zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A numerical method for Lane-Emden equations using hybrid functions and the collocation method. (English) Zbl 1235.65107
Summary: A numerical method to solve Lane-Emden equations as singular initial value problems is presented in this work. This method is based on the replacement of unknown functions through a truncated series of hybrid of block-pulse functions and Chebyshev polynomials. The collocation method transforms the differential equation into a system of algebraic equations. It also has application in a wide area of differential equations. Corresponding numerical examples are presented to demonstrate the accuracy of the proposed method.

65L99Numerical methods for ODE
34A34Nonlinear ODE and systems, general
Full Text: DOI
[1] S. Chandrasekhr, Introduction to Study of Stellar Structure, Dover, New York, NY, USA, 1967.
[2] H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, NY, USA, 1962. · Zbl 0217.54002 · doi:10.1145/368273.368557
[3] O. U. Richardson, The Emission of Electricity from Hot Bodies, Longman, Green and Co., London, UK, 1921.
[4] N. T. Shawagfeh, “Nonperturbative approximate solution for Lane-Emden equation,” Journal of Mathematical Physics, vol. 34, no. 9, pp. 4364-4369, 1993. · Zbl 0780.34007 · doi:10.1063/1.530005
[5] A.-M. Wazwaz, “A new algorithm for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 118, no. 2-3, pp. 287-310, 2001. · Zbl 1023.65067 · doi:10.1016/S0096-3003(99)00223-4
[6] A.-M. Wazwaz, “A new method for solving singular initial value problems in the second-order ordinary differential equations,” Applied Mathematics and Computation, vol. 128, no. 1, pp. 45-57, 2002. · Zbl 1030.34004 · doi:10.1016/S0096-3003(01)00021-2
[7] J. I. Ramos, “Series approach to the Lane-Emden equation and comparison with the homotopy perturbation method,” Chaos, Solitons and Fractals, vol. 38, no. 2, pp. 400-408, 2008. · Zbl 1146.34300 · doi:10.1016/j.chaos.2006.11.018
[8] M. Dehghan and F. Shakeri, “Approximate solution of differential equation arising in astrophysics using the variation method,” New Astronomy, vol. 13, pp. 53-59, 2008.
[9] S. A. Yousefi, “Legendre wavelets method for solving differential equations of Lane-Emden type,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1417-1422, 2006. · Zbl 1105.65080 · doi:10.1016/j.amc.2006.02.031
[10] K. Parand, M. Dehghan, A. R. Rezaei, and S. M. Ghaderi, “An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method,” Computer Physics Communications, vol. 181, no. 6, pp. 1096-1108, 2010. · Zbl 1216.65098 · doi:10.1016/j.cpc.2010.02.018
[11] H. Adibi and A. M. Rismani, “On using a modified Legendre-spectral method for solving singular IVPs of Lane-Emden type,” Computers & Mathematics with Applications, vol. 60, no. 7, pp. 2126-2130, 2010. · Zbl 1205.65201 · doi:10.1016/j.camwa.2010.07.056
[12] A.-M. Wazwaz, “A reliable treatment of singular Emden-Fowler initial value problems and boundary value problems,” Applied Mathematics and Computation, vol. 217, no. 24, pp. 10387-10395, 2011. · Zbl 1220.65095 · doi:10.1016/j.amc.2011.04.084
[13] S. Karimi Vanani and A. Aminataei, “On the numerical solution of differential equations of Lane-Emden type,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2815-2820, 2010. · Zbl 1193.65151 · doi:10.1016/j.camwa.2010.01.052
[14] M. Razzaghi and H.-R. Marzban, “Direct method for variational problems via hybrid of block-pulse and Chebyshev functions,” Mathematical Problems in Engineering, vol. 6, no. 1, pp. 85-97, 2000. · Zbl 0987.65055 · doi:10.1155/S1024123X00001265 · eudml:49243
[15] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zhang, Spectral Methods on Fluid Dynamics, Springer, New York, NY, USA, 1988. · Zbl 0658.76001
[16] K. Maleknejad and M. Tavassoli Kajani, “Solving linear integro-differential equation system by Galerkin methods with hydrid functions,” Applied Mathematics and Computation, vol. 159, no. 3, pp. 603-612, 2004. · Zbl 1063.65145 · doi:10.1016/j.amc.2003.10.046