput. Appl. Math., 166 (2004), 553--564
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] K. Parand, M. Dehghan, A. Z. Rezaei, S. M. Ghaderi, An approximation algorithm for the solution of the nonlinear Lane--Emden type equations arising in astrophysics using Hermite functions collocation method, Comput. Phys. Comm., 181 (2010), 1096--1108
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] J. Rashidinia, R. Mohammadi, R. Jalilian, The numerical solution of non-linear singular boundary value problems arising in physiology, Appl. Math. Comput., 185 (2007), 360--367
  • View Article
  • MathSciNet
  • Google Scholar


• [25] A. S. V. Ravi Kanth, Cubic spline polynomial for non-linear singular two-point boundary value problems, Appl. Math. Comput., 189 (2007), 2017--2022
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] A. S. V. Ravi Kanth, K. Aruna, He's variational iteration method for treating nonlinear singular boundary value problems, Comput. Math. Appl., 60 (2010), 821--829
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] R. D. Russell, L. F. Shampine, Numerical methods for singular boundary value problems, SIAM J. Numer. Anal., 12 (1975), 13--36
  • View Article
  • MathSciNet
  • Google Scholar


• [28] L. F. Shampine, Singular boundary value problems for ODEs, Appl. Math. Comput., 138 (2003), 99--112
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] S. Sharifi, J. Rashidinia, Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method, Appl. Math. Comput., 281 (2016), 28--38
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [30] R. Singh, J. Kumar, An efficient numerical technique for the solution of nonlinear singular boundary value problems, Comput. Phys. Commun., 185 (2014), 1282--1289
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] R. Singh, A.-M. Wazwaz, J. Kumar, An efficient semi-numerical technique for solving nonlinear singular boundary value problems arising in various physical models, Int. J. Comput. Math., 93 (2016), 1330--1346
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [32] W. Y. Wang, M. G. Cui, B. Han, A new method for solving a class of singular twopoint boundary value problems, Appl. Math. Comput., 206 (2008), 721--727
  • View Article
  • Google Scholar


• [33] Y. L. Wang, H. Yu, F. G. Tan, S. G. Li, Using an effective numerical method for solving a class of Lane-Emden equations, Abstr. Appl. Anal., 2014 (2014), 8 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [34] A. M. Wazwaz, The variational iteration method for solving nonlinear singular boundary value problems arising in various physical models, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 3881--3886
  • View Article
  • Google Scholar
  • MATH


• [35] L.-J. Xie, C.-L. Zhou, S. Xu, An effective numerical method to solve a class of nonlinear singular boundary value problems using improved differential transform method, Springer Plus, 5 (2016), 21 pages
  • View Article
  • Google Scholar