Abstract
In this paper, a new approach is proposed for solving a class of singular boundary value problems of Lane–Emden type. It is well known that the Adomian decomposition method (ADM) fails to provide a convergent series solution to strongly nonlinear boundary value problem in the wider region and the B-spline collocation method yields unsatisfactory approximation in the presence of singularity. To avoid these shortcomings of both methods, we propose a novel numerical method based on a combination of modified Adomian decomposition method and quintic B-spline collocation method to obtain more accurate solution of the problem under consideration. The principal idea of this approach is to decompose the domain of the problem \(D = [0, 1]\) into two subdomains as \(D = D_{1} U D_{2}= [0, \delta ] U [\delta , 1]\) (\(\delta \) is vicinity of the singularity). In the first domain \(D_1\), the underlying singular boundary value problem is efficiently tackled by a modified Adomian decomposition method. The intent is to apply the ADM in the smaller domain for finding a satisfactory solution. Finally, in the second domain \(D_2\), a collocation approach based on quintic B-spline basis function is designed for solving the resulting regular boundary value problem. The error estimation of the quintic B-spline interpolation is supplemented. In addition, six illustrative examples are presented to demonstrate the applicability and accuracy of the new method. It is shown that the resulting solutions appear to be higher accurate when compared to some existing numerical methods.
Similar content being viewed by others
References
P.L. Chambre, On the solution of the Poisson–Boltzmann equation with the application to the theory of thermal explosions. J. Chem. Phys. 20, 1795–1797 (1952)
H.S. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. J. Theor. Biol. 60, 449–457 (1976)
U. Flesch, The distribution of heat sources in the human head: a theoretical consideration. J. Theor. Biol. 54, 285–287 (1975)
B.F. Gray, The distribution of heat sources in the human head: a theoretical consideration. J. Theor. Biol. 82, 473–476 (1980)
H.S. Fogler, Elements of Chemical Reaction Engineering, 2nd edn. (Prentice-Hall Inc, Englewood Cliffs, 1997)
M. Chawla, R. Subramanian, H. Sathi, A fourth order method for a singular twopoint boundary value problem. BIT 28(1), 88–97 (1988)
J.A. Adam, A mathematical model of tumor growth II: effect of geometry and spatial nonuniformity on stability. Math. Biosci. 86, 183–211 (1987)
M.K. Kadalbajoo, V. Kumar, B-spline method for a class of singular two-point boundary value problems using optimal grid. App. Math. Comput. 188, 1856–1869 (2007)
J.B. Keller, Electrohydrodynamics I. The equilibrium of a charged gas in a container. J. Ration. Mech. Anal. 5, 715–724 (1956)
P. Roul, U. Warbhe, A novel numerical approach and its convergence for numerical solutions of nonlinear doubley singular boundary value problems. J. Comput. Appl. Math. 226, 661–676 (2016)
A. K. Verma, S. Kayenat, On the convergence of Mickens’ type nonstandard finite difference schemes on Lane-Emden type equations. J. Math. Chem. https://doi.org/10.1007/s10910-018-0880-y
R.K. Pandey, A note on a finite difference method for a class of singular boundary-value problems in physiology. Int. J. Comput. Math. 74(1), 127–132 (2000)
M.M. Chawla, S. Mckee, G. Shaw, Order \(h^2\) method for a singular two-point boundary value problem. BIT 26, 318–326 (1986)
M.M. Chawla, R. Subramanian, A new spline method for singular two-point boundary value problems. Int. J. Comput. Math. 24(3–4), 291–310 (1988)
S.R.K. Iyengar, P. Jain, Spline finite differnce methods for singular two point boundary value problem. Numer. Math. 50, 363–376 (1987)
A.S.V. Ravikanth, Cubic spline polynomial for non-linear singular two-point boundary value problems. Appl. Math. Comput. 189, 2017–2022 (2007)
H. Caglar, N. Caglar, M. Ozer, B-spline solution of non-linear singular boundary value problems arising in physiology. Chaos Solitons Fractals 39, 1232–1237 (2009)
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, 3881–3886 (2011)
P. Roul, U. Warbhe, New approach for solving a class of singular boundary value problem arising in various physical models. J. Math. Chem. 54, 1255–1285 (2016)
P. Roul, U. Warbhe, A new homotopy perturbation scheme for solving singular boundary value problems arising in various physical models. Z. Naturforschung A 72(8), 733–743 (2017)
P. Roul, K. Thula, A fourth-order B-spline collocation method and its error analysis for Bratu-type and Lane–Emden problems. Int. J. Comput. Math. 96(1), 85–104 (2019)
M. Inc, D.J. Evans, The decomposition method for solving of a class of singular two-point boundary value problems. Int. J. Comput. Math. 80, 869–882 (2003)
M. Kumar, N. Singh, Modified Adomian decomposition method and computer implementation for solving singular boundary value problems arising in various physical problems. Comput. Chem. Eng. 34(11), 1750–1760 (2010)
S.A. Khuri, A. Sayfy, A novel approach for the solution of a class of singular boundary value problems arising in physiology. Math. Comput. Model. 52, 626–636 (2010)
S.A. Khuri, A. Sayfy, A mixed decomposition-spline approach for the numerical solution of a class of singular boundary value problems. Appl. Math. Model 40, 4664–4680 (2016)
P. Roul, A fast and accurate computational technique for efficient numerical solution of nonlinear singular boundary value problems. Int. J. Comput. Math. 96(1), 51–72 (2019)
P. Roul, A new efficient recursive technique for solving singular boundary value problems arising in various physical models. Eur. Phys. J. Plus 131(4), 1–15 (2016)
P. Roul, D. Biswal, A new numerical approach for solving a class of singular two-point boundary value problems. Numer. Algorithms 75(3), 531–552 (2017)
P. Roul, K. Thula, A new high-order numerical method for solving singular two-point boundary value problems. J. Comput. Appl. Math. 343, 556–574 (2018)
D.L.S. McElwain, A re-examination of oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kineticsm. J. Theor. Biol. 71, 255–263 (1978)
S.H. Lin, Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics. J. Theor. Biol. 60, 449–457 (1976)
P. Hiltmann, P. Lory, On oxygen diffusion in a spherical cell with Michaelis–Menten oxygen uptake kinetics. Bull. Math. Biol. 45(5), 661–664 (1983)
M.J. Simpson, A.J. Ellery, An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell. Appl. Math. Model. 36, 3329–3334 (2012)
P. Roul, H. Madduri, A new highly accurate domain decomposition optimal homotopy analysis method and its convergence for singular boundary value problems. Math. Methods Appl. Sci. https://doi.org/10.1002/mma.5181 (2018)
Acknowledgements
The author is very grateful to anonymous referees for their valuable suggestions and comments which improved the paper and thankfully acknowledge the financial support provided by the SERB, Department of science and Technology, New Delhi, India in the form of Project No. SB/S4/MS/877/2014.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Roul, P. A new mixed MADM-Collocation approach for solving a class of Lane–Emden singular boundary value problems. J Math Chem 57, 945–969 (2019). https://doi.org/10.1007/s10910-018-00995-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-018-00995-x
Keywords
- Lane–Emden boundary value problem
- Quintic B-spline collocation method
- ADM
- Domain decomposition
- Error analysis
- Reaction–diffusion problem
- Oxygen-diffusion problem