A COLLOCATION SCHEME FOR SINGULAR BOUNDARY VALUE PROBLEMS ARISING IN PHYSIOLOGY

TitleA COLLOCATION SCHEME FOR SINGULAR BOUNDARY VALUE PROBLEMS ARISING IN PHYSIOLOGY
Publication TypeJournal Article
Year of Publication2018
AuthorsKUMAR D.
JournalNeural, Parallel, and Scientific Computations
Volume26
Issue1
Start Page95
Pagination24
Date Published2018
ISSN1056-2176
Keywords34B16, 65L10, 65L12
Abstract

A new collocation method for the solution of a class of second-order two-point boundary value problems associated with physiology and other areas with a singular point at one endpoint is constructed. The singularity of the differential equation is modified by L’Hôpital’s rule and the boundary condition ${ y′(0)=0}$ . Quintic B-spline functions on equidistant collocation points are used to approximate the solution. The quasi-linearization technique is used to reduce a non-linear problem to a sequence of linear problems. The system obtained on discretization is transformed to the system of linear algebraic equations which is easy to be solved. It is proved that the proposed algorithm converges to a smooth approximate solution of the singular boundary value problems and the error estimates are given. To check the theory and to demonstrate the efficiency of the proposed method, several numerical illustrations from physical model problems have been carried out. To show the effectiveness of the proposed method comparisons with several existing methods has also been done.

URLhttps://acadsol.eu/npsc/articles/26/1/6.pdf
DOI10.12732/npsc.v26i1.6
Refereed DesignationRefereed
Full Text

REFERENCES
[1] M. Abukhaled, S. A. Khuri and A. Sayfy, A numerical approach for solving a
class of singular boundary value problems arising in physiology, Int. J. Numer. Anal. Model., 8:353–363, 2011.
[2] J. A. Adam, A simplified mathematical model of tumor growth, Math. Biosci., 81:229–244, 1986.
[3] R. P. Agarwal and Y. M. Chow, Finite difference methods for boundary value
problems for differential equations with deviating arguments, Comput. Math.
Appl., 12:1143–1153, 1986.
[4] R. P. Agarwal, Boundary Value Problems For Higher-Order Differential Equations,
World Scientific, Singapore, 1986.
[5] N. Anderson and A. M. Arthurs, Analytical diffusion in a spherical cell with
michaelis-menten oxygen uptake kinetics, Bull. Math. Biol., 47:145–153, 1983.
[6] N. S. Asaithambi and J. B. Garner, Pointwise solution bounds for a class of
singular diffusion problems in physiology, Appl. Math. Comput., 30:215–222, 1989.
[7] N. C¸ a˘glar and H. C¸ a˘glar, B-spline solution of singular boundary value problems,
Appl. Math. Comput., 182:1509–1513, 2006.
[8] H. C¸ a˘glar, N. C¸ a˘glar and M. Ozer, B-spline solution of non-linear singular bound- ¨
ary value problems arising in physiology, Chaos Solutions and Fractals, 39:1232– 1237, 2009.
[9] H. C¸ a˘glar, C. Akkoyunlu, N. C¸ a˘glar and D. Dundar, The numerical solution of
the singular two-point boundary value problems by using non-polynomial spline
functions, Proceedings of the 9th WSEAS Int. Conference on Applied Computer
And Applied Computational Science. ISSN: 1790-5117, ISBN: 978-960-474-173- 1.
[10] M. M. Chawla and C. P. Katti, Finite difference methods and their convergence
for a class of singular two point boundary value problem, Numer. Math., 39:341– 350, 1982.
[11] M. M. Chawla, S. McKee and G. Shaw, Order h
2 method for a singular two point
boundary value problem, BIT., 26:318–326, 1986.
[12] M. M. Chawla and R. Subramanian, A new spline method for singular two point
boundary value problems, Int. J. Comput. Math., 24:291–310, 1988.
[13] M. M. Chawla, R. Subramanian and H. Sathi, A fourth order method for a
singular two point boundary value problem, BIT., 28:88–97, 1988.
[14] A. M. Cohen and D. E. Jones, A note on the numerical solution of some singular
second order differential equations, J. Inst. Math. Appl., 13:379–384, 1974.
[15] M. G. Cui and F. Z. Geng, Solving singular two-point boundary value problem
in reproducing kernel space, J. Comput. Appl. Math., 205:6–15, 2007.
[16] E. P. Doolan, J. J. H. Miller and W. H. A. Schilders, Uniform Numerical Methods
for Problems With Initial and Boundary Layers, Boole Press, Dublin, 1980.
[17] R. C. Duggan and A. M. Goodman, Point wise bounds for a nonlinear heat
conduction model of the human head, Bull. Math. Biol., 48:229–236, 1986.
[18] M. El-Gamel and A. Zayed, A comparison between the Wavelet-Galerkin and the
Sinc-Galerkin methods in solving nonhomogeneous heat equations, in: Contemporary
Mathematics of the American Mathematical Society, in: Zuhair Nashed,
Otmar Scherzer (Eds.), Series, Inverse Problem, Image Analysis, and Medical
Imaging, 2002, 313, AMS, Providence, 97–116.
[19] M. A. El-Gebeily and I. T. Abbu-Zaid, On a finite difference method for singular
two-point boundary value problem, IMA J. Numer. Anal., 18:179–190, 1998.
[20] U. Flesch, The distribution of heat sources in the human head: a theoretical
consideration, J. Theor. Biol., 54:285–287, 1975.
[21] J. B. Garner and R. Shivaji, Diffusion problems with mixed nonlinear boundary
condition, J. Math. Anal. Appl., 148:422–430, 1990.
[22] J. Goh, A. Majid and A. I. Ismail, A quartic B-spline for second-order singular
boundary value problems, Comput. Math. Appl., 64:115–120, 1983.
[23] P. Hiltmann and P. Lory, On oxygen diffusion in a spherical cell with michaelismenten
oxygen uptake kinetics, Bull. Math. Biol., 45:661–664, 1983.
[24] R. K. Jain and P. Jain, Finite difference methods for a class of singular two-point
boundary value problems, Int. J. Comput. Math., 27:113–120, 1989.
[25] M. K. Kadalbajoo and K. S. Raman, Numerical solution of singular boundary
value problems by invariant imbedding, J. Comput. Phys., 55:268–277, 1984.
[26] M. K. Kadalbajoo and V. K. Aggarwal, Numerical solution of singular boundary
value problems via Chebyshev polynomial and B-spline, Appl. Math. Comput.,
160:851–863, 2005.
[27] M. K. Kadalbajoo and V. K. Aggarwal, B-spline method for a class of singular
two-point boundary value problems using optimal grid, Appl. Math. Comput.,
188:1856–1869, 2007.
[28] H. B. Keller, Numerical Methods for Two-Point Boundary Value Problems, Blaisdell
Publishing Co., Waltham Massachusetts, 1968.
[29] S. A. Khuri and A. Sayfy, A Twofold spline Chebychev linearization approach
for a class of singular second-order nonlinear differential equation, Results Math.,
63:817–835, 2013.
[30] M. Kumar, A three-point finite difference method for a class of singular two-point
boundary value problems, J. Comput. Appl. Math., 145:89–97, 2002.
[31] H. S. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake
kinetics, J. Theor. Biol., 60:449–457, 1976.
[32] D. L. S. McElwain, A re-examination of oxygen diffusion in a spherical cell with
michaelis-menten oxygen uptake kinetics, J. Theor. Biol., 71:255–263, 1978.
[33] R. K. Pandey and 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., 166:553–564, 2004.
[34] P. M. Prenter, Spline and Variational Methods, John Wiley & Sons, New York, 1975.
[35] J. Rashidinia, R. Mohammadi and R. Jalilian, The numerical solution of nonlinear
singular boundary value problems arising in physiology, Appl. Math. Comput.,
185:360–367, 2007.
[36] A. S. V. Ravi Kanth and Y. N. Reddy, A numerical method for singular two-point
boundary value problems via Chebyshev economization, Appl. Math. Comput.,
146:691–700, 2003.
[37] A. S. V. Ravi Kanth and Y. N. Reddy, Higher order finite difference method for
a class of singular boundary value problems, Appl. Math. Comput., 55:249–258, 2004.
[38] A. S. V. Ravi Kanth and Y. N. Reddy, Cubic spline for a class of singular twopoint
boundary value problems, Appl. Math. Comput., 170:733–740, 2005.
[39] A. S. V. Ravi Kanth and V. Bhattacharya, Cubic spline polynomial for a class
of non-linear singular two-point boundary value problems arising in physiology,
Appl. Math Comput., 174:768–774, 2006.
[40] G. W. Reddien, On the collocation method for singular two point boundary value
problems, Numer. Math., 25:427–432, 1975.
[41] R. D. Russell and L. F. Shampine, Numerical methods for singular boundary
value problems, SIAM J. Numel. Anal., 12:13–36, 1975.
[42] A. Sayfy and S. Khuri, A generalized algorithm for the order verification of
numerical methods, Far East J. Appl. Math., 33:295–306, 2008.
[43] I. J. Schoenberg, On Spline Functions. MRC Report 625, University of Wisconsin (1966).
[44] S. Taliaferro, A nonlinear singular boundary value problem, Nonlinear Anal.,
3:897–904, 1979.
[45] W. Wang, M. Cui and B. Han, A new method for solving a class of singular
two-point boundary value problems, Appl. Math. Comput., 206:721–727, 2008.
[46] Y. Wang, M. Tadi and M. Radenkovic, A numerical method for singular and singularly
perturbed Dirichlet-type boundary-value problems, Int. J. Appl. Math.
Research., 3:292–300, 2014.
[47] X. Zhang, Modified cubic B-spline solution of singular two-point boundary value
problems, J. Inf. Comput. Sci., 11:3167–3176, 2014.