Numerical Solution of Singular Lane-Emden Equation

The Lane-Emden type equations are nonlinear ordinary differential equations on semi-infinite domain. They are categorized as singular initial value problems. These equations describe the temperature variation of a spherical gas cloud under themutual attraction of itsmolecules and subject to the laws of classical thermodynamics. The polytrophic 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. These equations are one of the basic equations in the theory of stellar structure and have been the focus of many studies.The general form of the Lane-Emden equations is the following form:


Introduction
The Lane-Emden type equations are nonlinear ordinary differential equations on semi-infinite domain.They are categorized as singular initial value problems.These equations describe the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of classical thermodynamics.The polytrophic 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.These equations are one of the basic equations in the theory of stellar structure and have been the focus of many studies.The general form of the Lane-Emden equations is the following form: () +     () +  (, ) =  () , 0 <  ≤ 1,  ≥ 0, with the following initial conditions: where (, ) is a continuous real-value function and () is an analytical function.Equation ( 1) was used to model several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas sphere, and theory of thermionic currents [1,2].The solution of the Lane-Emden equation, as well as those of a variety of nonlinear problems in quantum mechanics and astrophysics such as the scattering length calculations in the variable phase approach, is numerically challenging because of the singular point at the origin.Bender et al. [3] proposed a new perturbation technique based on an artificial parameter ; the method is often called -method.El-Gebeily and O'Regan [4] used the quasilinearization approach to solve the standard Lane-Emden equation.This method approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories, it is not based on the existence of some small parameters.Approximate solutions to the above problems were presented by Shawagfeh [5] and Wazwaz [6,7] by applying the Adomian method which provides a convergent series solution.Nouh [8] accelerated the convergence of a power series solution of the Lane-Emden equation by using an Euler-Abel transformation and Padé approximation.Mandelzweig and Tabakin [9] applied Bellman and Kalaba's quasilinearization method, and Ramos [10] used a piecewise linearization technique based on the piecewise linearization of the Lane-Emden equation.Bozkhov and Gilli Martins [11] and later Momoniat and Harley [12] applied the Lie Group method successfully to generalized Lane-Emden equations of the first kind.Exact solutions of generalized Lane-Emden solutions of the first kind are investigated by Goenner and Havas [13].Liao [14] solved Lane-Emden type equations by applying a homotopy analysis method.He [15] obtained an approximate analytical solution of the Lane-Emden equation by applying a variational approach which uses a semiinverse method.Ramos [16] presented a series approach to the Lane-Emden equation and gave the comparison with homotopy perturbation method.Özis and Yildirim [17,18] gave the solutions of a class of singular second-order IVPs of Lane-Emden type by using homotopy perturbation and variational iteration method.Parand et al. [19][20][21][22] presented three numerical techniques to solve higher ordinary differential equations such as Lane-Emden.Their approach was based on the rational Chebyshev, rational Legendre Tau, and Hermite functions collocation methods.In this paper, the new approximate analytical method will be introduced for exact solution of Lane-Emden equation.This paper is arranged as follows.
In Section 2, the properties of Legendre wavelets and the way to construct the wavelet technique for this type of equation are described.In Section 3, the proposed method is applied to some types of Lane-Emden equations, and a comparison is made with the existing analytic or exact solutions that were reported in other published works in the literature.Finally, we give a brief conclusion in the last section.

Legendre Wavelets Applied to Singular IVPs
of Lane-Emden Type Equation 2.1.Wavelets and Legendre Wavelets.Wavelets constitute a family of functions constructed from dilation and translation of single function called the mother wavelet.When the dilation parameter  and the translation parameter  vary continuously, we have the following family of continuous wavelets: If we restrict the parameter  and  to discrete values as  =  − 0 ,  =  0  − 0 ,  0 > 1,  0 > 0, and ,  positive integers, we have the following family of discrete wavelets: where  , () forms a wavelet basis for  2 (R).In particular, when  0 = 2 and  0 = 1 then  , () forms an orthonormal basis.
If the infinite series in ( 7) is truncated, then ( 7) can be written as where  and () are 2 −1  × 1 matrices given by The integration of the product of two Legendre wavelets vector functions is obtained as where  is an identity matrix.

Legendre Wavelets Operational Matrix of Integration.
The integration of the vector () defined in (11) can be obtained as where  is the 2 −1  × 2 −1  operational matrix of integration given by [23] as where , , and  are  ×  matrices given by The following property of the product of two Legendre wavelet vector functions will also be used: where  is a vector given in (10) and Ĉ is a 2 −1  × 2 −1  matrix, which is called the product operation of Legendre wavelet vector functions [23].For  = 3 and  = 1, the matrix Ĉ is obtained: 2.4.Solution of Lane-Emden Equations.We multiply both sides of (1) by , in order to use Legendre wavelets to approximate   () as Integrating (19) with respect to  twice from 0 to , we obtain where coefficients  0 and  1 are known and can be obtained from the initial conditions,  and () are defined similarly to (10) and (11), and  is 2 −1  × 2 −1  operational matrix for integration, defined in (14).Also consider the following approximations: where  1 and  2 are column vectors with the entries of the vectors  and coefficients of  known.

ISRN Mathematical Physics
Substitution of approximations ( 20) and ( 22) into (18) will be resulted to Simplifying () in ( 23), a nonlinear system in terms of  will be obtained: The element of vector functions  can be computed by solving these systems.

Numerical Examples
In this section, some examples of Lane-Emden equation are considered and will be solved by introduced method.
Example 1.Consider the following nonlinear Lane-Emden equation: subject to the initial conditions which has the following analytical solution: We solve (25) by the method discussed in this paper with  = 1 and  = 7.
We multiply both sides of (25) by , Let us consider the following approximations: Substitution into (28) and simplifying will be resulted to: Therefore, the approximate solution of (25) will be obtained as follows: Table 1 shows some values of the solutions and absolute errors at some , and plots of the exact and approximate solutions are shown in Figure 1.
Example 2. Consider the following nonlinear Lane-Emden equation: subject to the initial conditions where  ≥ 0 is constant.Substituting  = 0, 1, and 5 into (33) leads to the exact solution respectively.For  = 0, we solve (33) by the Legendre wavelet method with  = 1 and  = 3.For this equation, we find Therefore, we have which is the exact solution.
For  = 1, we solve (33) by the method discussed in this paper with  = 1 and  = 10.We have  Therefore, the following solution will result: Table 2 shows that the Legendre wavelet solution is very near to the exact solution.Figure 2(a) shows that Legendre wavelet solution coincides with the exact solution.
For solving (33) by Legendre wavelets with  = 1,  = 12, and  = 5, we find The approximate solution of () is as follows: Table 3 shows that the Legendre wavelet solution is very near to the exact solution.Figure 2(b) shows that Legendre wavelet solution coincides with the exact solution.
Example 3. Consider the following nonlinear Lane-Emden equation: subject to the initial conditions which has the following analytical solution:   Therefore, the solution of the Lane-Emden equation will be obtained as follows: Table 4 shows that the Legendre wavelet solution is very near to the exact solution.Figure 3 shows that Legendre wavelet solution coincides with the exact solution.Example 4. Consider the following nonlinear Lane-Emden equation: subject to the initial conditions which has the following analytical solution: We solve (47) by the method discussed in this paper with  = 1 and  = 6.We multiply both sides of (47) by ,   () + 2  () + 4 (2  +  /2 ) = 0, 0 <  ≤ 1.
Let us consider the following approximations: Substitution into (50) and simplifying will be resulted to By solving the system (52), we have The approximate solution of () is as follows: Table 5 shows some values of the solutions and absolute errors at some  , s, and plots of the exact and approximate solutions are shown in Figure 4.
Example 5. Consider the following nonlinear Lane-Emden equation: subject to the initial conditions which has the following analytical solution: We solve (55) by the method discussed in this paper with  = 1 and  = 6.This implies that (59) Table 6 shows that the Legendre wavelet solution is very near to the exact solution.Figure 5 shows that Legendre wavelet solution coincides with the exact solution.

Conclusion
In this research, we have presented the Legendre wavelet method for solving nonlinear singular Lane-Emden equation.The Legendre wavelets operational matrix of integration is used to solve Lane-Emden equation.The present method reduces Lane-Emden equation into a set of algebraic equations.Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.

Figure 1 :
Figure 1: The exact and LWM solution of Example 1.

Figure 3 :
Figure 3: The exact and LWM solution of Example 3.

2 Figure 4 :
Figure 4: The exact and LWM solution of Example 4.

Figure 5 :
Figure 5: The exact and LWM solution of Example 5.

Table 1 :
Numerical results of Example 1.

Table 4 :
Numerical results of Example 3.

Table 5 :
Numerical results of Example 4.

Table 6 :
Numerical results of Example 5.