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

Abstract

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.

1. 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:

$${\varsigma }^{″}+\frac{2}{\eta }{\varsigma }^{\prime }+{\varsigma }^3=0,0<\eta \le 10,$$

(1)

with the initial conditions

$$\varsigma \left(0\right)=1,{\varsigma }^{\prime }\left(0\right)=0,$$

(2)

where $\varsigma \left(\eta \right)$ 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]:

$$\sum _{i,j=1}^{n}k(x_i,y_j){\xi }_i{\xi }_j\ge 0.$$

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:

$$v\left(s\right)=⟨v\left(t\right),k(t,s)⟩.$$

In 1950, Aronszajn collected the works of the formers and studied a systematic reproducing kernel theory including the Bergman kernel function.

Reproducing kernel theory has valuable implementations in integral equations, differential equations, probability and statistics. This theory has been implemented for many model problems in recent years. The RKM, which accurately calculates the series solution, is of efficient interest to applied sciences. Recently, a lot of research work has been devoted to the application of RKM [6,7,8,9,10,11]. For more details, see [12,13,14,15,16,17,18,19,20,21,22].

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 ${\mathbb{R}}^k$. Furthermore, none of the methods above are considered in Minkowski space ${\mathcal{M}}^{k+1}$. One straight advantage of the formulation in ${\mathcal{M}}^{k+1}$ 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.

2. Reproducing Kernel Functions

We define some useful reproducing kernel spaces and find some reproducing kernel functions in this section.

Definition 1.

$W_2^1[0,1]$ is given as:

$$W_2^1[0,1]=\{u\in AC[0,1]:u^{\prime }\in L^2[0,1]\},$$

where $AC$ defines the space of absolutely continuous functions.

$${⟨u,g⟩}_{W_2^1}={\int }_0^1\left(u\left(\eta \right)g\left(\eta \right)+u^{\prime }\left(\eta \right)g^{\prime }\left(\eta \right)\right)\mathrm{d}\eta ,u,g\in W_2^1[0,1]$$

(3)

and

$${∥u∥}_{W_2^1}=\sqrt{{⟨u,u⟩}_{W_2^1}}u\in W_2^1[0,1]$$

(4)

are the inner product and the norm in $W_2^1[0,1],$ respectively. Reproducing kernel function $T_{\eta }\left(\varsigma \right)$ of $W_2^1[0,1]$ is given by [6]

$$T_{\eta }\left(\varsigma \right)=\frac{1}{2sinh\left(1\right)}\left[cosh(\eta +\varsigma -1)+cosh(\left|\eta -\varsigma \right|-1)\right].$$

(5)

Definition 2.

We describe the space ${}^o{W}_2^3[0,1]$ by

$$\begin{array}{c}\hfill {}^o{W}_2^3[0,1]=\{u\in AC[0,1]:u^{\prime },u^{″}\in AC[0,1],u^{\left(3\right)}\in L^2[0,1],u\left(0\right)=0=u^{\prime }\left(0\right)\}.\end{array}$$

$${⟨u,v⟩}_{{}^o{W}_2^3}=\sum _{i=0}^2{u}^{\left(i\right)}\left(0\right)v^{\left(i\right)}\left(0\right)+{\int }_0^1{u}^{\left(3\right)}\left(\eta \right)v^{\left(3\right)}\left(\eta \right)\mathrm{d}\eta ,u,v\in {}^o{W}_2^3[0,1]$$

and

$${∥u∥}_{{}^o{W}_2^3}=\sqrt{{⟨u,u⟩}_{{}^o{W}_2^3}},u\in {}^o{W}_2^3[0,1],$$

are the inner product and the norm in ${}^o{W}_2^3[0,1]$ respectively.

Theorem 1.

The reproducing kernel function $r_{\varsigma }$ of ${}^o{W}_2^3[0,1]$ is given as

$$r_{\varsigma }\left(\eta \right)=\left\{\begin{array}{c}{\sum }_{k=0}^5{c}_{k+1}\left(\varsigma \right){\eta }^k,0\le \eta <\varsigma \le 1,\hfill \\ {\sum }_{k=0}^5{d}_{k+1}\left(\varsigma \right){\eta }^k,0\le \varsigma <\eta \le 1,\hfill \end{array}\right.$$

(6)

where

$$\begin{array}{ccc}\hfill c_1\left(\varsigma \right)& =& 0,c_2\left(\varsigma \right)=0,c_3\left(\varsigma \right)=\frac{1}{4}{\varsigma }^2,c_4\left(\varsigma \right)=\frac{1}{12}{\varsigma }^2,\hfill \\ \hfill c_5\left(\varsigma \right)& =& -\frac{1}{24}\varsigma ,c_6\left(\varsigma \right)=\frac{1}{120},\hfill \\ \hfill d_1\left(\varsigma \right)& =& \frac{1}{120}{\varsigma }^5,d_2\left(\varsigma \right)=-\frac{1}{24}{\varsigma }^4,\hfill \\ \hfill d_3\left(\varsigma \right)& =& \frac{1}{12}{\varsigma }^3+\frac{1}{4}{\varsigma }^2,\hfill \\ \hfill d_4\left(\varsigma \right)& =& 0,d_5\left(\varsigma \right)=0,d_6\left(\varsigma \right)=0.\hfill \end{array}$$

Proof. 

Let $u\in {}^o{W}_2^3[0,1]$ and $0\le \varsigma \le 1$. Define $r_{\varsigma }$ by Equation (6). We have

$$r_{\varsigma }^{\prime }\left(\eta \right)=\left\{\begin{array}{c}{\sum }_{k=0}^4(k+1)c_{k+1}\left(\varsigma \right){\eta }^k,0\le \eta <\varsigma \le 1,\hfill \\ {\sum }_{k=0}^4(k+1)d_{k+1}\left(\varsigma \right){\eta }^k,0\le \varsigma <\eta \le 1,\hfill \end{array}\right.$$

$$r_{\varsigma }^{\prime \prime }\left(\eta \right)=\left\{\begin{array}{c}{\sum }_{k=0}^3(k+1)(k+2)c_{k+2}\left(\varsigma \right){\eta }^k,0\le \eta <\varsigma \le 1,\hfill \\ {\sum }_{k=0}^3(k+1)(k+2)d_{k+2}\left(\varsigma \right){\eta }^k,0\le \varsigma <\eta \le 1,\hfill \end{array}\right.$$

$$r_{\varsigma }^{\left(3\right)}\left(\eta \right)=\left\{\begin{array}{c}{\sum }_{k=0}^2(k+1)(k+2)(k+3)c_{k+3}\left(\varsigma \right){\eta }^k,0\le \eta <\varsigma \le 1,\hfill \\ {\sum }_{k=0}^2(k+1)(k+2)(k+3)d_{k+3}\left(\varsigma \right){\eta }^k,0\le \varsigma <\eta \le 1,\hfill \end{array}\right.$$

$$r_{\varsigma }^{\left(4\right)}\left(\eta \right)=\left\{\begin{array}{c}{\sum }_{k=0}^1(k+1)(k+2)(k+3)(k+4)c_{k+4}\left(\varsigma \right){\eta }^k,0\le \eta <\varsigma \le 1,\hfill \\ {\sum }_{k=0}^1(k+1)(k+2)(k+3)(k+4)d_{k+4}\left(\varsigma \right){\eta }^k,0\le \varsigma <\eta \le 1,\hfill \end{array}\right.$$

and

$$r_{\varsigma }^{\left(5\right)}\left(\eta \right)=\left\{\begin{array}{c}120c_5\left(\varsigma \right),0\le \eta <\varsigma \le 1,\hfill \\ 120d_5\left(\varsigma \right),0\le \varsigma <\eta \le 1.\hfill \end{array}\right.$$

We get

$$\begin{array}{ccc}\hfill {⟨u,r_{\varsigma }⟩}_{{}^o{W}_2^3}& =& \sum _{i=0}^2{u}^{\left(i\right)}\left(0\right)r_{\varsigma }^{\left(i\right)}\left(0\right)+{\int }_0^1{u}^{\left(3\right)}\left(\eta \right)r_{\varsigma }^{\left(3\right)}\left(\eta \right)\mathrm{d}\eta \hfill \\ & =& u^{\prime }\left(0\right)r_{\varsigma }^{\prime }\left(0\right)+u^{″}\left(0\right)r_{\varsigma }^{″}\left(0\right)+u^{″}\left(1\right)r_{\varsigma }^{\left(3\right)}\left(1\right)-u^{″}\left(0\right)r_{\varsigma }^{\left(3\right)}\left(0\right)\hfill \\ & & -u^{\prime }\left(1\right)r_{\varsigma }^{\left(4\right)}\left(1\right)+u^{\prime }\left(0\right)r_{\varsigma }^{\left(4\right)}\left(0\right)+{\int }_0^1{u}^{\prime }\left(\eta \right)r_{\varsigma }^{\left(5\right)}\left(\eta \right)\mathrm{d}\eta \hfill \\ & =& c_1\left(\varsigma \right)u^{\prime }\left(0\right)+2c_2\left(\varsigma \right)u^{″}\left(0\right)\hfill \\ & & +6(d_3\left(\varsigma \right)+4d_4\left(\varsigma \right)+10d_5\left(\varsigma \right))u^{″}\left(1\right)-6c_3\left(\varsigma \right)u^{″}\left(0\right)\hfill \\ & & -24(d_4\left(\varsigma \right)+5d_5\left(\varsigma \right))u^{\prime }\left(1\right)+24c_4\left(\varsigma \right)u^{\prime }\left(0\right)\hfill \\ & & +{\int }_0^{\varsigma }120c_5\left(\varsigma \right)u^{\prime }\left(\eta \right)\mathrm{d}\eta +{\int }_{\varsigma }^{1}120d_5\left(\varsigma \right)u^{\prime }\left(\eta \right)\mathrm{d}\eta \hfill \\ & =& (c_1\left(\varsigma \right)+24c_4\left(\varsigma \right))u^{\prime }\left(0\right)+2(c_2\left(\varsigma \right)-3c_3\left(\varsigma \right))u^{″}\left(0\right)\hfill \\ & & +6(d_3\left(\varsigma \right)+4d_4\left(\varsigma \right)+10d_5\left(\varsigma \right))u^{″}\left(1\right)-24(d_4\left(\varsigma \right)+5d_5\left(\varsigma \right))u^{\prime }\left(1\right)\hfill \\ & & +120(c_5\left(\varsigma \right)-d_5\left(\varsigma \right))u\left(\varsigma \right)\hfill \\ & =& u\left(\varsigma \right).\hfill \end{array}$$

 ☐

2.1. Solutions in ${}^o{W}_2^3[0,1]$

The solution of Equation (1) is considered in the reproducing kernel space ${}^o{W}_2^3[0,1]$. On describing the operator

$$L:{}^o{W}_2^3[0,1]\to W_2^1[0,1]$$

as

$$Lv\left(\eta \right)=v^{″}\left(\eta \right)+\frac{2}{\eta }{v}^{\prime }\left(\eta \right),$$

(7)

model problems (1) and (2) convert to the following problem:

$$\left\{\begin{array}{c}Lv=M(\eta ,v),\eta \in [0,1],\hfill \\ v\left(0\right)=0=v^{\prime }\left(0\right).\hfill \end{array}\right.$$

(8)

Theorem 2.

L defined by Equation (7) is a bounded linear operator.

Proof. 

We need to prove ${∥Lv∥}_{W_2^1}^2\le M{∥v∥}_{{}^o{W}_2^3}^2$, where $P>0$. By Equations (3) and (4), we obtain

$${∥Lv∥}_{W_2^1}^2={⟨Lv,Lv⟩}_{W_2^1}={\int }_0^1\left(Lv{\left(\eta \right)}^2+L{v}^{\prime }{\left(\eta \right)}^2\right)\mathrm{d}\eta .$$

We get

$$v\left(\eta \right)={⟨v(·),r_{\eta }(·)⟩}_{{}^o{W}_2^3}$$

by reproducing property and

$$Lv\left(\eta \right)={⟨v(·),L{r}_{\eta }(·)⟩}_{{}^o{W}_2^3},$$

so

$$\left|Lv\left(\eta \right)\right|\le {∥v∥}_{{}^o{W}_2^3}{∥L{r}_{\eta }∥}_{{}^o{W}_2^3}=P_1{∥v∥}_{{}^o{W}_2^3},$$

where $P_1>0$. Therefore, we get

$${\int }_0^1{\left[\left(Lv\right)\left(\eta \right)\right]}^2\mathrm{d}\eta \le P_1^2{∥v∥}_{{}^o{W}_2^3}^2.$$

Since

$${\left(Lv\right)}^{\prime }\left(\eta \right)={⟨v(·),{\left(L{r}_{\eta }\right)}^{\prime }(·)⟩}_{{}^o{W}_2^3},$$

then

$$\left|{\left(Lv\right)}^{\prime }\left(\eta \right)\right|\le {∥v∥}_{{}^o{W}_2^3}{∥{\left(L{r}_{\eta }\right)}^{\prime }∥}_{{}^o{W}_2^3}=P_2{∥v∥}_{{}^o{W}_2^3},$$

where $P_2>0$. Thus, we acquire

$${\left[{\left(Lv\right)}^{\prime }\left(\tau \right)\right]}^2\le P_2^2{∥v∥}_{{}^o{W}_2^3}^2$$

and

$${\int }_0^1{\left[{\left(Lv\right)}^{\prime }\left(\eta \right)\right]}^2\mathrm{d}\eta \le P_2^2{∥v∥}_{{}^o{W}_2^3}^2,$$

that is

$${∥Lv∥}_{W_2^1}^2\le {\int }_0^1\left({\left[\left(Lv\right)\left(\eta \right)\right]}^2+{\left[{\left(Lv\right)}^{\prime }\left(\eta \right)\right]}^2\right)\mathrm{d}\eta \le \left(P_1^2+P_2^2\right){∥v∥}_{{}^o{W}_2^3}^2=P{∥v∥}_{{}^o{W}_2^3}^2,$$

where $P=P_1^2+P_2^2>0$ is a positive constant. ☐

2.2. The Main Results

Let ${\phi }_i\left(\eta \right)=T_{{\eta }_i}\left(\eta \right)$ and ${\psi }_i\left(\eta \right)=L^{*}{\phi }_i\left(x\right)$; $L^{*}$ is a conjugate operator of L. The orthonormal system ${\left\{{\overline{\mathsf{\Psi }}}_i\left(\eta \right)\right\}}_{i=1}^{\infty }$ of ${}^o{W}_2^3[0,1]$ can be achieved from Gram–Schmidt orthogonalization operation of ${\left\{{\psi }_i\left(\eta \right)\right\}}_{i=1}^{\infty }$,

$${\overline{\psi }}_i\left(\eta \right)=\sum _{k=1}^i{\beta }_{ik}{\psi }_k\left(\eta \right),({\beta }_{ii}>0,i=1,2,\dots ).$$

(9)

Theorem 3.

Let ${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ be dense in $[0,1]$ and ${\psi }_i\left(\eta \right)={\left.L_{\varsigma }{r}_{\eta }\left(\varsigma \right)\right|}_{\varsigma ={\eta }_i}$. Then, the sequence ${\left\{{\psi }_i\left(\eta \right)\right\}}_{i=1}^{\infty }$ is a complete system in ${}^o{W}_2^3[0,1]$.

Proof. 

By reproducing property and property of the operator, we get

$${\psi }_i\left(\eta \right)=\left(L^{*}{\phi }_i\right)\left(\eta \right)=⟨\left(L^{*}{\phi }_i\right)\left(\varsigma \right),r_{\eta }\left(\varsigma \right)⟩=⟨{\phi }_i)\left(\varsigma \right),L\varsigma r_{\eta }\left(\varsigma \right)⟩={\left.L_{\varsigma }{r}_{\eta }\left(\varsigma \right)\right|}_{\varsigma ={\eta }_i}.$$

It is clear that ${\psi }_i\left(\eta \right)\in {}^o{W}_2^3[0,1]$. For each fixed $u\left(\eta \right)\in {}^o{W}_2^3[0,1]$, let $⟨u\left(\eta \right),{\psi }_i\left(\eta \right)⟩=0,$ $(i=1,2,\dots ),$

$$⟨u\left(\eta \right),\left(L^{*}{\phi }_i\right)\left(\eta \right)⟩=⟨Lu(·),{\phi }_i(·)⟩=\left(Lu\right)\left({\eta }_i\right)=0.$$

${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ is dense in $[0,1]$. Therefore, $\left(Lu\right)\left(\eta \right)=0$. $u\equiv 0$ by the $L^{-1}$. ☐

Theorem 4.

If $u\left(\eta \right)$ is the exact solution of Equation (8), then

$$u=\sum _{i=1}^{\infty }\sum _{k=1}^i{\beta }_{ik}M({\eta }_k,u_k){\widehat{\mathsf{\Psi }}}_i\left(\eta \right),$$

(10)

where ${\left\{\left({\eta }_i\right)\right\}}_{i=1}^{\infty }$ is dense in $[0,1]$.

Proof. 

We obtain

$$\begin{array}{ccc}\hfill u\left(\eta \right)& =& \sum _{i=1}^{\infty }{⟨u\left(\eta \right),{\widehat{\mathsf{\Psi }}}_i\left(\eta \right)⟩}_{{}^o{W}_2^3}{\widehat{\mathsf{\Psi }}}_i\left(\eta \right)\hfill \\ & =& \sum _{i=1}^{\infty }\sum _{k=1}^i{\beta }_{ik}{⟨u\left(\eta \right),{\mathsf{\Psi }}_k\left(\eta \right)⟩}_{{}^o{W}_2^3}{\widehat{\mathsf{\Psi }}}_i\left(\eta \right)\hfill \\ & =& \sum _{i=1}^{\infty }\sum _{k=1}^i{\beta }_{ik}{⟨u\left(\eta \right),L^{*}{\phi }_k\left(\eta \right)⟩}_{{}^o{W}_2^3}{\widehat{\mathsf{\Psi }}}_i\left(\eta \right)\hfill \\ & =& \sum _{i=1}^{\infty }\sum _{k=1}^i{\beta }_{ik}{⟨Lu\left(\eta \right),{\phi }_k\left(\eta \right)⟩}_{W_2^1}{\widehat{\mathsf{\Psi }}}_i\left(\eta \right)\hfill \\ & =& \sum _{i=1}^{\infty }\sum _{k=1}^i{\beta }_{ik}Lu\left({\eta }_k\right){\widehat{\mathsf{\Psi }}}_i\left(\eta \right)\hfill \\ & =& \sum _{i=1}^{\infty }\sum _{k=1}^i{\beta }_{ik}M({\eta }_k,u_k){\widehat{\mathsf{\Psi }}}_i\left(\eta \right),\hfill \end{array}$$

from Equation (9) and the uniqueness of solution Equation (8). ☐

The approximate solution $u_n$ can be achieved as:

$$u_n=\sum _{i=1}^n\sum _{k=1}^i{\beta }_{ik}M({\eta }_k,u_k){\widehat{\mathsf{\Psi }}}_i\left(\eta \right).$$

(11)

Lemma 1.

If ${∥u_n-u∥}_{{}^o{W}_2^3}\to 0$, ${\eta }_n\to \eta$, $(n\to \infty )$ and $M(\eta ,u)$ is continuous for $\eta \in [0,1]$, then [6]

$$M({\eta }_n,u_{n-1}\left({\eta }_n\right))\to M(\eta ,u\left(\eta \right))asn\to \infty .$$

Theorem 5.

For any fixed $u_0\left(\eta \right)\in {}^o{W}_2^3[0,1],$ assume the following conditions are satisfied:

(i) 

$$u_n\left(\eta \right)=\sum _{i=1}^n{A}_i{\overline{\psi }}_i\left(\eta \right),$$

(12)

$$A_i=\sum _{k=1}^i{\beta }_{ik}M({\eta }_k,u_{k-1}\left({\eta }_k\right)),$$

(13)

(ii) 

${∥u_n∥}_{{}^o{W}_2^3}$ is bounded;

(iii) 

${\left\{{\eta }_i\right\}}_{i=1}^{\infty }$ is dense in $[0,1]$;

(iv) 

$M(\eta ,u)\in$ $W_2^1[0,1]$ for any $u\left(\eta \right)\in {}^o{W}_2^3[0,1]$.

Then, $u_n\left(\eta \right)$ in Equation (13) converges to the exact solution of Equation (10) in ${}^o{W}_2^3[0,1]$ and

$$u\left(\eta \right)=\sum _{i=1}^{\infty }{A}_i{\overline{\psi }}_i\left(\eta \right),$$

where $A_i$ is given by (13).

Proof. 

Let us demonstrate the convergence of $u_n\left(\eta \right)$ firstly. By Equation (12), we obtain

$$u_{n+1}\left(\eta \right)=u_n\left(\eta \right)+A_{n+1}{\widehat{\mathsf{\Psi }}}_{n+1}\left(x\right).$$

(14)

From the orthonormality of ${\left\{{\widehat{\mathsf{\Psi }}}_i\right\}}_{i=1}^{\infty }$, we acquire

$${∥u_{n+1}∥}^2={∥u_n∥}^2+A_{n+1}^2={∥u_{n-1}∥}^2+A_n^2+A_{n+1}^2=\dots =\sum _{i=1}^{n+1}{A}_i^2.$$

(15)

From boundedness of ${∥u_n∥}_{{}^o{W}_2^3}$, we get

$$\sum _{i=1}^{\infty }{A}_i^2<\infty ,$$

i.e.,

$$\left\{A_i\right\}\in l^2,(i=1,2,\dots ).$$

Let $m>n$, by $\left(u_m-u_{m-1}\right)\perp \left(u_{m-1}-u_{m-2}\right)\perp \dots \perp \left(u_{n+1}-u_n\right)$, we acquire

$$\begin{array}{ccc}\hfill {∥u_m-u_n∥}_{{}^o{W}_2^3}^2& =& {∥u_m-u_{m-1}+u_{m-1}-u_{m-2}+\dots +u_{n+1}-u_n∥}_{{}^o{W}_2^3}^2\hfill \\ & \le & {∥u_m-u_{m-1}∥}^2+\dots +{∥u_{n+1}-u_n∥}_{{}^o{W}_2^3}^2\hfill \\ & =& \sum _{i=n+1}^m{A}_i^2\to 0,m,n\to \infty ,\hfill \end{array}$$

where ⊥ denotes the orthogonality. Taking into consideration the completeness of ${}^o{W}_2^3[0,1]$, there exists $u\left(\eta \right)\in {}^o{W}_2^3[0,1]$, such that

$$u_n\left(\eta \right)\to u\left(\eta \right)asn\to \infty .$$

Taking limits in Equation (9) gives

$$u\left(\eta \right)=\sum _{i=1}^{\infty }{A}_i{\overline{\psi }}_i\left(\eta \right).$$

Since

$$\left(Lu\right)\left({\eta }_j\right)=\sum _{i=1}^{\infty }{A}_i{⟨L{\overline{\psi }}_i\left(\eta \right),{\phi }_j\left(\eta \right)⟩}_{W_2^1}=\sum _{i=1}^{\infty }{A}_i{⟨{\overline{\psi }}_i\left(\eta \right),L^{*}{\phi }_j\left(\eta \right)⟩}_{{}^o{W}_2^3}=\sum _{i=1}^{\infty }{A}_i{⟨{\overline{\psi }}_i\left(\eta \right),{\overline{\psi }}_j\left(\eta \right)⟩}_{{}^o{W}_2^3},$$

it follows that

$$\sum _{j=1}^n{\beta }_{nj}\left(Lu\right)\left({\eta }_j\right)=\sum _{i=1}^{\infty }{A}_i{⟨{\overline{\psi }}_i\left(\eta \right),\sum _{j=1}^n{\beta }_{nj}{\overline{\psi }}_j\left(\eta \right)⟩}_{{}^o{W}_2^3}=\sum _{i=1}^{\infty }{A}_i{⟨{\overline{\psi }}_i\left(\eta \right),{\overline{\psi }}_n\left(\eta \right)⟩}_{{}^o{W}_2^3}=A_n.$$

If $n=1$, then

$$Lu\left({\eta }_1\right)=M({\eta }_1,u_0\left({\eta }_1\right)).$$

(16)

If $n=2$, then

$${\beta }_{21}\left(Lu\right)\left({\eta }_1\right)+{\beta }_{22}\left(Lu\right)\left({\eta }_2\right)={\beta }_{21}M({\eta }_1,u_0\left({\eta }_1\right))+{\beta }_{22}M({\eta }_2,u_1\left({\eta }_2\right)).$$

(17)

From Equations (16) and (17),

$$\left(Lu\right)\left({\eta }_2\right)=M({\eta }_2,u_1\left({\eta }_2\right)).$$

Additionally, it is simple to show by induction that

$$\left(Lu\right)\left({\eta }_j\right)=M({\eta }_j,u_{j-1}\left({\eta }_j\right)).$$

(18)

Therefore, we get

$$\left(Lu\right)\left(\varsigma \right)=M(\varsigma ,u(\varsigma \left)\right),$$

that is, $u\left(\eta \right)$ is the solution of Equation (8) and

$$u\left(\eta \right)=\sum _{i=1}^{\infty }{A}_i{\overline{\psi }}_i,$$

where $A_i$ are given by Equation (13). This completes the proof. ☐

Theorem 6.

If $u\in {}^o{W}_2^3[0,1],$ then

$${∥u_n-u∥}_{{}^o{W}_2^3}\to 0,n\to \infty .$$

Moreover, a sequence ${∥u_n-u∥}_{{}^o{W}_2^3}$ is monotonically decreasing in n.

Proof. 

We acquire

$${∥u_n-u∥}_{{}^o{W}_2^3}={∥\sum _{i=n+1}^{\infty }\sum _{k=1}^i{\beta }_{ik}f({\eta }_k,u_k,){\widehat{\mathsf{\Psi }}}_i∥}_{{}^o{W}_2^3},$$

by Equations (10) and (11). Thus, we get

$${∥u_n-u∥}_{{}^o{W}_2^3}\to 0,n\to \infty .$$

$$\begin{array}{ccc}\hfill {∥u_n-u∥}_{{}^o{W}_2^3}^2& =& {∥\sum _{i=n+1}^{\infty }\sum _{k=1}^i{\beta }_{ik}f({\eta }_k,u_k){\widehat{\mathsf{\Psi }}}_i∥}_{{}^o{W}_2^3}^2\hfill \\ & =& \sum _{i=n+1}^{\infty }{\left(\sum _{k=1}^i{\beta }_{ik}M({\eta }_k,u_k){\widehat{\mathsf{\Psi }}}_i\right)}^2.\hfill \end{array}$$

Clearly, ${∥u_n-u∥}_{{}^o{W}_2^3}$ is monotonically decreasing in n. ☐

3. 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:

$${\mathbf{y}}^{\prime }=\mathsf{\Psi }(\eta ,\mathbf{y}),\mathbf{y}\in {\mathbb{R}}^k,\eta \in \mathbb{R}.$$

(19)

Then, by using a definition for a unit vector of the orientation of the state vector y for Equation (19), we have:

$$\mathbf{n}:=\frac{\mathbf{y}}{\parallel \mathbf{y}\parallel },$$

(20)

where $\parallel \mathbf{y}\parallel =\sqrt{\mathbf{y}·\mathbf{y}}>0$ is the Euclidean norm of y. Equations (19) and (20) conclude:

$$\dot{\mathbf{n}}:=\frac{\mathsf{\Psi }(\eta ,\mathbf{y})}{\parallel \mathbf{y}\parallel }-\left(\frac{\mathsf{\Psi }(\eta ,\mathbf{y})}{\parallel \mathbf{y}\parallel }·\mathbf{n}\right)\mathbf{n}.$$

(21)

In addition, upon utilizing Equations (19) and (20), we can write:

$$\frac{d}{d\eta }\parallel \mathbf{y}\parallel =\frac{d}{d\eta }\sqrt{\mathbf{y}·\mathbf{y}}=\dot{\mathbf{y}}·\mathbf{n}=\mathsf{\Psi }(\eta ,\mathbf{y})·\mathbf{n}.$$

(22)

From Equations (21) and (22), it follows:

$$\frac{d}{d\eta }\left[\begin{array}{c}\mathbf{y}\\ \parallel \mathbf{y}\parallel \end{array}\right]=\left[\begin{array}{cc}{\mathbf{0}}_{k\times k}& \frac{\mathsf{\Psi }(\eta ,\mathbf{y})}{\parallel \mathbf{y}\parallel }\\ \frac{{\mathsf{\Psi }}^T(\eta ,\mathbf{y})}{\parallel \mathbf{y}\parallel }& 0\end{array}\right]\left[\begin{array}{c}\mathbf{y}\\ \parallel \mathbf{y}\parallel \end{array}\right].$$

(23)

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 $\mathbf{Y}:=({\mathbf{y}}^T{,\parallel \mathbf{y}\parallel )}^T\in {\mathcal{M}}^{k+1}\left(\mathbb{R}\right),$ which describes an inner product on ${\mathbb{R}}^{k+1}$ given by:

$$⟨U,V⟩=U^T\mathsf{\Lambda }V=u_1{v}_1+\cdots +u_k{v}_k-u_{k+1}{v}_{k+1},$$

(24)

where

$$\mathsf{\Lambda }=\left[\begin{array}{cc}{I}_k& {\mathbf{0}}_{k\times 1}\\ {\mathbf{0}}_{1\times k}& -1\end{array}\right],$$

(25)

and

$$U^T={(u_1,\dots ,u_k,u_{k+1})}^T,V^T={(v_1,\dots ,v_k,v_{k+1})}^T.$$

This is the Lorentz inner product on ${\mathbb{R}}^{k+1}$.

Actually, the null vector in ${\mathcal{M}}^{k+1}\left(\mathbb{R}\right)$ lies in the set

$${\mathcal{H}}_{k,1}\left(0\right)=\{\mathbf{X}\in {\mathbb{R}}^{k+1}:\mathbf{X}\ne 0,⟨\mathbf{X},\mathbf{X}⟩=\mathbf{x}·\mathbf{x}-t^2=0\}.$$

It is easy to investigate that, in the Minkowskian structure, the augmented variable $\mathbf{Y}:=({\mathbf{y}}^T{,\parallel \mathbf{y}\parallel )}^T$ is a null vector and, from the Lorentz inner product, satisfy the cone condition:

$$⟨\mathbf{Y},\mathbf{Y}⟩={\mathbf{Y}}^T\mathsf{\Lambda }\mathbf{Y}=0.$$

(26)

Equation (23) can be written in the abstract form:

$${\mathbf{Y}}^{\prime }=\mathsf{\Omega }\mathbf{Y},\mathbf{Y}\in {\mathcal{H}}_{k,1}\left(0\right),$$

(27)

where

$$\mathsf{\Omega }:=\left[\begin{array}{cc}{\mathbf{0}}_{k\times k}& \frac{\mathsf{\Psi }(\eta ,\mathbf{y})}{\parallel \mathbf{y}\parallel }\\ \frac{{\mathsf{\Psi }}^T(\eta ,\mathbf{y})}{\parallel \mathbf{y}\parallel }& 0\end{array}\right].$$

(28)

Definition 3.

Let A be a real square matrix. Then,

$$Sk\_Sy{m}_k\left({\mathcal{M}}^k\left(\mathbb{R}\right)\right)=\{A:A^T\mathsf{\Lambda }+\mathsf{\Lambda }A=0\}$$

is the space of skew symmetric matrices in Minkowskian structure.

We have to note that, in Equation (27), $\mathsf{\Omega }\in Sk\_Sy{m}_{k+1}\left({\mathcal{M}}^{k+1}\left(\mathbb{R}\right)\right)$.

There is a group of real square matrices that is well-known as a global linear group, defined by:

$$G{L}_k\left(\mathbb{R}\right)=\{G\in M_{k,k}:det\left(G\right)\ne 0\}.$$

Moreover, we can consider the closed subgroup

$$O(k,1)=\{G\in G{L}_{k+1}\left(\mathbb{R}\right):G^T\mathsf{\Lambda }G=\mathsf{\Lambda }\}.$$

We have to note that $G\in O(k,1)$ if and only if for all $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^{k+1}$, $⟨G\mathbf{x},G\mathbf{y}⟩=⟨\mathbf{x},\mathbf{y}⟩$. Thus, $O(k,1)$ consist of all the Lorentzian isometries of ${\mathbb{R}}^{k+1}$. Notice that, for $G\in O(k,1),$ we have $det\left(G\right)=\pm 1$.

Another useful subgroup of $O(k,1)$ is

$$S{O}_0(k,1)=\{G\in O(k,1):det\left(G\right)=1\},$$

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 $S{O}_0(k,1)$, then

$$exp:so(k,1)\to S{O}_0(k,1).$$

(29)

Moreover, we know that $so(k,1)=Sk\_Sy{m}_{k+1}\left({\mathcal{M}}^{k+1}\left(\mathbb{R}\right)\right)$ (See reference [34], p. 82). Therefore, in Equation (27), $\mathsf{\Omega }\in so(k,1)$ and the corresponding discretized $G\in S{O}_0(k,1)$, obtained from the exponential map (29), have the following properties:

$$G^T\mathsf{\Lambda }G=\mathsf{\Lambda },det\left(G\right)=1.$$

(30)

Now, we are ready to develop our desired numerical scheme in the form:

$${\mathbf{Y}}_{n+1}=G\left(n\right){\mathbf{Y}}_n,$$

(31)

where ${\mathbf{Y}}_n$ interprets the numerical value of $\mathbf{Y}$ at a discrete $t_n$, and the discretized group element $G\left(n\right)$ is obtained through a Cayley transform as follows:

$$\begin{array}{cc}\hfill G\left(n\right)& ={[I_k-\Delta \eta \mathsf{\Omega }\left(n\right)]}^{-1}[I_k+\Delta \eta \mathsf{\Omega }\left(n\right)]\hfill \\ & =\left[\begin{array}{cc}{I}_k+\frac{2\Delta {\eta }^2{\mathsf{\Psi }}_n{\mathsf{\Psi }}_n^T}{\parallel {\mathbf{y}}_n{\parallel }^2-\Delta {\eta }^2{\parallel {\mathsf{\Psi }}_n\parallel }^2}& \frac{2\Delta \eta \parallel {\mathbf{y}}_n\parallel {\mathsf{\Psi }}_n}{\parallel {\mathbf{y}}_n{\parallel }^2-\Delta {\eta }^2{\parallel {\mathsf{\Psi }}_n\parallel }^2}\\ \frac{2\Delta \eta \parallel {\mathbf{y}}_n\parallel {\mathsf{\Psi }}_n^T}{\parallel {\mathbf{y}}_n{\parallel }^2-\Delta {\eta }^2{\parallel {\mathsf{\Psi }}_n\parallel }^2}& \frac{\parallel {\mathbf{y}}_n{\parallel }^2+\Delta {\eta }^2{\parallel {\mathsf{\Psi }}_n\parallel }^2}{\parallel {\mathbf{y}}_n{\parallel }^2-\Delta {\eta }^2{\parallel {\mathsf{\Psi }}_n\parallel }^2}\end{array}\right].\hfill \end{array}$$

(32)

Substituting Equation (32) into Equation (31) and taking its first row, we get

$${\mathbf{y}}_{n+1}={\mathbf{y}}_n+2\Delta \eta \frac{\parallel {\mathbf{y}}_n{\parallel }^2+\Delta \eta {\mathsf{\Psi }}_n·{\mathbf{y}}_n}{\parallel {\mathbf{y}}_n{\parallel }^2-\Delta {\eta }^2{\parallel {\mathsf{\Psi }}_n\parallel }^2}{\mathsf{\Psi }}_n={\mathbf{y}}_n+{\sigma }_n{\mathsf{\Psi }}_n.$$

(33)

Now, we are ready to use the GPS for solving Equation (1) with initial conditions (2). According to Equation (19), we have:

$$\begin{array}{c}\mathsf{\Psi }(\eta ,\mathbf{y}):=\left(\begin{array}{c}{y}_2\left(\eta \right)\\ -\frac{2}{\eta }{y}_2\left(\eta \right)-y_1^3\left(\eta \right)\end{array}\right),\hfill \\ \mathbf{y}=\left(\begin{array}{c}{y}_1\left(\eta \right)\\ y_2\left(\eta \right)\end{array}\right)=\left(\begin{array}{c}\varsigma \left(\eta \right)\\ {\varsigma }^{\prime }\left(\eta \right)\end{array}\right).\hfill \end{array}$$

Results of this example are obtained by fixing $\Delta \eta ={10}^{-7}$. 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

$${\eta }_i=\frac{i}{m},i=1,2,3,\cdots ,m$$

for our numerical results. Using the reproducing kernel method, we choose 100 points. It is possible to improve the results by increasing the points.

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

Table 1. Comparison of approximate solutions obtained by reproducing kernel method (RKM) and GPS for Equation (1).

η	RKM	GPS
0.1	0.9983360948	0.998335829543602
0.5	0.9598395393	0.959839062543164
1.0	0.8550592570	0.855057541543122
2.0	0.5829639252	0.582850462212463
3.0	0.3592354020	0.359226444051538
4.0	0.2091578370	0.209281565659890
5.0	0.1106289100	0.110819798197543
6.0	0.0435212480	0.043737947433237
7.0	−0.004536310	−0.00431221951973
8.0	−0.040571182	−0.04034773735436
9.0	−0.068517970	−0.06829954400156
10.0	−0.090565560	−0.09035595601487

4. 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 $\eta$ 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.