Variational Iteration Method for Analytical Solution of the Lane-Emden Type Equation with Singular Initial and Boundary Conditions

In this paper, a well-known equation used in astrophysics and mathematical physics called the Lane-Emden equation is to be solved by a variational iteration method. The main purpose of this approach is to solve the singular initial value problems and also boundary value problem of Lane-Emden type equations. This technique overcomes its singularity at origin rapidly. It gives the approximate and exact solution with easily computable terms. The approach is illustrated with some examples to show its reliability and compactness.


Introduction
Lane-Emden type equations have been the focal point of various publications because of their frequent appearance in astrophysics, mathematical physics, engineering, mathematical biology and other fields.Recently, many mathematicians and physicists have been greatly attracted towards the study of singular initial value problems (IVPs) in second order ordinary differential equations of Lane-Emden type problems.
One of these type equations is formulated as 128 ( ) under the initial conditions where A and B are constants and ( ) is a real-valued continuous function.This equation was used to model various phenomena such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres and the theory of thermionic currents [1,2,3,4].
While, another class of singular initial value problems of Lane-Emden type can also be given in the form under the initial conditions (2), where ( ) is a real-valued continuous function and (3) shows the standard Lane-Emden type equation.The numerical solution of the Lane-Emden equation (3) as well as other types of linear and nonlinear singular IVPs in quantum mechanics and astrophysics [5,6], is numerically challenging because of the singularity behavior at the origin.Hasan and Zhu [7,8] have solved such a singular initial value problem by the Taylor series and modified Adomian decomposition methods.Gupta and Sharma [9] have also used the Taylor series method to solve Lane-Emden and Emden-Fowler equations.The approximate analytical solutions to the Lane-Emden equations were presented by Shawagfeh [10] and Wazwaz [11,12,13] using the Adomian decomposition method (ADM) [14].He [15] obtained an approximate analytical solution of the Lane-Emden equation by applying a variational approach which uses a semi inverse method.Laio [16] solved Lane-Emden type equations by applying a homotopy analysis method.Chowdhury and Hashim [17] gave the solutions of class of singular second-order IVPs of Lane-Emden type by using He's homotopy perturbation method.Ramos [18] presented a series approach to the Lane-Emden equation and gave the comparison with He's homotopy perturbation method.Recently, Dehghan and Shakeri [19] first applied an exponential transformation to the Lane-Emden equation to overcome the difficulty of a singular point at 0 = x and solved the resulting nonsingular problem by the variational 129 iteration method.Exact solutions of generalized Lane-Emden solutions of the first kind are investigated by Goenner and Havas [20].But analytical solutions are more needed to understand physical better.
Recently, a lot of attention has been devoted to the study of VIM such as study about the variational iteration method to find out the numerical solution of Sine-Gordon [21], Fokker-Plank equation [28].[22] use the variational iteration method for solving biharmonic equation, [23,24,25,19,26] investigate various models, singular and nonsingular, linear and nonlinear, and ODEs and PDEs as well.
Our aim in this work to apply the VIM for these two types of Emden-Fowler equations with initial values and boundary values.In particular, we will first examine the Lane-Emden initial value problem given by [27] ( ) under the initial conditions Next, we will discuss the Lane-Emden boundary value problem ( ) under the initial conditions It is seen that for 1 = β and , 2 = β both (4) and (6) shows cylindrical and spherical models [27].

VIM and Lagrange Multipliers
In this section, we will discuss about the variational iteration method (VIM) and find out the distinct Lagrange multipliers ( ) for different values of .β Consider the following nonlinear differential equation where L is a linear operator, N being a non-linear operator and ( ) is known as analytical function.According to VIM, we can construct the following correction functional.
where ( ) η λ is general Lagrange multiplier which can be identified optimally via variational theory.
( ) Taking the variation δ on both sides, we get as, This in turn gives the stationary conditions Here we will discuss the following three cases to find out the value of ( ), η λ which shows for cylindrical problems. 131 which shows for spherical problems.
which shows for general problems.
Furthermore, by the successive approximation upon ( ) for the solution of ( ), x y n will be readily obtained by using the obtained values of Lagrange multiplier and by using selective function .

Lane-Emden IVPs
In this section, initial value model of Lane-Emden with singular behavior ( ) and 2 > will be presented.Three initial value problems are discussed here.

Example 1
Consider the Lane-Emden type problem (4) with with initial conditions the correctional functional for (17) is, 132 where ( ) x ln as defined in above ( 14), considering the given initial value, we can select ( ) .
Using this selection in (19), we obtain the following successive approximation.

Example 2
Consider the Lane-Emden type problem (4) with , 2 = β with initial conditions the correctional functional for ( 21) is,

Lane-Emden BVPs
In this section, boundary value model of Lane-Emden with singular behavior ( ) and 2 > will be presented.Three boundary value problems are discussed here.

Example 4
Consider the Lane-Emden type problem (6) with , 1 = β where ( ) x ln as defined in above ( 14), considering the given initial value, we can select ( ) where f being any arbitrary constant.
Using this selection in (31), we obtain the following successive approximation.

Example 5
Consider the Lane-Emden type problem ( ) with conditions the correctional functional for (34) is, where ( ) ( ) as defined in above (15), considering the given initial value, we can select ( ) where g being any arbitrary constant.
where ( ) ( ) ( ) x as defined in above ( 16), considering the given initial value, we can select ( )

Conclusion
In this paper, we applied He's VIM to attain an approximate-exact solution of the singular Lane-Emden type equations.This method is applied to both initial value problems and boundary value problems.As this technique is more efficient and reliable but the difficulty of these type problem is due to the existence of singularity at .0 = x Our major work is to examine the singular point of Lane-Emden type problems with initial value problems as well as boundary value problems with different Lagrange multipliers.These Lagrange multipliers for each case are different.We demonstrate six distinct examples some of them are linear, nonlinear, homogenous and non-homogenous.Finally, we conclude that the proposed technique is well suit for numerical and analytic solutions.

0y
Consequently, the exact solution can be obtained by using

⋮
Using this selection in(23), we obtain the following successive approximation.Since noise term vanish in limit which gives the exact solution of(21)

⋮
Using this selection in(27), we obtain the following successive approximation.Since noise term vanish in limit which gives the exact solution of (25),

⋮
every value of f, we obtain the following sequences for values of f.Since noise term vanish in limit which gives the exact solution of (29) ( )

⋮
Since noise term vanish in limit which gives the exact solution of (37) ( )