An Approximation Algorithm for the Solution of Astrophysics Equations Using Rational Scaled Generalized Laguerre Function Collocation Method Based on Transformed Hermite-Gauss Nodes

In this paper we propose a collocation method for solving Lane-Emden type equation which is nonlinear ordinary differential equation on the semi-infinite domain. This equation is categorized as singular initial value problems. We solve this equation by the generalized Laguerre polynomial collocation method based on Hermite-Gauss nodes. This method solves the problem on the semi-infinite domain without truncating it to a finite domain and transforming domain of the problem to a finite domain. In addition, this method reduces solution of the problem to solution of a system of algebraic equations.


Introduction
There are many problems in science and engineering arising in unbounded domains.
Spectral methods are famous ways to solve these kinds of problems.The most common approach on spectral methods, that is used in this paper too, is through the use of functions that are orthogonal over unbounded domains, such as the Hermite and the Laguerre functions [1][2][3][4][5][6][7][8].
The second approach is reformulating the original problem in the semi-infinite domain to a singular problem in a bounded domain by variable transformation and then using the Jacobi polynomials to approximate the resulting singular problem [9][10][11].A third approach of spectral method is based on rational orthogonal functions, for example, Christov [12] and Boyd [13,14] developed some spectral methods on unbounded intervals by using mutually orthogonal systems of rational functions.Boyd [14] defined a new spectral basis, named rational Chebyshev functions on the semi-infinite interval, by mapping it to the Chebyshev polynomials.Guo et al. [15] proposed and analyzed a set of Legendre rational func-tions which are mutually orthogonal in . A forth approach is replacing the semi-infinite domain with interval by choosing L, sufficiently large, this method is named as the domain truncation [16].
[0, ] L In this paper, we investigate the Generalized Laguerrecollocation method based on Hermite-Gauss Nodes which is another approach for solving ODEs on the half line.In [7] proposed spectral methods using Laguerre functions and analyzed for model elliptic equations on regular unbounded domains.It is shown that spectral-Galerkin approximations based on Laguerre functions are stable and convergent with spectral accuracy in the Sobolev spaces.Siyyam [8] applied two numerical methods for solving initial value problem differential equations using the Laguerre Tau method.He generated linear systems and solved them.Maday, et al. [6] proposed a Laguerre type spectral method for solving partial differential equations.They introduced a general presentation of the method and a description of the derivation discretization matrices and then determined the optimum estimations in the adapted Hilbert norms.

The Lane-Emden Equation
This equation is one of the basic equations in the theory of stellar structure and has been the focus of many studies [17][18][19][20][21].This equation describes the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of classical thermodynamics.The polytropic theory of stars essentially follows out of thermodynamic considerations, that deal with the issue of energy transport, through the transfer of material between different levels of the star.We simply begin with the Poisson equation and the condition for hydrostatic equilibrium: where is the gravitational constant, is the pressure, is the mass of a star at a certain radius , and r  is the density, at a distance from the center of a spherical star.Combination of these equations yields the following equation, which as should be noted, is an equivalent form of the Poisson Equation.
From these equations one can obtain the Lane-Emden equation through the simple supposition that the density is simply related to the density, while remaining independent of the temperature.We already know that in the case of a degenerate electron gas that the pressure and density are 3 5 P   , assuming that such a relation exists for other states of the star we are led to consider a relation of the following form: where K and m are constants, at this point it is important to note that m is the polytropic index which is related to the ratio of specific heats of the gas comprising the star.
Based upon these assumptions we can insert this relation into our first equation for the hydrostatic equilibrium condition and from this rewrite equation to: where the additional alteration to the expression for density has been inserted with  representing the central density of the star and that of a related dimensionless quantity that are both related to y  through the follow-= .
Additionally, if place this tio (7) result into the Poisson equan, we obtain a differential equation for the mass, with a dependance upon the polytropic index m.Though the differential equation is seemingly difficult to solve, this problem can be partially alleviated by the introduction of an additional dimensionless variable x, given by the following: Inserting these relations into our previous relations we obtain the famous form of the Lane-Emden equation, given below: Taking this simple relation we will have the Lane-Emden equation: At this point it is also important to introduce the bo (11) As a result an additional condition must b in undary conditions, which are based upon the following boundary conditions for hydrostatic equilibrium, and normalization consideration of the newly introduced quantities x and y.What follows for 0 r  is:   e introduced order to maintain the condition of Equation ( 11) simultaneously: In other words, the boundary conditions are as follow: Physically interesting value of m li 5] r is arranged as : in Section 3, we expl last section.e in the interval [0, .Exact soloution for Equation (??) are known only for = 0,1 m and 5.For other value of m the Lane-Emden is to be integrated numerically.In this paper, we solve it for = 1.5,2,2.5,3m and 4.This pape follows equation ain the formulation of rational scaled generalized Laguerre polynomials and Hermite functions required for our subsequent development.In Section 4, we summarize the application of the method for solving Lane-Emden equation and compare it with the existing methods in the literature.Finally we give a brief conclusion in the ing relation

Rational Scaled Generalized Laguerre
Polynom Properties is section is d n scaled generalized Laguerre polynomials and later we present some properties of Hermite function and Hermite-Gauss nodes.

Properties of
The generalized Laguerre with the following recurrence 1 polynomials are defined formula: With the normalizing condition: Let denotes a non-ne valued n over the interval , we define Is the norm induced by the inner prod , These are orthogonal polynomials for tion uct of the space Let be an integer and we define 1  and the po 0. It can be show th x  , j = 0,…,N-1 and e corresponding weights are: The following ormula is known: In particular term on the right ha vanishes when f nomial of degree at most  [24] introduced a scaling function and appropriate numerical procedures in order to limit these unpleasant phenomena.
We define scaled Laguerre functions   n  as follows: where d is gen > 0 k Laguerre polynom is a constant an ials for   We denote scal functions with (SLF).Boyd [26] offer es for optimizing the map pa From Eq as follo rameter k where > 0 k is the scaling parameter.uations ( 21) and ( 22), for = 1


, we obtain the following formula: This system is an al basis with weight func- generalized Laguerre-Gauss-type interpolation were inced by [24,25].It is nt to define the weights of Some of the relations of scaled Laguerre functions and trodu convenie   ; n x k  as follows: where we noted that, for = 1, 2,..., 1. j N

Properties of Hermite Functions
The Hermite function is defined for all and can R x  be written in recursive formula as follows [27][28][29]: where N P at mo sociated is the set of all Hermite polynomials of degree st We now introduce ture as with the Hermite functio Then we have:

ane-Emden Equation
To apply rational scaled generalized Laguerre collocaon method to the standard Lane-Emden Equation initions Eqation (13), we define residual function by substituting Solving the L ti troduced in Equation (10) with boundary cond u And we construct the  in the Lane-Emden Equation ( 10): To find the unknown coefficients and we equalize boundary conditions in E tio quan (13) too, therefore we have: But as mentioned before Lane-Emden equations are defined on the interval   0,  deri and we know properties of Hermite functions are ved in the infinite domain   ,   str .Also we kn oximations can be con-ow appr ct ucted for infinite, semi-infinite and finite intervals.One of the approaches to constru Hermite-Gauss nodes on the interval   0,  which is used in the current paper, is apping, that is a change of variable of the form: Where is the inverse map of

Summary and Conclusions
ials and Hermite FunctionsTh evoted to the introduction of the basic otions and working tools concerning orthogonal rational Rational Scaled Generalized Laguerre Polynomials polynomial) is the th ction of the Sturm-Liouville problem [2,22, z j is transformed root of th ite-Gauss nodes which can be N + to obtain convergence of e method, consequently,   y x given in Equation (10) can be calculated.The resulting graph of Lane-Laguerre functions obtained by present for the Lane-Emden equation with m .

A
set of rational scaled generalized laguerre orthogonal functions are proposed to solve Lane-Em n w gularity at x = 0, by collocation metho m location method we must equalize   Res x to zero at suitable points in   0,  interval.Since Hermite functions are derived in the infinite domain   ,   , we can't apply Hermite-Gauss nodes for equalizing   Res x to zero, therefore we transform this nodes from   ,

Figure 1 .
Figure 1.Lane-Emden equation graph obtained y present method for various m.