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Lane-Emden's equation has fundamental importance in the recent analysis of many problems in relativity and astrophysics including some models of density profiles for dark matter halos. An efficient numerical method is presented for linear and nonlinear Lane-Emden-type equations using the Bernstein polynomial operational matrix of integration. The proposed approach is different from other numerical techniques as it is based on the Bernstein polynomial integration matrix. Some illustrative examples are given to demonstrate the efficiency and validity of the proposed algorithm.

In recent years, the studies of singular initial value problems in some special second-order ordinary differential equations (ODEs) have attracted the attention of many mathematicians and physicists. One of the most intriguing equations is the Lane-Emden-type equations which models many phenomena in mathematical physics and astrophysics. It is a nonlinear ordinary differential equation which describes the equilibrium density distribution in self-gravitating sphere of polytrophic isothermal gas and has a singularity at the origin. This equation has fundamental importance in the field of radiative cooling and modeling of clusters of galaxies. It has also proven to be most versatile in the examination of a variety of situations, including the analysis of isothermal cores, convective stellar interiors, and fully degenerate stellar configurations. Moreover, it has been recently observed [

Lane-Emden’s equations [

It has been shown [

In the special case, where

The parameter

Similarly, by choosing

A numerical method based on conversion into integral equations solved by Legendre wavelets is given in [

In [

Legendre’s spectral method for solving only singular IVPs is given in [

A collocation method based on Chebyshev’s polynomials is proposed in [

The aim of the present paper is to apply the Bernstein polynomial operational matrix of integration for the first time, to propose a reliable numerical technique for solving linear and nonlinear Lane-Emden’s equations. Some special cases of the problem are solved to show its validity and efficiency in comparison with other existing numerical methods. The approximate solution, obtained by the proposed method, shows its superiority on the other existing numerical solution.

A Bernstein polynomial [

If (

a set of block pulse functions (BPFs) is defined on

The Bernstein polynomial can also be expanded and approximated into an

In this section, the method presented in Section

Since exact solutions for the case

Let

In this section some special cases of (

Consider the standard Lane-Emden equation:

Comparison of the numerical solution and error obtained by present method for

Our method |
Series method [ | |||

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0.1 | ||||

0.5 | ||||

1.0 |

Graph of standard Lane-Emden’s equation and its approximate solution at different values of

Here we can see that the function basically follows the same form as that for an index

Letting

Comparison of the numerical solution and error obtained by present method with series solution [

Our method |
Series method [ | |||

0.0 | ||||

0.1 | ||||

0.2 | ||||

0.5 | ||||

1.0 |

Graph of approximate solution in comparison with [

Let

Comparison of the numerical solution and error obtained by present method with series solution [

Present method |
Series method [ | |||

0.0 | ||||

0.1 | ||||

0.5 | ||||

1.0 |

Graph of exact solution and approximate solution (at

Absolute error for

The Bernstein polynomial operational matrix of integrations has been applied for solving one of the most popular and intriguing differential equations, that is, the Lane-Emden equations. These results are useful in a few respects and deal with some actual state equation for stars. Though these two solutions for