Research ArticlesNo Access

An Iterative Numerical Method for Solving the Lane–Emden Initial and Boundary Value Problems

International Center for Applied Mathematics and Computational Bioengineering, Department of Mathematics and Natural Sciences, Gulf University for Science and Technology, P.O. Box 7207, Hawally 32093, Kuwait

E-mail Address: romdhane.m@gust.edu.kw

Corresponding author.

Search for more papers by this author

and

Helmi Temimi

E-mail Address: temimi.h@gust.edu.kw

Search for more papers by this author

https://doi.org/10.1142/S0219876218500202Cited by:13 (Source: Crossref)

Abstract

In this paper, we propose fast iterative methods based on the Newton–Raphson–Kantorovich approximation in function space [Bellman and Kalaba, (1965)] to solve three kinds of the Lane–Emden type problems. First, a reformulation of the problem is performed using a quasilinearization technique which leads to an iterative scheme. Such scheme consists in an ordinary differential equation that uses the approximate solution from the previous iteration to yield the unknown solution of the current iteration. At every iteration, a further discretization of the problem is achieved which provides the numerical solution with low computational cost. Numerical simulation shows the accuracy as well as the efficiency of the method.

Keywords:

Remember to check out the Most Cited Articles!
Check out these titles in finite element methods!

References

Abd-Elhameed, W. M. [2015] “ New Galerkin operational matrix of derivatives for solving Lane–Emden singular-type equations,” Eur. Phys. J. Plus 130, 52. https://doi.org/10.1140/epjp/i2015-15052-2 Crossref, Web of Science, Google Scholar
Abd-Elhameed, W. M. and Doha, E. H. and Saad, A. S. and Bassuony, M. A. [2016] “ New Galerkin operational matrices for solving Lane–Emden type equations,” Rev. Mex. Astron. Astron 52, 83–92. Web of Science, Google Scholar
Bellman, R. E. and Kalaba, R. E. [1965] Quasilinearisation and nonlinear boundary-value problems (Elsevier, New York). Google Scholar
Bengochea, G. [2014] “ Algebraic approach to the Lane–Emden equation,” Appl. Math. Comput. 232, 424–430. Web of Science, Google Scholar
Boubaker, K. and Bhrawy, A. H. [2012] “ Polytropic star structure analysis under Bonnor-Ebert gas sphere astrophysical configuration thorough investigating analytical solutions to the related Lane–Emden equation,” Adv. Space Res. 49(6), 1062–1066. Crossref, Web of Science, Google Scholar
Boubaker, K. and Van Gorder, R. A. [2012] “ Application of the BPES to Lane–Emden equations governing polytropic and isothermal gas spheres,” New Astron. 17(6), 565–569. Crossref, Web of Science, Google Scholar
Caglar, H., Caglar, N. and Ozer, M. [2009] “ B-spline solution of non-linear singular boundary value problems arising in physiology,” Chaos, Solitons Fractals 39, 1232–1237. Crossref, Web of Science, Google Scholar
Doha, E. H., Abd-Elhameed, W. M. and Youssri, Y. H. [2013] “ Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane–Emden type,” New Astron. 23–24, 113–117. Crossref, Web of Science, Google Scholar
Doha, E. H., Abd-Elhameed, W. M. and Bassuony, M. A. [2015] “ On using third and fourth kinds Chebyshev operational matrices for solving Lane–Emden type equations, Rom. J. Phys. 60(3–4), 281–292. Web of Science, Google Scholar
Emden, R. [1907] Gaskugeln Anwendungen der Mechanischen Warmtheorie (Druck und Verlag Von B. G. Teubner, Leipzig and Berlin). Google Scholar
Gurbuz, B. and Sezer, M. [2014] “ Laguerre polynomial approach for solving Lane–Emden type functional differential equations,” Appl. Math. Comput. 242, 255–264. Web of Science, Google Scholar
Harley, C. and Momoniat, E. [2008] “ First integrals and bifurcations of a Lane–Emden equation of the second kind,” J. Math. Anal. Appl. 344(2), 757–764. Crossref, Web of Science, Google Scholar
Kazemi Nasab, A., Kilicman, A., Pashazadeh Atabakan, Z. and Leong, W. J. [2015] “ A numerical approach for solving singular nonlinear Lane–Emden type equations arising in astrophysics,” New Astron. 34, 178–186. Crossref, Web of Science, Google Scholar
Khuri, S. A. and Sayfy, A. [2010] “ A novel approach for the solution of a class of singular boundary value problems arising in physiology,” Math. Comput. Model. 52, 626–636. Crossref, Web of Science, Google Scholar
Kumar, M. and Singh, N. [2010] “ 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. Crossref, Web of Science, Google Scholar
Lane, J. H. [1870] “ On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment,” Am. J. Sci. Arts 50(148), 57–74. Crossref, Google Scholar
Lima, P. M. and Morgado, L. [2010] “ Numerical modeling of oxygen diffusion in cells with Michaelis-Menten uptake kinetics,” J. Math. Chem. 48(1), 145–158. Crossref, Web of Science, Google Scholar
Mall, S. and Chakraverty, S. [2014] “ Chebyshev neural network based model for solving Lane–Emden type equations,” Appl. Math. Comput. 247, 100–114. Web of Science, Google Scholar
Motsa, S. S. and Sibanda, P. [2010] “ A new algorithm for solving singular IVPs of Lane–Emden type,” In Proc. 4th Int. Conf. Appl. Mathematics, Simulation, Modelling, pp. 176–180. Google Scholar
Parand, K., Shahini, M. and Dehghan, M. [2009] “ Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type,” J. Comput. Phys. 228(23), 8830–8840. Crossref, Web of Science, Google Scholar
Van Gorder, R. A. [2011] “ Exact first integrals for a Lane–Emden equation of the second kind modeling a thermal explosion in a rectangular slab,” New Astron. 16(8), 492–497. Crossref, Web of Science, Google Scholar
Wazwaz, A. M. [2001] “ A new algorithm for solving differential equations of Lane–Emden type,” Appl. Math. Comput. 118(2), 287–310. Web of Science, Google Scholar
Wazwaz, A. M. [2011] “ A reliable treatment of singular Emden-Fowler initial value problems and boundary value problems,” Appl. Math. Comput. 217(24), 10387–10395. Crossref, Web of Science, Google Scholar
Wazwaz, A. M. [2011] “ 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. Crossref, Web of Science, Google Scholar
Wazwaz, A. M., Rach, R. and Duan, J.-S. [2013] “ Adomian decomposition method for solving the Volterra integral form of the Lane–Emden equations with initial values and boundary conditions,” Appl. Math. Comput. 219(10), 5004–5019. Web of Science, Google Scholar