Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 850170, 9 pages
http://dx.doi.org/10.1155/2013/850170
Research Article
A New Tau Method for Solving Nonlinear Lane-Emden Type
Equations via Bernoulli Operational Matrix of Differentiation
E. Tohidi,
1
Kh. Erfani,
1
M. Gachpazan,
2
and S. Shateyi
3
1
Department of Mathematics, Islamic Azad University, Zahedan Branch, Zahedan, Iran
2
Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran
3
Department of Mathematics, University of Venda, Private Bag X5050, Tohoyandou 0950, South Africa
Correspondence should be addressed to S. Shateyi; stanford.shateyi@univen.ac.za
Received 18 February 2013; Accepted 24 April 2013
Academic Editor: Mehmet Sezer
Copyright © 2013 E. Tohidi et al. Tis is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
A new and efcient numerical approach is developed for solving nonlinear Lane-Emden type equations via Bernoulli operational
matrix of diferentiation. Te fundamental structure of the presented method is based on the Tau method together with the Bernoulli
polynomial approximations in which a new operational matrix is introduced. Afer implementation of our scheme, the main
problem would be transformed into a system of algebraic equations such that its solutions are the unknown Bernoulli coefcients.
Also, under several mild conditions the error analysis of the proposed method is provided. Several examples are included to illustrate
the efciency and accuracy of the proposed technique and also the results are compared with the diferent methods. All calculations
are done in Maple 13.
1. Introduction
In recent years, the researches on the singular initial value
problems (SIVPs) in several special second-order ordinary
diferential equations (ODEs) have received considerable
attention among mathematicians and physicists. One of the
most well-known classes of such equations are the Lane-
Emden type equations which model many phenomena in
mathematical physics and astrophysics. Tey are nonlinear
ordinary diferential equations which describe the equi-
librium density distribution in self-gravitating sphere of
polytrophic isothermal gas and have a singularity at the
origin [1]. It must be noted that these equations have
fundamental importance in the feld of radiative cooling
and modeling of clusters of galaxies. Moreover, it has been
recently observed that the density profles of dark matter
halos are ofen modeled by the isothermal Lane-Emden
equation with suitable boundary conditions at the origin
[2]. Since getting the analytic solution of these equations
is a difcult task in many cases, robust numerical schemes
must be constructed for obtaining the approximated solu-
tions. In this paper, we will present an efcient method for
computing the numerical solution of the Lane-Emden type
equations [3, 4]
()+
()+ (, ())=(), ∈[0,1],>0,
(1)
with the initial conditions
(0)=,
(0)=0, (2)
where the prime denotes the diferentiation with respect to ,
is a constant, and are nonlinear continuous functions.
Selecting =2, (,()) = (())
, () = 0 and =1
yields [5, 6]
()+
2
()+
()=0, ∈[0,1], (3)
which has another form
1
2
(
2
)+
()=0, ∈[0,1], (4)