Solving the Lane – Emden Equation within a Reproducing Kernel Method and Group Preserving Scheme

We apply the reproducing kernel method and group preserving scheme for investigating the Lane–Emden equation. The reproducing kernel method is implemented by the useful reproducing kernel functions and the numerical approximations are given. These approximations demonstrate the preciseness of the investigated techniques.


Introduction
The work of singular initial value problems modeled by second order nonlinear ordinary differential equations (ODEs) have captivated many mathematicians and physicists.One of the equations in this class is the Lane-Emden equation [1].We use the reproducing kernel method (RKM) and the group preserving scheme (GPS) to investigate this equation in this paper.We have investigated solutions of the following problem: with the initial conditions where ς(η) is a sufficiently smooth function.We recall that this problem is in the class of Astrophysics equations [2][3][4][5].
We recall that there are many papers on the solution of the nonlinear problems with a reproducing kernel method.The notion of the reproducing kernel can be traced back to the paper of Zaremba in 1908.It was presented to discuss the boundary value problems of the harmonic functions.In the early development stage of the reproducing kernel theory, most of the works were implemented by Bergman.This researcher obtained the corresponding kernels of the harmonic functions with one or several variables, and the corresponding kernel of the analytic function in squared metric, and implemented them in the research of the boundary value problem of the elliptic partial differential equation.This is the first stage in the development history of reproducing kernel.The second stage of the reproducing kernel theory was started by Mercer who discovered that the continuous kernel of the positive definite integral equation has the positive definite property as [6]: He named the kernel with this property positive definite Hermite matrix.He presented a Hilbert space with inner product f , g , and showed the reproducibility of the kernel as: In 1950, Aronszajn collected the works of the formers and studied a systematic reproducing kernel theory including the Bergman kernel function.
The GPS in the present paper is based on the group invariant schemes, introduced by Liu [23].The most important difference between GPS andthe conventional techniques, such as the Runge-Kutta method, is that these techniques are all formulated directly in the usual Euclidean R k .Furthermore, none of the methods above are considered in Minkowski space M k+1 .One straight advantage of the formulation in M k+1 is that the new techniques can avoid the lacking of spurious solutions and ghost fixed points.Some interesting papers in GPS are [24][25][26][27][28][29][30][31][32][33].
This work is prepared as follows.Section 2 presents some useful reproducing kernel functions.The approximate solutions of Lane-Emden equations are presented in this section.In addition, some numerical experiments are shown.We explained the GPS and apply it to our investigated equation in Section 3. Conclusions are discussed in the final section.

Reproducing Kernel Functions
We define some useful reproducing kernel spaces and find some reproducing kernel functions in this section.
where AC defines the space of absolutely continuous functions.
Theorem 1.The reproducing kernel function r ς of o W 3 2 [0, 1] is given as where Define r ς by Equation ( 6).We have We get ς (1) + u (0)r ( 4) The solution of Equation ( 1) is considered in the reproducing kernel space o W 3 2 [0, 1].On describing the operator model problems ( 1) and (2) convert to the following problem: Theorem 2. L defined by Equation ( 7) is a bounded linear operator.
Proof.We need to prove Lv 2 , where P > 0. By Equations ( 3) and ( 4), we obtain by reproducing property and where P 1 > 0. Therefore, we get where P 2 > 0. Thus, we acquire where

The Main Results
Let Proof.By reproducing property and property of the operator, we get where Proof.We obtain from Equation ( 9) and the uniqueness of solution Equation ( 8).
The approximate solution u n can be achieved as: Theorem 5.For any fixed u 0 (η) ∈ o W 3 2 [0, 1], assume the following conditions are satisfied: (i) 13) converges to the exact solution of Equation (10) where A i is given by (13).
Proof.Let us demonstrate the convergence of u n (η) firstly.By Equation ( 12), we obtain From the orthonormality of { Ψ i } ∞ i=1 , we acquire From boundedness of u n o W 3 2 , we get where ⊥ denotes the orthogonality.Taking into consideration the completeness Taking limits in Equation ( 9) gives If n = 2, then From Equations ( 16) and ( 17), Additionally, it is simple to show by induction that Therefore, we get that is, u (η) is the solution of Equation ( 8) and where A i are given by Equation ( 13).This completes the proof.
is monotonically decreasing in n.
Proof.We acquire by Equations ( 10) and (11).Thus, we get Clearly, is monotonically decreasing in n.

Group Preserving Scheme
Internal symmetry group of a system, especially dynamical systems obtained from Equation (1), preserves using the GPS and when we do not have the symmetry group of nonlinear Lane-Emden equation, it is possible to embed them into the augmented dynamical systems.Consider a dynamical system corresponding to a differential equation as follows: Then, by using a definition for a unit vector of the orientation of the state vector y for Equation ( 19), we have: where y = √ y • y > 0 is the Euclidean norm of y.Equations ( 19) and (20) conclude: In addition, upon utilizing Equations ( 19) and ( 20), we can write: From Equations ( 21) and ( 22), it follows: Obviously, the first equation in Equation ( 23) is the same as the original Equation ( 19), but the addition of the second equation presents us a Minkowskian structure of the augmented state variables of Y := (y T , y ) T ∈ M k+1 (R), which describes an inner product on R k+1 given by: where and This is the Lorentz inner product on R k+1 .
Actually, the null vector in M k+1 (R) lies in the set It is easy to investigate that, in the Minkowskian structure, the augmented variable Y := (y T , y ) T is a null vector and, from the Lorentz inner product, satisfy the cone condition: Equation ( 23) can be written in the abstract form: where Definition 3. Let A be a real square matrix.Then, is the space of skew symmetric matrices in Minkowskian structure.
There is a group of real square matrices that is well-known as a global linear group, defined by: Moreover, we can consider the closed subgroup We have to note that G ∈ O(k, 1) if and only if for all x, y ∈ R k+1 , Gx, Gy = x, y .Thus, O(k, 1) consist of all the Lorentzian isometries of R k+1 .Notice that, for G ∈ O(k, 1), we have det(G) = ±1.Another useful subgroup of O(k, 1) is which is well-known as the proper Orthochronous Lorentz group.Connections between the Lie groups and Lie algebras are specified by the exponential map.That is, if so(k, 1) is the Lie algebra of SO 0 (k, 1), then exp : so(k, 1) → SO 0 (k, 1).
Moreover, we know that so(k, 1) = Sk_Sym k+1 (M k+1 (R)) (See reference [34], p. 82).Therefore, in Equation (27), Ω ∈ so(k, 1) and the corresponding discretized G ∈ SO 0 (k, 1), obtained from the exponential map (29), have the following properties: Now, we are ready to develop our desired numerical scheme in the form: where Y n interprets the numerical value of Y at a discrete t n , and the discretized group element G(n) is obtained through a Cayley transform as follows: Substituting Equation (32) into Equation (31) and taking its first row, we get Now, we are ready to use the GPS for solving Equation (1) with initial conditions (2).According to Equation (19), we have: Results of this example are obtained by fixing ∆η = 10 −7 .Figure 1 shows the graph of the approximate solution obtained by GPS.Moreover, the approximate solutions of Equation (1) obtained by the reproducing kernel method and group preserving scheme are reported in Table 1.Results of this paper show that two investigated methods are in good agreement and approximate solutions are reliable.We calculated all our results with Maple 13 (Siirt University).We used for our numerical results.Using the reproducing kernel method, we choose 100 points.It is possible to improve the results by increasing the points.

Conclusions
We discussed the RKM and the GPS for solving the Lane-Emden equation with initial conditions expressed given by Equation ( 1).An example depicted in Equation ( 1) was presented and the computational accuracy was illustrated.We found the approximate solutions for different values of η by using RKM and GPS, respectively.As it is shown in Table 1, these two investigated methods are very accurate.In addition, we reported very useful reproducing kernel functions and a geometric approach in this work.

Figure 1 .
Figure 1.Numerical solution of Equation (1) obtained by the group preserving scheme (GPS).