Numerical Study of Astrophysics Equations by Meshless Collocation Method Based on Compactly Supported Radial Basis Function
Name\corref{cor1
In this paper, we propose compactly supported radial basis functions for solving some well-known classes of astrophysics problems categorized as non-linear singular initial ordinary differential equations on a semi-infinite domain. To increase the convergence rate and to decrease the collocation points, we use the compactly supported radial basis function through the integral operations. Afterwards, some special cases of the equation are presented as test examples to show the reliability of the method. Then we compare the results of this work with some results and show that the new method is efficient and applicable.
CSRBF.tex
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\journal{Int. J. Appl. Comput. Math}

\begin{document}

\begin{frontmatter}

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\title{Numerical Study of Astrophysics Equations by Meshless Collocation Method Based on Compactly Supported Radial Basis Function}

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\author[mymainaddress]{K. Parand\corref{mycorrespondingauthor}}
\cortext[mycorrespondingauthor]{Corresponding author. Member of research group of Scientific Computing. Fax:
+98 2122431653.}
\ead{k\_parand@sbu.ac.ir}

\author[mymainaddress]{M. Hemami}
\ead{mohammadhemami@yahoo.com}

\address[mymainaddress]{Department of Computer Sciences, Shahid Beheshti University, G.C. Tehran 19697-64166, Iran}

\begin{abstract}

In this paper, we propose compactly supported radial basis  functions for solving some well-known classes of
astrophysics problems categorized as non-linear singular initial ordinary differential equations
on a semi-infinite domain. To increase the convergence rate and to decrease the collocation
points, we use the compactly supported radial basis function through the integral operations. Afterwards,
some special cases of the equation are presented as test examples to show the reliability of the
method. Then we compare the results of this work with some results and show that the
new method is efficient and applicable.

\end{abstract}

\begin{keyword}
Lane-Emden type equation\sep Compact support radial basis functions\sep Isothermal Gas sphere\sep White Dwarf equation\sep Non-linear ordinary differential equation
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\end{keyword}

\end{frontmatter}

 %%\linenumbers
 \section{Introduction}
\label{intro}
In recent decades, the so-called meshless methods have been extensively used to find approximate solutions of various types of linear and non-linear equations \citep{MeshfreeFasshhauer} such as differential equations (DEs) and integral equations (IEs). Unlike the other methods which were used to mesh the domain of the problem, meshless  methods don't require a structured grid and only make use of a scattered set of collocation points regardless of the connectivity information between the collocation points.\par
For the last years, the radial basis functions (RBFs) method was known as a powerful tool for the scattered data interpolation problem.
One of the domain-type meshless methods is given in \citep{Kansa} in 1990, which directly collocates RBF, particularly the multiquadric (MQ) to find an approximate solution of linear and non-linear DEs. The RBFs can be compactly and globally supported. The global RBF's are infinitely differentiable and contain a free parameter $c$, called the shape parameter \citep{Islam,Buhman2,Sara}. The interested reader is referred to the recent books and paper by Buhmann \citep{Buhman2,Buhman3} and Wendland \citep{Wendland2} for more basic details about RBFs, compactly and globally supported and the convergence rate of the RBFs. In RBF method  using globally RBFs, if $c$ increases, the system of equations to be solved becomes  ill-conditioned. Cheng et al. \citep{Cheng} showed that when $c$ is very large then the RBFs system error is of exponential convergence. To overcome the problems of globally RBF, alternative ways suggested such as domain decomposition\citep{Li}, LU decomposition\citep{KansaHon}, good matrix pre-conditioners\citep{KansaHon} change the global support RBFs with compactly support RBFs (CSRBFs)\citep{Shen} which are local support and have not any parameter. \par
There are two basic approaches for obtaining basis functions from RBFs, namely direct approach (DRBF) based on differential process \citep{Kansa2} and indirect approach (IRBF) based on an integration process \citep{Duy1,Duy2,Duy3}. Both approaches were tested on the solution of second order DEs and the indirect approach was found to be superior to the direct approach \citep{Duy1}.\par
In this paper, we use the indirect CSRBF (ICSRBF) for finding the solution of Lane-Emden type equations and also Isothermal gas sphere, White  Dwarf equation.\par
This paper is arranged as following:\\
 In Section 2 we describe the Lane-Emden equations. In Section 3  we survey several methods that have been used to solve Lane-Emden type equations. In Section 4, the properties of CSRBF and the way to construct the  ICSRBF method for this type of equations are described. In Section 5 the proposed method is applied to some types of Lane-Emden equations, and a comparison is made with the existing solutions that were reported in other published works. Finally we give a brief conclusion in the last section.
 \section{Lane-Emden type equations}
\label{sec:1}
The Lane-Emden equation describes a variety of phenomena in theoretical physics and astrophysics, including aspects of stellar structure, the thermal history of a spherical cloud of gas, isothermal gas spheres, and thermionic currents.\\\par
Let $P(r)$ denote the total pressure at a distance $r$ from the center of spherical gas cloud. The total pressure is due to the usual gas pressure and a contribution from radiation:
\begin{equation}
P=\frac{1}{3}\varsigma T^4+\frac{RT}{v},\nonumber
\end{equation}
where $\varsigma$ , $T$, $R$  and $v$ are respectively the radiation constant, the absolute temperature, the gas constant, and the specific volume\citep{Ravi}. Let $M(r)$ be the mass within a sphere of radius $r$ and $G$ the constant of gravitation. The equilibrium equations for the configuration are
\begin{equation}
\label{dens}
\frac{dP}{dr}=-\frac{GM(r)}{r^2}, \\
\end{equation}
\begin{equation}
\frac{dM(r)}{dr}=4\pi \rho r^2,\nonumber
\end{equation}
where $\rho$ is the density, at a distance $r$ from the center of a spherical star.\\
Eliminating $M$ yields:
\begin{equation}
\frac{1}{r^2}\frac{d}{dr}\Bigg(\frac{r^2}{\rho}\frac{dP}{dr}\Bigg)=-4\pi G\rho .\nonumber
\end{equation}

Pressure and density $\rho=v^{-1}$ vary with $r$ and $P=K\rho^{1+\frac{1}{m}} $ where $K$ and $m$ are constants.\\
We can insert this relation into Eq. (\ref{dens}) for the hydrostatic equilibrium condition and from this rewrite equation to:
\begin{equation}
\Bigg[\frac{K(m+1)}{4\pi G}\lambda ^{\frac{1}{m}-1}\Bigg]\frac{1}{r^2}\frac{d}{dr}\Bigg(r^2\frac{dy}{dr}\Bigg)=-y^m,\nonumber
\end{equation}
where $\lambda$ representing the central density of the star and $y$ is dimensionless quantity that are both related to $\rho$ through the following relation:
\begin{equation}
\rho=\lambda y^m,\nonumber
\end{equation}
and let:
\begin{eqnarray}
&&r=ax,\nonumber\\
&&a=\Bigg[\frac{K(m+1)}{4\pi G}\lambda^{\frac{1}{m}-1}\Bigg]^{\frac{1}{2}}.\nonumber
\end{eqnarray}
Inserting these relations into our previous relations we obtain the Lane-Emden equation \citep{Dehghan1}:
\begin{equation}
\frac{1}{x^2}\frac{d}{dx}\Bigg(x^2\frac{dy}{dx}\Bigg)=-y^m,\nonumber
\end{equation}
and with simplifying previous equation we have:
\begin{equation}
y''(x)+\frac{2}{x}y'(x)+y^m(x)=0,~~~x>0,\nonumber
\end{equation}
with the boundary conditions:
\begin{equation}
y(0)=1,~~~y'(0)=0.\nonumber
\end{equation}
It has been claimed in the literature that only for $m=0,~1$ and $5$ the solutions of the Lane-Emden equation could be exact. For the other values of $m$, the Lane-Emden equation is to be integrated numerically.
 In this paper, we solve it for $m=1,1.5,2,2.5,3,4$ and $5$.\\
 \section{Methods have been proposed to solve Lane-Emden type equation}
\label{sec:3}
in years, many analytical and numerical methods have been used to solve Lane-Emden equations, the main difficulty arises in the singularity of the equations at $x=0$. Currently, most techniques which were used in handling the Lane-Emden type problems are based on either series solutions or perturbation techniques.\\
Bender et al. \citep{Bender}, proposed a new perturbation technique based on an artificial parameter $\delta$, the method is often called $\delta-$method.\par
Mendelzweig and Tabakin \citep{Mendelzweig} used quasi-linearization approach to solve the standard Lane-Emden equation. This method approximates the solution of a non-linear differential equation by treating the non-linear terms as a perturbation about the linear ones, and unlike perturbation theories is not based on the existence of some small parameter. He showed that the quasi-linearization method gives excellent results when applied to different no-nlinear ordinary differential equations in physics, such as the Blasius, Duffing, Lane-Emden and Thomas-Fermi equations.\par
Shawagfeh \citep{Shawagfeh} applied a non-perturbative approximate analytical solution for the Lane-Emden type equation using the Adomian decomposition method. His solution was in the form of a power series. He used Pad\'e approximants method to accelerate the convergence of the power series.\par
In \citep{Wazwaz1}, Wazwaz employed the Adomian decomposition method with an alternate framework designed to overcome the difficulty of the singular point. It was applied to the differential equations of Lane-Emden type. In future \citep{Wazwaz2} he used the modified decomposition method for solving analytical treatment of non-linear differential equations such as Lane-Emden equations. The modified method accelerates the rapid convergence of the series solutions, dramatically reduces the size of work, and provides the solution by using few iterations only without any need to the so-called Adomian polynomials.\par
Liao \citep{LiaO} provided a reliable, easy-to-use analytical algorithm for Lane-Emden type equations. This algorithm logically contained the well-known Adomian decompositions method. Different from all other analytical techniques, this algorithm itself provides us with a convenient way to adjust convergence regions even without Pad\'e technique.\par
He \citep{He} employed Ritz's method to obtain an analytical solution of the problem. By the semi-inverse method, a variational principle is obtained for the Lane-Emden equation, which he gave much numerical convenience when applied to finite element methods or the Ritz method.\par
Parand et al. \citep{Parand1, Parand2,Parand15} presented some numerical techniques to solve higher ordinary differential equations such as Lane-Emden. Their approach was based on a rational Chebyshev and rational Legendre tau method. They presented the derivative and product operational matrices of rational Chebyshev and rational Legendre functions.\par
These matrices together with the tau method were utilized to reduce the solution of these physical problems to the solutions of systems of algebraic equations. Also, in some paper's \citep{Parand7,Parand8,Parand9,Parand12}, Parand et al. applied Hermite function collocation method (HFC) , Bessel function collocation method and meshless collocation method based on Radial basis function (RBFs) to solving the Lane-Emden type equations.\par
Ramos \citep{Ramos1,Ramos2,Ramos3,Ramos4} solved Lane-Emden equation through different methods. In \citep{Ramos1} he presented linearzation methods for singular initial value problems in second order ordinary differential equation such as Lane-Emden. These methods result in linear constant-coefficient ordinary differential equations which can be integrated analytically, thus they yield piecewise analytical solutions and globally smooth solutions \citep{Ramos2}. Later, he developed piecewise-adaptive decomposition methods for the solution of non-linear ordinary differential equations. Piecewise-decomposition methods provide series solutions in intervals which are subject to continuity conditions at the end points of each interval and their adoption is based on the use of either a fixed number of approximants and a variable step size, a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size. In \citep{Ramos3}, series solutions of the Lane-Emden equation, based on either a volterra integral equation formulation or the expansion of the dependent variable in the original ordinary differential equation are presented and compared with series solutions obtained by means of integral or differential equations based on a transformation of the dependent variables.\par
Yousefi \citep{Yousefi} used integral operator and converted Lane-Emden equations to integral equations and then applied Legendre Wavelet approximations. In this work properties of Legendre wavelet together with the Gaussian integration method were utilized to reduce the integral equations to the solution of algebraic equations. By his method, the equation was formulated on [0, 1].\par
Chowdhury and Hashim \citep{Hashim} obtained analytical solutions of the generalized Emden-Fowler type equations in the second order ordinary differential equations by homotopy-perturbation method (HPM). This method is a coupling of the perturbation method and the homotopy method. The main feature of the HPM \citep{Dehghan2} is that it deforms a difficult problem into a set of problems which are easier to solve. HPM yields solution in convergent series forms with easily computable terms.\par
Aslanov \citep{Aslanov} constructed a recurrence relation for the components of the approximate solution and investigated the convergence conditions for the Emden-Fowler type of equations. He improved the previous results on the convergence radius of the series solution.\par
Dehghan and Shakeri \citep{Dehghan3} investigated Lane-Emden equation using the variational iteration method 
and showed the efficiency and applicability of their procedure for solving this equation. Their technique does not require any discretization, linearization or small perturbations and therefore reduces the volume of computations.\par
Bataineh et al. \citep{Bataineh} obtained analytical solutions of singular initial value problems (IVPs) of the Emden-Fowler type by the homotopy analysis method (HAM). Their solutions contained an auxiliary parameter which provided a convenient way of controlling the convergence region of the series solutions. It was shown that the solutions obtained by the Adomian decomposition method (ADM) and the homotopy-perturbation method (HPM) are only special cases of the HAM solutions.\par
%Marzban et al. \citep{Marzban} used a method based upon hybrid function approxmations. He used properties of hybrid of block-pulse functions and Lagrange interpolating polynomials together with the operational integration matrix for solving non-linear second-order, initial value problems and the Lane-Emden equation.\par
 singh et al. \citep{Singh} used the modified Homotopy analysis method for solving the Lane-Emden-equation and White Dwarf equation.\par
Adibi and Rismani in \citep{Adibi} proposed the approximate solutions of singular IVPs of the Lane-Emden type in second-order ordinary differential equations by improved Legendre-spectral method. The Legendre-Gauss point used as collocation nodes and Lagrange interpolation is employed in the Volterra term.\par
Karimi vanani and Aminataei \citep{Vanani}  provide a numerical method which produces an approximate polynomial solution for solving Lane-Emden equations as singular initial value problem. They are first, used as an integral operator and convert Lane-Emden equations into integral equations, then convert the acquired integral equation into a power series and finally, they transforming the power series into pade series form.\par
Kaur et al. \citep{Kaur}, obtained the Haar wavelet approximate solutions for the generalized Lane-Emden equations. This method was based on the quasi-linearization approximation and replacement of an unknown function through a truncated series of Haar wavelet series of the function.\par
Other researchers try to solve the Lane-Emden type equations with several methods, For example, Y{\i}ld{\i}r{\i}m and \"{O}zi\c{s} \citep{Ozis1,Ozis2} by using HPM and VIM methods, Benko et al. \citep{Benko} by using Nystr\"{o}m method, Iqbal and javad \citep{IQbal} by using Optimal HAM, Boubaker and Van Gorder \citep{Boubaker} by using Boubaker polynomials expansion scheme, Da\c{s}c{\i}o\u{g}lu and Yaslan \citep{Dascioglu} by using Chebyshev collocation method, Y\"{u}zba\c{s}{\i} \citep{Yuzbasi1,Yuzbasi2} by using Bessel matrix and improved Bessel collocation method, Boyd \citep{Boyd} by using Chebyshev spectral method, Bharwy and Alofi \citep{Alofi} by using Jacobi-Gauss collocation method, Pandey et al. \citep{Pandey1,Pandey2} by using Legendre and Brenstein operation matrix, Rismani and monfared \citep{Rismani} by using Modified Legendre spectral method, Nazari-Golshan et al. \citep{Golshan} by using Homotopy perturbation with Fourier transform, Doha et al. \citep{Doha} by using second kind Chebyshev operation matrix algorithm, Carunto and bota \citep{bota} by using Squared reminder minimization method, Mall and Chakaraverty \citep{Mall} by using Chebyshev Neural Network based model, G\"{u}rb\"{u}z and sezer \citep{Gurbuz} by using Laguerre polynomial, Kazemi-Nasab et al. \citep{Kazemi} by using Chebyshev wavelet finite difference method, Hosseini and Abbasbandy \citep{Abbas1} by using combination of spectral method and ADM method and Azarnavid et al. \citep{Abbas2} by using Picard-Reproducing Kernel Hilbert Space Method\par
\section{ ICSRBF method}
\label{sec:4}
\subsection{CSRBF}
\label{ssec:41}
Many problems in science and engineering arise in infinite and semi-infinite domains. Different numerical methods have been proposed for solving problems on various domains such as FEM\citep{DH10,DH11}, FDM\citep{DH9,DH10}, Spectral\citep{DH5,Parand8,Parand13,Parand14,Parand17} methods and meshfree methods\citep{DH3,DH4,dehghan2009meshless,Abbas3}.
The use of the RBF is one of the popular meshfree method for solving the differential equations  \citep{DH1,DH2,radi1,radi2}. For many years the global radial basis functions such as Gaussian, Multi quadric, Thin plate spline, Inverse multiqudric and  others were used  \citep{DH6,DH7,DH8} to solve different DEs and interpolation. These functions are globally supported and generate a system of equations  with ill-condition full matrix. To convert the ill-condition matrix to a well-condition matrix, CSRBFs can be used  instead of global RBFs. CSRBFs can convert the global scheme into a local one with banded matrices, Which makes the RBF method more feasible for solving large-scale problem \citep{Wong}.
\subsection*{Wendland's functions}
\label{ssec:42}
The most popular family of CSRBF are Wendland functions. These functions were introduced by Holger Wendland in 1995 \citep{wendland}. he started with the truncated power function $\phi_l(r)=(1-r)^l_+$ which is strictly positive definite and radial on $\mathbb{R}^s$ for $l\geq \lfloor\frac{s}{2}\rfloor+1$ , and then he walks through dimensions by repeatedly applying the mont\'{e} operator (I).\\
\textbf{Definition 1 \citep{MeshfreeFasshhauer}}~~
with $\phi_l(r)=(1-r)^l_+$ he defines
 \begin{equation}
 \phi_{s,k}=I^k\phi_{\lfloor\frac{s}{2}\rfloor +k+1},
 \end{equation}
 it turns out that the functions $\phi_{s,k}$ are all supported on [0,1].\\
\textbf{Theorem 1 \citep{MeshfreeFasshhauer}}~~
 The functions $\phi_{s,k}$ are strictly positive definite (SPD)  and radial on $\mathbb{R}^s$ and are of the form
 \begin{equation}
\phi _{s,k}(r)=\begin{cases}
p_{s,k}(r)&   r\in [0,1],\nonumber\\
0&       r>1,
\end{cases}
\end{equation}
with a univariate polynomial $p_{s,k}$ of degree $\lfloor \frac{s}{2}\rfloor+3k+1$. Moreover, ،$\phi_{s,k}\in C^{2k}(R)$ are unique up to a constant factor, and the polynomial degree is minimal for given space dimension $s$ and smoothness $2k$ \citep{MeshfreeFasshhauer}.
Wendland gave recursive formulas for the functions $\phi_{s,k}$ for all $s, k$.\\
\textbf{Theorem 2 \citep{MeshfreeFasshhauer}}
~~~~The functions  $\phi_{s,k}$, $k=0,~1,~ 2,~ 3,$ have form
 \begin{eqnarray}
&&\phi_{s,0}=(1-r)^l_+,\nonumber\\
&&\phi_{s,1}\doteq(1-r) _+^{l+1}[(l+1)r+1],\nonumber\\
&&\phi_{s,2}\doteq(1-r)_+^{l+2}[(l^2+4l+3)r^2+(3l+6)r+3],\nonumber\\
&&\phi_{s,3}\doteq(1-r)_+^{l+3}[(l^3+9l^2+23l+15)r^3+(6l^2+36l+45)r^2\nonumber\\
&&+(15l+45)r+15],\nonumber
\end{eqnarray}
where $l=\lfloor \frac{s}{2}\rfloor +k+1$, and the symbol $\doteq$ denotes equality up to a multiplicative positive constant.\\
~~~~~~The case $k=0$ directly follows  from the definition 1. application of the definition 1 for the case $k=1$ yields
\begin{eqnarray}
&&\phi_{s,1}(r)=(I\phi_l)(r)\nonumber
=\int^\infty_r t\phi_l(t)dt  \nonumber\\
&&=\int^\infty _r t(1-t)^l_+dt\nonumber
=\int^1_r t(1-t)^ldt\nonumber\\
&&=\frac{1}{(l+1)(l+2)} (1-r)^{l+1}[(l+1)r+1],\nonumber
\end{eqnarray}
where the compact support of $\phi_l$ reduces the improper integral to a definite integral which can be evaluated using integration by parts. The other two cases are obtained similarly by repeated application of $I$.\citep{MeshfreeFasshhauer}
We showed most of the wendland functions in table \ref{Table. 1.} .
\begin{table}[htbp]
% table caption is above the table
\caption{Wendland's compactly supported radial function for various choices of k and s=3.}
\label{Table. 1.}       % Give a unique label
% For LaTeX tables use
\centering  \begin{tabular}{lll}
\hline\noalign{\smallskip}
 $\phi_{s,k}$ & smoothness & SPD   \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 $\phi_{3,0}(r) =(1-r)^2_+$ & $C^0$ & $\mathbb{R}^3$  \\
 $\phi_{3,1}(r)\doteq(1-r)^4_+(4r+1)$ & $C^2$ & $\mathbb{R}^3$ \\
 $\phi_{3,2}(r)\doteq(1-r)^6_+(35r^2+18r+3)$ & $C^4$ & $\mathbb{R}^3$  \\
 $\phi_{3,3}(r)\doteq(1-r)^8_+(32r^3+25r^2+8r+1)$ & $C^6$ & $\mathbb{R}^3$  \\
 $\phi_{3,4}(r)\doteq(1-r)^{10}_+(429r^4+450r^3+210r^2+50r+5)$ & $C^8$ & $\mathbb{R}^3$ \\
 $\phi_{3,5}(r)\doteq(1-r)^{12}_+(2048r^5+2697r^4+1644r^3+566r^2+108r+9)$ & $C^{10}$ & $\mathbb{R}^3$  \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\subsection{Interpolation by CSRBFs}
\label{ssec:43}
To interpolate or approximate one dimentional function $y(x)$, we can represent it by a CSRBF as

\begin{equation}
y(x)\approx y_n(x)=\sum_{i=1}^N \xi_i \phi_i (x)=\Phi ^T(x)\Xi\nonumber,
\label{systmat}
\end{equation}
where
\begin{eqnarray}
&&\phi_i (x)=\phi (\dfrac{\|x-x_i\|}{r_\omega}),\nonumber\\
&&\Phi ^T(x)=[\phi_1 (x),\phi_2 (x),\cdots ,\phi _N(x)],\nonumber\\
&&\Xi =[\xi_1,\xi_2, \cdots , \xi_N ]^T,\nonumber
\end{eqnarray}
\begin{equation}
\end{equation}
$x$ is the input, $r_\omega$ is the local support domain and $\xi_i$s are the set of coefficients to be determined. By using the local support domain, we mapped the domain of problem to CSRBF local domain. By choosing $N$ interpolate points $(x_j,~ j=1, 2,\cdots,  N)$ in domain:
\begin{equation}
y_j=\sum_{i=1}^N\xi_i\phi_i (x_j)   (j=1, 2, \cdots, N).\nonumber
\end{equation}
To summarize the discussion on the coefficients matrix, we define
\begin{equation}
A\Xi = Y,
\label{system}
\end{equation}
where :
\begin{eqnarray}
&&Y=[y_1, y_2, \cdots,  y_N]^T,\nonumber\\
&&A=[\Phi ^T(x_1), \Phi ^T(x_2), \cdots , \Phi ^T(x_N)]^T\nonumber\\
&&=\begin{pmatrix} 
\phi_1(x_1)&\phi_2(x_1)&\cdots &\phi_N(x_1)\cr \phi_1(x_2)&\phi_2(x_2)&\cdots &\phi_N(x_2)\cr \vdots &\vdots &\ddots &\vdots \cr \phi_1(x_N)&\phi_2(x_N)&\cdots &\phi_N(x_N)\nonumber
\end{pmatrix}.
\end{eqnarray}
Note that $\phi_i(x_j)=\phi(\dfrac{\|x_i-x_j\|}{r_\omega})$, by solving the system (\ref{system}), the unknown coefficients $\xi_i$  will be achieved.
\subsection{ICSRBF method}
\label{ssec:44}
In this paper, we construct the $\phi_i(x)$ in Eq. (\ref{Original}) by using the Wendland function with parameter $k=3$ and $l=5$.\\
At first, we approximate the highest order derivative in the problem by expansion of CSRBFs:\\
\begin{equation}
\dfrac{d^2}{dx^2}y(x)\simeq y^{(2)}_n(x)=\sum_{i=1}^N \xi_i \phi_i(x), 
\label{Original}
\end{equation}
where $\phi_i (x)$ is the CSRBF and $r_\omega$ is the local support domain and  coefficients $\xi_i$ must be determined. Now we define the lower order of derivatives and unknown function $y(x)$ using Gauss-Legendre integration as follows:
\begin{eqnarray}
&&\dfrac{d}{dx}y(x)\simeq\int^x_0 y^{(2)}_n(t)dt-\dfrac{d}{dx}y(x)|_{x=0}=\dfrac{x}{2}\sum_{j=1}^q\sum_{i=1}^N w[j]\xi_i \phi_i (\dfrac{x}{2}\eta[j]+\dfrac{x}{2})=(y^{(1)}_n(x)),\\
&&y(x)\simeq   \int_0^x (y^{(1)}_n(t))dt-y(0)=(y_n(x)),
\end{eqnarray}
where $w[j]$ are weighted coefficients of Gauss-Legendre integration and define as follow:
\begin{equation}
w[j]=\frac{2}{(1-\eta[j]^2)(\frac{d^2}{dx^2}P_q(x))|_{x=\eta[j]}},\nonumber
\end{equation}
$P_q(x)$ is Legendre polynomial of order $q$ and $\eta[j]$ is $j$-th root of $P_{q-1}(x)$.\\
To satisfy the initial conditions of problem, we have:
\begin{eqnarray}
&&(y^{(1)}_n(x))=  \int_0^x y^{(2)}_n(t)dt,\\
&&(y_n(x))=  \int_0^x (y^{(1)}_n(t))dt+1,
\end{eqnarray}
then, we produce the residual function as follows:
\begin{equation}
Res(x)=xy^{(2)}_n(x)+\alpha (y^{(1)}_n(x))+x(y_n(x))^m,
\end{equation}
now, obtain $N$-nodes as follows:
\begin{equation}
\label{formula}
x_j=L(\frac{i}{N})^\gamma ,~~~j=1, 2, \cdots,  N,
\end{equation}
where $L$ is the last collocate node in domain and $\gamma$ is arbitrary parameter. For  Lane-Emden type equations, we selected $\gamma$ between 1.5 and 1.8, because with these values we found the best $\|Res\|_2$ for solution equations.\\
By using ${x_j},  j=1, 2, ..., N $ and residual function, we obtain $N$ equation, so by solving these equations we obtain the unknown coefficients ${\xi_i}_{i=1}^N$.\\
The result of this section can be summarized in the following algorithm for the IVP:
\begin{equation}
F(x,y(x),y'(x),y''(x))=0,~~~~y(0)=a,~~~y'(0)=b.
\end{equation}
\textbf{Algorithm} The algorithm works in the following manner:\\
(1) Choose $N$ center points $\{X_j\}^N_{j=1}$ from domain $[0, \infty)$.\\
(2) Approximate $y''(x)$ as the from $u''_N(x)=\sum_{i=1}^N \xi_i\phi_i(x)$.\\
(3) Obtain $y'(x)$ by using defined integral operation $I_\chi(h(x))=\int_0^x h(t)dt$ in the form $u'_N(x)= \sum_{i=1}^N \xi_i \int _0^x \phi_i(t) dt+b$.\\
(4) Obtain $y(x)$ by using defined integral operation $I_\chi(h(x))=\int_0^x h(t)dt$ in the form $u_N(x)=  \int _0^x u'_N(t) dt+a$.\\
(4) Substitiute $u_N(x)$, $u'_N(x)$ and $u''_N(x)$ into the main problem and create residual function Res(x).\\
(5) Substitiute collocation points $\{X_j\}_{j=1}^{N}$ into the Res(x).\\
(6) Solve the $N$ equations with $N$ unknown coefficients $\{\xi_i\}_{i=1}^N$ and  find the numerical solution.

\begin{equation}
Res(x_j)=0,~~~j=1, 2, \cdots,  N.\nonumber
\end{equation}

\section{Application}
\label{sec:5}
In this section, we apply ICSRBF method to solve the Lane-Emden type equations. In general the Lane-Emden type equation are formulated as follows:
\begin{equation}
y''(x)+\frac{\alpha}{x}y'(x)+p(x)q(y(x))=h(x),~~~\alpha x\geq 0,
\label{lane-eq}
\end{equation}
with initial conditions :
\begin{equation}
y(0)=A ,~~~y'(0)=B,
\end{equation}\\

where $\alpha$, $A$ and $B$ are real constants and $p(x)$, $q(y(x))$ and $h(x)$ are some given function. To apply the collocation method, we construct the residual function by substituting $y_n(x)$  in the Lane-Emden type Eq. (\ref{lane-eq}):
\begin{equation}
Res(x)=y^{(2)}_n(x)+\frac{\alpha}{x}y^{(2)}_n(x)+p(x)q(y_n(x))-h(x)\nonumber.
\end{equation}
The equation for obtaining the coefficient $\xi_is$ arise from equalizing $Res(x)$ to zero at $N$ point (\ref{formula}):
\begin{equation}
Res(x_j)=0,~~~j=0,1,2,...,N.
\end{equation}
By solving this set of equations we obtain a approximate function $y_n(x)$. Note that these $N$ equations generate a set of $N$ nonlinear equations which can be solved by a well-known method such as Newton method for unknown coefficient $\xi_is$. 
%\subsection{The homogeneous Lane-Emden type equations}
%\label{ssec:51}
\subsection{Example 1 (The standard Lane-Emden equation)}
\label{sssec:51}
\begin{table}[htbp]
\caption{Comparison of the first zeros of standard Lane Emden equations between the present method and exact numerical value given by Horedt\citep{Horedt}.}
\label{Table. 2.}   
\centering   \begin{tabular}{lllllll}
\hline\noalign{\smallskip}
 m &  N & $r_\omega$ & L & The present method & Horedt & Error   \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 0 & 40 & 6.5 & 10 & 2.44948974  & 2.44948974 & 0.00e-00 \\
 1 & 40 & 6.5 & 10 & 3.14159265 & 3.14159265 & 0.00e-00 \\
 1.5 & 30 & 2 & 4 & 3.65375388 & 3.65375374 & 1.40e-07 \\
 2 & 20 & 4 & 6 & 4.35287462 & 4.35287460 & 2.00e-08 \\
 2.5 & 20 & 4 & 6 & 5.35527531 & 5.35527546 & 1.50e-07 \\
 3 & 20 & 6.5 & 8 & 6.89684855 & 6.89684862 & 7.00e-08 \\
 4 & 20 & 14 & 16 & 14.9715808 & 14.9715463 & 3.45e-05 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\begin{table}[htbp]
% table caption is above the table
\caption{Comparison of $y(x)$ of standard Lane-Emden equation between present method and exact values given by Horedt\citep{Horedt}, for $m=2$.}
\label{Table. 3.}       % Give a unique label
% For LaTeX tables use
\centering \begin{tabular}{llll}
\hline\noalign{\smallskip}
 x & ICSRBF & Horedt & Error    \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 0.00 & 1.0000000 & 1.0000000 & 0.00e+00 \\
 0.1 & 0.9983350 & 0.9983349 & 0.00e+00 \\
 0.5 & 0.9593527 & 0.9593527 & 0.00e+00 \\
 1.0 & 0.8486541 & 0.8486541 & 0.00e+00 \\
 3.0 & 0.2418240 & 0.2418241 & 1.00e-07 \\
 4.0 & 0.0488398 & 0.0488401 & 3.00e-07 \\
 4.3 & 0.0068107 & 0.0068109 & 2.00e-07 \\
 4.35 & 0.0003660 & 0.0003660 & 0.00e+00 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\begin{table}[htbp]
% table caption is above the table
\caption{Comparison of $y(x)$ of standard Lane-Emden equation between present method and exact values given by Horedt\citep{Horedt}, for $m=3$.}
\label{Table. 4.}       % Give a unique label
% For LaTeX tables use
\centering \begin{tabular}{llll}
\hline\noalign{\smallskip}
 x & ICSRBF & Horedt & Error  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 0.00 & 1.0000000 & 1.0000000 & 0.00e+00 \\
 0.1 & 0.9983358 & 0.9983358 & 0.00e+00 \\
 0.5 & 0.9598391 & 0.9598391 & 0.00e+00 \\
 1.0 & 0.8550575 & 0.8550576 & 1.00e-07 \\
 5.0 & 0.1108198 & 0.1108198 & 0.00e+00 \\
 6.0 & 0.0437379 & 0.0437380 & 1.00e-07 \\
 6.8 & 0.0041677 & 0.0041678 & 1.00e-07 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\begin{table}[htbp]
% table caption is above the table
\caption{Comparison of $y(x)$ of standard Lane-Emden equation between present method and exact values given by Horedt\citep{Horedt}, for $m=4$.}
\label{Table. 5.}       % Give a unique label
% For LaTeX tables use
\centering \begin{tabular}{llll}
\hline\noalign{\smallskip}
 x & ICSRBF  & Horedt & Error   \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 0.00 & 1.0000000 & 1.0000000 & 0.00e+00 \\
 0.1 & 0.9983401 & 0.9983367 & 3.40e-06 \\
 0.2 & 0.9933921 & 0.9933862 & 5.90e-06 \\
 0.5 & 0.9603107 & 0.9603109 & 2.00e-07 \\
 1.0 & 0.8608195 & 0.8608138 & 5.70e-06 \\
 5.0 & 0.2359598 & 0.2359227 & 3.71e-06 \\
 10 & 0.05965343 & 0.0596727 & 1.92e-05 \\
 14 & 0.0083590 & 0.0083305 & 2.85e-05 \\
 14.9 & 0.00058404 & 0.0005764 & 7.64e-06 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\begin{table}[htbp]
% table caption is above the table
\caption{$\|Res\|_2$ of the standard Lane-Emden equations for $m=1.5, 2, 2.5, 3, 4$ respectively.}
\label{Table. 6.}       % Give a unique label
% For LaTeX tables use
\centering  \begin{tabular}{llllll}
%\toprule
\hline\noalign{\smallskip}
n & m=1.5 & m=2 & m=2.5 & m=3 & m=4  \\
%\midrule
\noalign{\smallskip}\hline\noalign{\smallskip}
 5 & 5.516e-03 & 1.434e-02 & 1.338e-03 & 9.996e-02 & 1.869e-00 \\
 10 & 7.250e-05 & 1.646e-04 & 7.148e-06 & 1.851e-04 & 6.436e-03 \\
 15 & 1.050e-06 & 1.379e-05 & 5.919e-09 & 1.699e-08 & 5.362e-03 \\
 20 & 1.731e-08 & 1.869e-06 & 1.268e-12 &  9.786e-10 & 3.049e-05 \\
%\bottomrule
\noalign{\smallskip}\hline
\end{tabular}
\end{table}



\begin{table}[htbp]
% table caption is above the table
\caption{maximum error of the standard Lane-Emden equations for $m=0, 1, 5$ respectively.}
\label{Table. 7.}       % Give a unique label
% For LaTeX tables use
\centering \begin{tabular}{llll}
\hline\noalign{\smallskip}
 n & m=0 & m=1 & m=5    \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 5 & 2.70199e-01 & 7.87765e-02 & 3.54822e-01 \\
 10 & 6.90811e-03 & 2.30291e-03 & 6.61006e-03 \\
 15 & 1.15061e-03 & 2.87683e-04 & 9.08909e-04 \\
 20 & 2.41206e-04 & 4.78916e-05 & 1.71788e-05 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

For  $p(x)=1,~q(y(x))=y^m(x),~\alpha=2,~A=1,~h(x)=0,$ and $B=0$, Eq. (\ref{lane-eq}) is the standard Lane-Emden eqution that was used to model the thermal behaviour of a spherical cloud of gas acting under the mutual attracting of its molecules and subject to the classical laws of thermodynamics\citep{Shawagfeh,Davis}.
\begin{equation}
y''(x)+\frac{2}{x}y'(x)+y^m(x)=0,   x\geq 0,
\label{Gen-LE}
\end{equation}
subject to the boundary conditions\\
\begin{equation}
y(0)=1,~~~y'(0)=0,\nonumber
\end{equation}
where $m\geq  0$ is constant. Substituting $m=0,~1,~5$ into Eq. (\ref{Gen-LE}) leads to the exact solution
\begin{eqnarray*}
&&y(x)=1-\frac{1}{3!}x^2, \\ 
&&y(x)=\frac{\sin(x)}{x}, \\  
&&y(x)=\Bigg(1+\frac{x^2}{3}\Bigg)^{-\frac{1}{2}}, \\ 
\end{eqnarray*}
respectively.\\
In other cases there aren't any analytic exact solutions. Therefore, we apply ICSRBF method to solve the standard Lane-Emden Eq. (\ref{Gen-LE}) for $m=0,~ 1,~ 1.5,~ 2,~ 2.5,~ 3,~ 4$ and $5$. To this way now we can construct the residual functions as follows:
\begin{equation}
Res(x)=xy^{(2)}_n(x)+2 (y^{(1)}_n(x))+x(y_n(x))^m,
\end{equation}
As said before, to obtain the coefficient $\xi_is,~Res(x)$ is equalized to zero at $N$ points by (\ref{formula}) :
\begin{equation}
Res(x_j)=0,~~~j=0, 1, 2, ..., N.\nonumber
\end{equation}
By solving this set of equations, we can find the approximating function $y_n(x)$.\\
Table \ref{Table. 2.} shows the comparison of the first zeros of  standard Lane-Emden equations, from the present method and exact given by Horedt\citep{Horedt} for $m=0,~ 1,~ 1.5,~ 2,~ 2.5,~ 3$ and 4, respectively.\\
Table \ref{Table. 3.}, \ref{Table. 4.}, \ref{Table. 5.} shows the approximation of $y(x)$ for the standard Lane-Emden equation for $m=2,~ 3,~4$ respectively obtained by the method proposed in this paper and those obtained by Horedt\citep{Horedt}.\\
Table \ref{Table. 6.} represents the $\|Res\|_2$ by the present method for $m=1.5,~ 2,~2.5,~ 3,~ 4$ for several points and Table \ref{Table. 7.} represent the maximum error by the present method for $m=0, ~1,~ 5$ for several points to how that the new method has appropriate convergence rate. The result graph of the standard Lane-Emden equation for $m=0, ~1, ~1.5, ~2, ~2.5, ~3, ~4$ and 5 is shown in Fig. \ref{Fig. 1.}.\\
\begin{figure}[htbp!]
\center\includegraphics[scale=0.7]{Fig1.jpg}
\caption{Graph of standard Lane-Emden equation for m=0, 1, 1.5, 2, 2.5, 3, 4 and 5.}
\label{Fig. 1.}
\end{figure}

\subsection{Example 2 (The isothermal gas sphere equation)}
\label{sssec52}
\begin{table}[htbp]
% table caption is above the table
\caption{Comparison of $y(x)$, between present method and series solution given by Wazwaz\citep{Wazwaz1} and HFC method\citep{Parand7} for isothermal gas sphere.}
\label{Table. 8.}       % Give a unique label
% For LaTeX tables use
\centering \begin{tabular}{llll}
\hline\noalign{\smallskip}
 x  & ICSRBF & HFC & Adomain  \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 0.00 & 0.0000000000 & 0.0000000000 & 0.0000000000 \\
 0.1 & -0.0016658338 & -0.0016664188 & -0.0016658339 \\
 0.2 & -0.006533671 & -0.0066539713 & -0.0066533671 \\
 0.5 & -0.0411539573 & -0.0411545150 & -0.0411539568 \\
 1.0 & -0.1588276775 & -0.1588281737 & -0.1588273537 \\
 1.5 & -0.3380194248 & -0.3380198308 & -0.3380131103 \\
 2.0 & -0.5598230044 & -0.5598233120 & -0.5599626601 \\
 2.5 & -0.8063408707 & -0.8063410846 & -0.8100196713 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
For $p(x)=1,~q(y(x))=e^{y(x)},~h(x)=0,~\alpha=2,~A=0$ and $B=0$, Eq.(\ref{lane-eq}) is the isothermal gas sphere equation
\begin{equation}
\label{Eq17}
y''(x)+\frac{2}{x}y'(x)+e^{y(x)}=0,   x\geq 0,
\end{equation}\\
subject to the boundary conditions\\
\begin{equation}
y(0)=0,~~~y'(0)=0,\nonumber\\
\end{equation}
 Wazwaz\citep{Wazwaz1} by using ADM has produced a series solution as follow:
\begin{equation}
\label{Eq18}
y(x)\simeq -\frac{1}{6}x^2+\frac{2}{5.4!}x^4-\frac{8}{21.6!}x^6+\frac{122}{81.8!}x^8-\frac{61.67}{495.10!}x^{10}.
\end{equation}
We applied the ICSRBF method to solve this equation (\ref{Eq17}). We construct the residual function as follows:\\
\begin{equation}
Res(x)=xy^{(2)}_n(x)+2 (y^{(1)}_n(x))+xe^{(y_n(x))},
\end{equation}
To obtain the coefficients $\xi_is,~Res(x)$ is equalized to zero at $N$ point by (\ref{formula}) with $\gamma=1.7$:\\
\begin{equation}
Res(x_j)=0,~~~j=0,1,2,...,N.\nonumber
\end{equation}
By solving this set of equations, we have the approximating function $y(x)$. Table \ref{Table. 8.} shows the comparison of $y(x)$ obtained by method proposed in this paper with $(N=40,~r_\omega=6.5)$, and those obtained by Wazwaz\citep{Wazwaz1} (\ref{Eq18}) and HFC method by Parand et al\citep{Parand7}.\\
In order to compare the present method with those obtained by Wazwaz\citep{Wazwaz1} the resulting graph of Eq.(\ref{Eq16}) is shown in Fig. \ref{Fig. 2.}. The graph of residual for this equation for $N= 5, ~10, ~15, ~20, ~25, ~30, ~35$ and 40 is shown in Fig. \ref{Fig. 3.}.

\begin{figure}
\center\includegraphics[scale=0.7]{Fig2.jpg}
\caption{\scriptsize Graph of Isothermal gas sphere equation in comparison with Wazwaz solution\citep{Wazwaz1}}
\label{Fig. 2.}
\end{figure}

\begin{figure}
\center\includegraphics[scale=0.7]{Fig3.jpg}
\caption{\scriptsize The $\|Res\|_2$  of Isothermal gas sphere equation in for N=5, 10, 15, 20, 25, 30, 35 and 40}
\label{Fig. 3.}
\end{figure}

\subsection{Example 3 (The White Dwarf equation)}
\label{sssec53}
\begin{table}
% table caption is above the table
\caption{Comparison of $y(x)$, between present method and series solution given by Singh\citep{Singh} and Haar method\citep{Kaur} for White Dwarf equation.}
\label{Table. 9.}       % Give a unique label
% For LaTeX tables use
\centering \begin{tabular}{lllll}
\hline\noalign{\smallskip}
 x &$\sigma$& ICSRBF & MHAM & Haar    \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 0.0001 & 0.1 & 0.999999 & 0.999999 & 1 \\
 0.01 &   & 0.999985 & 0.999985 & 0.999986 \\
 0.1 &   & 0.998579 & 0.998578 & 0.998581 \\
 0.2 &   & 0.994340 & 0.994340 & 0.994379 \\
 0.4 &   & 0.977738 & 0.977738 & 0.978345 \\
 0.6 &   & 0.951270 & 0.951263 & 0.953273 \\
 0.7 &   & 0.934823 & 0.934801 & 0.936497 \\
 0.9 &   & 0.896673 & 0.896522 & 0.895488 \\
\noalign{\smallskip}
 0.0001 & 0.2 & 0.999999 & 0.999999 & 1 \\
 0.01 &   & 0.999988 & 0.999988 & 0.999988 \\
 0.1 &   & 0.998809 & 0.998809 & 0.998580 \\
 0.2 &   & 0.995255 & 0.995251 & 0.995296 \\
 0.4 &   & 0.981320 & 0.981320 & 0.981931 \\
 0.6 &   & 0.959049 & 0.959045 & 0.961128 \\
 0.7 &   & 0.945179 & 0.945165 & 0.947116 \\
 0.9 &   & 0.912913 & 0.912812 & 0.912833 \\
\noalign{\smallskip}
 0.0001 & 0.3 & 0.999999 & 0.999999 & 1 \\
 0.01 &   & 0.999990 & 0.999990 & 0.999999 \\
 0.1 &   & 0.999025 & 0.999025 & 0.999028 \\
 0.2 &   & 0.996115 & 0.996115 & 0.996154 \\
 0.4 &   & 0.984690 & 0.984690 & 0.985295 \\
 0.6 &   & 0.966384 & 0.966381 & 0.968464 \\
 0.7 &   & 0.954953 & 0.954944 & 0.956942 \\
 0.9 &   & 0.928280 & 0.928216 & 0.928599 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}

For $p(x)=1,~q(y(x))=(y^2-\sigma)^{\frac{3}{2}},~h(x)=0,~\alpha=2,~A=1$ and $B=0$, Eq.(\ref{lane-eq})\\
will be one of the white Dwarf
 equation that is absorbing to solve
\begin{equation}
\label{Eq15}
y''(x)+\frac{2}{x}y'(x)+(y(x)^2-\sigma)^{\frac{3}{2}}=0,   x\geq 0,
\end{equation}\\
subject to the boundry conditions\\
\begin{equation}
y(0)=1,~~~y'(0)=0,\nonumber
\end{equation}
which was introduced by Chandraskhar \citep{Chandrasekhar} in his study of the gravitational potential of the degenerate White Dwarf stars.
A series solution obtained by Singh\citep{Singh} by using MHAM as follows:
\begin{equation}
\label{Eq16}
y(x)\simeq 1-\frac{1}{6}\varpi^3x^2+\frac{1}{40}\varpi^4x^4-\frac{1}{7!}\varpi^5[5\varpi^2+14]x^6.
\end{equation}
where $\varpi=\sqrt{1-\sigma}$.\\
We applied the ICSRBF method to solve this equation (\ref{Eq15}). We construct the residual function as follows:\\
\begin{equation}
Res(x)=xy^{(2)}_n(x)+2 (y^{(1)}_n(x))+x({(y_n(x))}^2-\sigma^2)^\frac{3}{2},
\end{equation}
To obtain the coefficients $\xi_is,~Res(x)$ is equalized to zero at $N$ point by (\ref{formula}) with $\gamma=1.5$:\\
\begin{equation}
Res(x_j)=0,~~~j=0,1,2,...,N.\nonumber
\end{equation}
By solving this set of equations,we have the approximating function $y(x)$. Table \ref{Table. 9.} shows the comparison of $y(x)$ obtained by method proposed in this paper with $(N=20,~r_\omega=0.5)$, and those obtained by singh\citep{Singh} (\ref{Eq16}) and Haar method by Kaur\citep{Kaur}.\\
 The result graph of the White Dwarf equation for $\sigma$= 0.1, 0.2 and 0.3  is shown in Fig. \ref{Fig. 4.}.\\



\begin{figure}
\center\includegraphics[scale=0.7]{Fig4.jpg}
\caption{\scriptsize Graph of White Dwarf equation for $\sigma$=0.1, 0.2 and 0.3}
\label{Fig. 4.}
\end{figure}

\subsection{Example 4}
\label{sssec54}
\begin{table}
% table caption is above the table
\caption{Comparison of $y(x)$, between present method and series solution given by Wazwaz\citep{Wazwaz1} and HFC method\citep{Parand7} for Example 4.}
\label{Table. 10.}       % Give a unique label
% For LaTeX tables use
\centering \begin{tabular}{llll}
\hline\noalign{\smallskip}
 x & ICSRBF & HFC & Adomain   \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 0.00 & 1.0000000000 & 1.0000000000 & 1.0000000000 \\
 0.1 & 0.9980430038 & 0.9981138095 & 0.9980428414 \\
 0.2 & 0.9921896287 & 0.9922758837 & 0.9921894348 \\
 0.5 & 0.9519612468 & 0.9520376245 & 0.9519611019 \\
 1.0 & 0.8182430031 & 0.8183047481 & 0.8182516669 \\
 1.5 & 0.6254386159 & 0.6254886192 & 0.6258916077 \\
 2.0 & 0.4066218732 & 0.4066479695 & 0.4136691039 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
For $p(x)=1,~q(y(x))=\sinh(y(x)),~h(x)=0,~\alpha=2,~A=1$ and $B=0$, Eq.(\ref{lane-eq}) will be one of the Lane-Emden type equations that is absorbing to solve
\begin{equation}
\label{Eq13}
y''(x)+\frac{2}{x}y'(x)+\sinh(y)=0,   x\geq 0,
\end{equation}\\
subject to the boundry conditions\\
\begin{equation}
y(0)=1,~~~y'(0)=0,\nonumber
\end{equation}

A series solution obtained by Wazwaz\citep{Wazwaz1} by using ADM is:
\begin{eqnarray}\footnotesize
\label{Eq14}
&&y(x)\simeq 1-\frac{(e^2-1)x^2}{12e}+\frac{1}{480}\frac{(e^4-1)x^4}{e^2}-\nonumber\\
&&\frac{1}{30240}\frac{(2e^6+3e^2-3e^4-2)x^6}{e^3}+\nonumber\\
&&\frac{1}{26127360}\frac{(16e^8-104e^6+104e^2-61)x^8}{e^4}.
\end{eqnarray}
We applied the ICSRBF method to solve this equation (\ref{Eq13}). We construct the residual function as follows:\\
\begin{equation}
Res(x)=xy^{(2)}_n(x)+2(y^{(1)}_n(x))+x\sinh({y_n(x)}),
\end{equation}
To obtain the coefficients $\xi_is,~Res(x)$ is equalized to zero at $N$ point by (\ref{formula}) with $\gamma=1.7$:\\
\begin{equation}
Res(x_j)=0,~~~j=0,1,2,...,N.\nonumber
\end{equation}
By solving this set of equations,we have the approximating function $y(x)$. Table \ref{Table. 10.} shows the comparison of $y(x)$ obtained by method proposed in this paper with $(N=20,~r_\omega=1)$,and those obtained by Wazwaz\citep{Wazwaz1} (\ref{Eq14}) and HFC method by Parand et al.\citep{Parand7}.\\
In order to compare the present method with those obtained by Wazwaz\citep{Wazwaz1} the resulting graph of Eq.(\ref{Eq16}) is shown in Fig. \ref{Fig. 5.}. The graph of $\|Res\|_2$ for this equation for $N= 5, ~6, ~7,~ 8,~ 9,~ 10,~ 13,~ 16$ and 20 is shown in Fig. \ref{Fig. 6.}.\\
\begin{figure}
\center\includegraphics[scale=0.7]{Fig5.jpg}
\caption{\scriptsize Graph of equation of example 4 in comparing the present method and Wazwaz solution \citep{Wazwaz1}.}
\label{Fig. 5.}
\end{figure}

\begin{figure}
\center\includegraphics[scale=0.7]{Fig6.jpg}
\caption{\scriptsize The $\|Res\|_2$   of example 4  for $N= 5,~ 6,~ 7,~ 8,~ 9,~ 10,~ 13,~ 16$ and 20 }
\label{Fig. 6.}
\end{figure}

\subsection{Example 5}
\label{sssec55}
\begin{table}
% table caption is above the table
\caption{Comparison of $y(x)$, between present method and series solution given by Wazwaz\citep{Wazwaz1} and HFC method\citep{Parand7} for Example 5.}
\label{Table. 11.}       % Give a unique label
% For LaTeX tables use
\centering \begin{tabular}{llll}
\hline\noalign{\smallskip}
 x & ICSRBF & HFC & Adomian   \\
\noalign{\smallskip}\hline\noalign{\smallskip}
 0.0 & 1.0000000000 & 1.0000000000 & 1.0000000000 \\
 0.1 & 0.9985979436 & 0.9986051425 & 0.9985979358 \\
 0.2 & 0.9943962892 & 0.9944062706 & 0.9943962733 \\
 0.5 & 0.9651778048 & 0.9651881683 & 0.9651777886 \\
 1.0 & 0.8636811302 & 0.8636881301 & 0.8636811027 \\
 1.5 & 0.7050451897 & 0.7050524103 & 0.7050419247 \\
 2.0 & 0.5064632371 & 0.5064687568 & 0.5063720330 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
For $p(x)=1,~q(y(x))=\sin(y(x)),~h(x)=0,~\alpha=2,~A=1$ and $B=0$, Eq.(\ref{lane-eq}) will be one of the Lane-Emden type equations that is absorbing to solve
\begin{equation}
\label{Eq11}
y''(x)+\frac{2}{x}y'(x)+\sin(y)=0,   x\geq 0,
\end{equation}\\
subject to the boundry conditions\\
\begin{equation}
y(0)=1,~~~ y'(0)=0,\nonumber
\end{equation}
A series solution obtained by Wazwaz\citep{Wazwaz1} by using ADM is:
\begin{equation}
\label{Eq12}
y(x)\simeq 1-\frac{1}{6}k_1x^2+\frac{1}{120}k_1k_2x^4+k_1(\frac{1}{3024}k_1^2.-\frac{1}{5040}k_2^2)x^6+
k_1k_2(-\frac{113}{3265920}k^2_1+ \\  \frac{1}{362880}k^2_2)x^8+...
%k_1(\frac{1781}{898128000}k^2_1k^2_2-\frac{1}{399168000}k^4_2-\frac{19}{23950080}k^4_1)x^{10},\nonumber
\end{equation}
where $k_1=\sin(1)$ and $k_2=\cos(1)$.\\
We applied the ICSRBF method to solve this equation (\ref{Eq11}). We construct the residual function as follows:\\
\begin{equation}
Res(x)=xy^{(2)}_n(x)+2(y^{(1)}_n(x))+x\sin({y_n(x)}),
\end{equation}
To obtain the coefficients $\xi_is,~Res(x)$ is equalized to zero at $N$ point by (\ref{formula}) with $\gamma=1.6$:\\
\begin{equation}
Res(x_j)=0,~~~j=0,1,2,...,N.\nonumber
\end{equation}
By solving this set of equations, we have the approximating function $y(x)$. Table \ref{Table. 11.} shows the comparison of $y(x)$ obtained by method proposed in this paper with $(N=20,~r_\omega=2)$, and those obtained by Wazwaz\citep{Wazwaz1} (\ref{Eq12}) and HFC method by Parand et al. \citep{Parand7}.\\
In order to compare the present method with those obtained by Wazwaz\citep{Wazwaz1} the resulting graph of Eq.(\ref{Eq11}) is shown in Fig. \ref{Fig. 7.}.\\ The graph of $\|Res\|_2$ for this equation for $N= 5, ~6, ~7, ~8, ~9, ~10, ~13, ~16$ and 20 is shown in Fig. \ref{Fig. 8.}.\\

\begin{figure}
\center\includegraphics[scale=0.7]{Fig7.jpg}
\caption{\scriptsize Graph of equation of example 5 in comparing the present method and Wazwaz solution \citep{Wazwaz1}.}
\label{Fig. 7.}
\end{figure}

\begin{figure}
\center\includegraphics[scale=0.7]{Fig8.jpg}
\caption{\scriptsize The $\|Res\|_2$  graph of example 5  for $N= 5,~ 6,~ 7,~ 8,~ 9,~ 10,~ 13,~ 16$ and 20 }
\label{Fig. 8.}
\end{figure}

%\subsubsection{Example 6}
%\label{sssec56}
%\begin{table}[htbp]
% table caption is above the table
%\caption{Comparison of $y(x)$, between present method and exact solution for Example 6.}
%\label{Table. 12.}       % Give a unique label
% For LaTeX tables use
%\centering \begin{tabular}{llll}
%\hline\noalign{\smallskip}
% x & ICSRBF & Exact value & Error    \\
%\noalign{\smallskip}\hline\noalign{\smallskip}
% 0.00 & 0.0000000000 & 0.0000000000 & 0.00e+00 \\
% 0.01 & -0.0001999863 & -0.0001999900 & 3.70e-09 \\
% 0.10 & -0.0199006171 & -0.0199006617 & 4.46e-08 \\
%0.50 & -0.4462870236 & -0.4462871026 & 7.90e-08 \\
% 1.00 & -1.3862943522 & -1.3862943611 & 8.90e-09 \\
% 2.00 & -3.2188753419 & -3.2188758249 & 4.83e-07 \\
% 3.00 & -4.6051713735 & -4.6051701860 & 1.18e-06 \\
% 4.00 & -5.6664298279 & -5.6664266881 & 3.13e-06 \\
% 5.00 & -6.5161906344 & -6.5161930760 & 2.44e-06 \\
% 6.00 & -7.2218415102 & -7.2218358253 & 5.68e-06 \\
% 7.00 & -7.8240453867 & -7.8240460109 & 6.24e-07 \\
% 8.00 & -8.3487742634 & -8.3487745398 & 2.76e-07 \\
% 9.00 & -8.8134413237 & -8.8134384945 & 9.29e-06 \\
% 10.00 & -9.2302314578 & -9.2302410337 & 9.57e-06 \\
%\noalign{\smallskip}\hline
%\end{tabular}
%\end{table}
%For $p(x)=1,~q(y(x))=4(2e^{y(x)}+e^{y(x)/2}),h(x)=0,~\alpha=2,~A=0$ and $B=0$, Eq.(\ref{lane-eq}) will be one of the Lane-Emden type equations that is absorbing to solve
%\begin{equation}
%\label{Eq9}
%y''(x)+\frac{2}{x}y'(x)+4(2e^{y(x)}+e^{y(x)/2})=0,   x\geq 0,
%\end{equation}\\
%subject to the boundary conditions\\
%\begin{equation}
%y(0)=0,~~~y'(0)=0,\nonumber
%\end{equation}
%which has the following analytical solution:
%\begin{equation}
%\label{Eq10}
%y(x)=-2\ln(1+x^2).
%\end{equation}
%This type of equation has been solved by \citep{Ozis2,Hashim,Parand7} with VIM, HPM and HFC method respectively. We applied the ICSRBF method to solve this equation (\ref{Eq9}). We construct the residual function as follows:\\
%\begin{equation}
%Res(x)=xy_n(x)+2I_\vartheta(y_n(x))-4x(2e^{I_\vartheta^{(2)}(y_n(x))}+e^{I_\vartheta^{(2)}(y_n(x))/2}),\nonumber
%\end{equation}
%To obtain the coefficients $\xi_is,~Res(x)$ is equalized to zero at $N$ point by (\ref{formula}) with $\gamma=1.6$:\\
%\begin{equation}
%Res(x_j)=0,~~~j=0,1,2,...,N.\nonumber
%\end{equation}
%By solving this set of equations, we have the approximating function $y(x)$. Table \ref{Table. 12.} shows the comparison of $y(x)$ obtained by method proposed in this paper with $(N=40,~r_\omega=8,~D_\Omega\in[0,..,10])$, and analytic solution (\ref{Eq10}). The resulting graph of Eq.(\ref{Eq9}) with present method in comparison to the analytic solution is shown in Fig, \ref{Fig. 9.}. The graph of error for this equation for $N= 5, ~10, ~15, ~20, ~25, ~30, ~35,$ and 40 is shown in Fig. \ref{Fig. 10.}.\\


%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig9.jpg}
%\caption{\scriptsize Graph of equation of example 6 in comparing the present method and analytic solution .}
%\label{Fig. 9.}
%\end{figure}

%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig10.jpg}
%\caption{\scriptsize The error  of example 6  for $N= 5,~ 10,~ 15,~ 20,~ 25,~ 30,~ 35$ and 40 }
%\label{Fig. 10.}
%\end{figure}

%\subsubsection{Example 7}
%\label{sssec57}
%\begin{table}
% table caption is above the table
%\caption{Comparison of $y(x)$, between present method and exact solution for Example 7.}
%\label{Table. 13.}       % Give a unique label
% For LaTeX tables use
%\centering \begin{tabular}{llll}
%\hline\noalign{\smallskip}
 %x & ICSRBF & Exact value & Error     \\
%\noalign{\smallskip}\hline\noalign{\smallskip}
% 0.00 & 1.0000000000 & 1.0000000000 & 0.00e+00 \\
% 0.01 & 1.0001000656 & 1.0001000050 & 6.06e-08 \\
% 0.02 & 1.0004002279 & 1.0004000800 & 1.49e-07 \\
% 0.05 & 1.0025033777 & 1.00250311276 & 2.64e-07 \\
%0.10 & 1.0100504528 & 1.0100501671 & 2.85e-07 \\
% 0.20 & 1.0408111162 & 1.0408107742 & 1.34e-06 \\
% 0.50 & 1.2840263764 & 1.2840254167 & 9.59e-07 \\
% 0.70 & 1.6323161614 & 1.6323162200 & 4.60e-06 \\
% 0.80 & 1.8964775415 & 1.8964808793 & 3.33e-06 \\
% 0.90 & 2.2479129022 & 2.2479079867 & 4.91e-06 \\
% 1.00 & 2.7182915446 & 2.7182818285 & 9.71e-06 \\
%\noalign{\smallskip}\hline
%\end{tabular}
%\end{table}
%For $p(x)=1,~q(y(x))=-6y-4y\ln(y(x)),~h(x)=0,~\alpha=2,~A=1$ and $B=0$, Eq.(\ref{lane-eq}) will be one of the Lane-Emden type equations that is absorbing to solve
%\begin{equation}
%\label{Eq7}
%y''(x)+\frac{2}{x}y'(x)-6y(x)\ln(y(x))=0,   x\geq 0,
%\end{equation}\\
%subject to the boundry conditions\\
%\begin{equation}
%y(0)=1,~~~y'(0)=0,\nonumber
%\end{equation}

%which has the following analytical solution:
%\begin{equation}
%\label{Eq8}
%y(x)=e^{x^2}.
%\end{equation}
%This type of equation has been solved by \citep{Ramos1,Ozis2,Parand7} with linearization, VIM, and HFC method respectively. We applied the ICSRBF method to solve this equation (\ref{Eq7}). We construct the residual function as follows:\\
%\begin{equation}
%Res(x)=xy_n(x)+2I_\vartheta(y_n(x))-6xI_\vartheta^{(2)}(y_n(x))\ln(I_\vartheta^{(2)}(y_n(x))),\nonumber
%\end{equation}
%To obtain the coefficients $\xi_is,~Res(x)$ is equalized to zero at $N$ point by (\ref{formula}) with $\gamma=1.6$:\\
%\begin{equation}
%Res(x_j)=0,~~~j=0,1,2,...,N.\nonumber
%\end{equation}
%By solving this set of equations,we have the approximating function $y(x)$. Table \ref{Table. 13.} shows the comparison of $y(x)$ obtained by method proposed in this paper with $(N=33,~r_\omega=1.3,~D_\Omega\in[0,..,2])$, and analytic solution (\ref{Eq8}). the resulting graph of Eq.(\ref{Eq7}) with present method in comparison to the analytic solution is shown in Fig. \ref{Fig. 11.}. The graph of error for this equation for $N= ~10, ~15, ~20, ~25,$ and 33 is shown in Fig. \ref{Fig. 12.}.\\



%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig11.jpg}
%\caption{\scriptsize Graph of equation of example 7 in comparing the present method and analytic solution .}
%\label{Fig. 11.}
%\end{figure}

%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig12.jpg}
%\caption{\scriptsize The error of example 7  for  $N= 10,~ 15,~ 20,~ 25$ and 33 }
%\label{Fig. 12.}
%\end{figure}

%\subsubsection{Example 8}
%\label{sssec58}
%\begin{table}
% table caption is above the table
%\caption{Comparison of $y(x)$, between present method and exact solution for Example 8.}
%\label{Table. 14.}       % Give a unique label
% For LaTeX tables use
%\centering \begin{tabular}{llll}
%\hline\noalign{\smallskip}
 %x & ICSRBF & Exact value & Error   \\
%\noalign{\smallskip}\hline\noalign{\smallskip}
% 0.00 & 1.0000000000 & 1.0000000000 & 0.00e+00 \\
% 0.01 & 1.0000999020 & 1.0001000050 & 1.03e-07 \\
% 0.02 & 1.0003998549 & 1.0004000800 & 2.25e-07 \\
% 0.05 & 1.0025027913 & 1.0025031276 & 3.36e-07 \\
% 0.10 & 1.0100497997 & 1.0100501671 & 3.67e-07 \\
% 0.20 & 1.0408103322 & 1.0408107742 & 4.42e-07 \\
% 0.50 & 1.2840245722 & 1.2840254167 & 8.44e-07 \\
% 0.70 & 1.6323152724 & 1.6323162200 & 9.47e-07 \\
% 0.80 & 1.8964797383 & 1.8964808793 & 1.14e-06 \\
% 0.90 & 2.2479086659 & 2.2479079867 & 6.79e-07 \\
% 1.00 & 2.7182808699 & 2.7182818285 & 9.58e-07 \\
%\noalign{\smallskip}\hline
%\end{tabular}
%\end{table}

%For $p(x)=-2(2x^2+3),~q(y(x))=y,~h(x)=0,~\alpha=2,~A=1$ and $B=0$, Eq.(\ref{lane-eq}) will be one of the Lane-Emden type equations that is absorbing to solve
%\begin{equation}
%\label{Eq5}
%y''(x)+\frac{2}{x}y'(x)-2(2x^2+3)y(x)=0,   x\geq 0,
%\end{equation}\\
%subject to the boundry conditions\\
%\begin{equation}
%y(0)=1,~~~y'(0)=0,\nonumber
%\end{equation}
%which has the following analytical solution:
%\begin{equation}
%\label{Eq6}
%y(x)=e^{x^2}.
%\end{equation}
%This type of equation has been solved by \citep{Ramos1,Ozis2,Hashim,Parand7} with linearization, VIM, HPM and HFC method respectively. We applied the ICSRBF method to solve this equation (\ref{Eq5}). We construct the residual function as follows:\\
%\begin{equation}
%Res(x)=xy_n(x)+2I_\vartheta(y_n(x))-x2(2x^2+3)I_\vartheta^{(2)}(y_n(x)),\nonumber
%\end{equation}
%To obtain the coefficients $\xi_is,~Res(x)$ is equalized to zero at $N$ point by (\ref{formula}) with $\gamma=1.7$:\\
%\begin{equation}
%Res(x_j)=0,~~~j=0,1,2,...,N.\nonumber
%\end{equation}

%By solving this set of equations,we have the approximating function $y(x)$. Table \ref{Table. 14.} shows the comparison of $y(x)$ obtained by method proposed in this paper with $(N=40,~r_\omega=1.5,~D_\Omega\in[0,..,2])$, and analytic solution (\ref{Eq6}). the resulting graph of Eq.(\ref{Eq5}) with present method in comparison to the analytic solution is shown in Fig. \ref{Fig. 13.}. The graph of error for this equation for $N= ~10, ~15, ~20, ~25, ~30, ~35,$ and 40 is shown in Fig. \ref{Fig. 14.}.\\

%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig13.jpg}
%\caption{\scriptsize Graph of equation of example 8 in comparing the present method and analytic solution.}
%\label{Fig. 13.}
%\end{figure}


%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig14.jpg}
%\caption{\scriptsize The error  of example 8  for   $N= 10,~ 15,~ 20,~ 25,~ 30,~ 35$ and 40 }
%\label{Fig. 14.}
%\end{figure}

%\subsection{ The non-homogeneous Lane-Emden type equations}
%\label{ssec52}
%\subsubsection{Example 9}
%\label{sssec59}
%\begin{table}
% table caption is above the table
%\caption{Comparison of $y(x)$, between present method and exact solution for Example 9.}
%\label{Table. 15.}       % Give a unique label
% For LaTeX tables use
%\centering \begin{tabular}{llll}
%\hline\noalign{\smallskip}
 %x  & ICSRBF & Exact value & Error  \\
%\noalign{\smallskip}\hline\noalign{\smallskip}
 %0.00 & 0.0000000000 & 0.0000000000 & 0.00e+00 \\
 %0.01 & -0.0000009900 & -0.0000009900 & 0.00e+00 \\
 %0.10 & -0.0008999987 & -0.0009000000 & 1.31e-09 \\
 %0.50 & -0.0624999848 & -0.0625000000 & 1.52e-08 \\
% 1.0 & 0.0000000438 & 0.0000000000 & 4.38e-08 \\
% 2.0 & 7.9999999841 & 8.0000000000 & 1.59e-08 \\
% 3.0 & 53.9999998530 & 54.0000000000 & 1.47e-07 \\
% 4.0 & 191.9999978495 & 192.0000000000 & 2.15e-06 \\
 %5.0 & 499.9999978260 & 500.0000000000 & 2.17e-06 \\
 %6.0 & 1079.9999987320 & 1080.0000000000 & 1.26e-06 \\
 %7.0 & 2058.000022713 & 2058.0000000000 & 2.22e-05 \\
 %8.0 & 3583.99997612 & 3584.0000000000 & 2.38e-05 \\
 %9.0 & 5831.9998053810 & 5832.0000000000 & 1.94e-04 \\
 %10.0 & 9000.0078236982 & 9000.0000000000 & 7.80e-03 \\
%\noalign{\smallskip}\hline
%\end{tabular}
%\end{table}
%For $p(x)=x,~ q(y(x))=y(x),~ h(x)=x^5-x^4+44x^2-30x,~ \alpha=8,~ A=0$ and $B=0$, Eq.(\ref{lane-eq}) will be one of the Lane-Emden type equations that is absorbing to solve
%\begin{equation}
%\label{Eq3}
%y''(x)+\frac{8}{x}y'(x)+xy(x)=x^5-x^4+44x^2-30x,   x\geq 0,
%\end{equation}\\
%subject to the boundry conditions\\
%\begin{equation}
%y(0)=0,~~~  y'(0)=0,\nonumber
%\end{equation}
%which has the following analytical solution:
%\begin{equation}
%\label{Eq4}
%y(x)=x^4-x^3.
%\end{equation}
%This type of equation has been solved by \citep{Ramos1,Ozis2,Bataineh,Parand7} with linearization, HPM, HAM and HFC method respectively. We applied the ICSRBF method to solve this equation (\ref{Eq3}). We construct the residual function as follows:\\
%\begin{equation}
%Res(x)=y_n(x)+\frac{8}{x}I_\vartheta(y_n(x))+I_\vartheta^{(2)}(y_n(x))-x^5+x^4-44x^2+30x,\nonumber
%\end{equation}
%To obtain the coefficients $\xi_is,~Res(x)$ is equalized to zero at $N$ point by (\ref{formula}) with $\gamma=1.7$:\\
%\begin{equation}
%Res(x_j)=0,~~~j=0,1,2,...,N.\nonumber
%\end{equation}
%By solving this set of equations, we have the approximating function $y(x)$. Table \ref{Table. 15.} shows the comparison of $y(x)$ obtained by method proposed in this paper with $(N=40,r_\omega=8,~D_\Omega\in[0,..,10])$, and analytic solution (\ref{Eq4}). The resulting graph of Eq.(\ref{Eq3}) with present method in comparison to the analytic solution is shown in Fig. \ref{Fig. 15.}. The graph of error for this equation for $N= 10, ~15, ~20, ~25, ~30,~ 35,$ and 40 is shown in Fig. \ref{Fig. 16.}.\\

%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig15.jpg}
%\caption{\scriptsize Graph of equation of example 9 in comparing the present method and analytic solution.}
%\label{Fig. 15.}
%\end{figure}


%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig16.jpg}
%\caption{\scriptsize The error of example 9  for   $N= 10,~ 15,~ 20,~ 25,~ 30,~ 35,$ and 40 }
%\label{Fig. 16.}
%\end{figure}

%\subsubsection{Example 10}
%\label{sssec510}
%\begin{table}
% table caption is above the table
%\caption{Comparison of $y(x)$, between present method and exact solution for Example 10.}
%\label{Table. 16.}       % Give a unique label
% For LaTeX tables use
%\centering \begin{tabular}{llll}
%\hline\noalign{\smallskip}
 %x & ICSRBF & Exact value & Error  \\
%\noalign{\smallskip}\hline\noalign{\smallskip}
% 0.00 & 0.0000000000 & 0.0000000000 & 0.00e+00 \\
%0.01 & 0.0001010000 & 0.0001010000 & 0.00e+00 \\
% 0.10 & 0.0110000005 & 0.0110000000 & 5.00e-11 \\
% 0.50 & 0.3749999991 & 0.3750000000 & 9.00e-10 \\
% 1.0 & 1.9999999866 & 2.0000000000 & 1.34e-08 \\
% 2.0 & 11.9999999628 & 12.0000000000 & 3.72e-08 \\
% 3.0 & 36.0000000079 & 36.0000000000 & 7.90e-09 \\
% 4.0 & 80.0000001539 & 80.0000000000 & 1.54e-08 \\
% 5.0 & 149.9999999861 & 150.0000000000 & 1.39e-08 \\
% 6.0 & 251.9999995886 & 252.0000000000 & 4.11e-07 \\
 %7.0 & 391.9999990896 & 392.0000000000 & 9.10e-07 \\
 %8.0 & 575.9999989802 & 576.0000000000 & 1.01e-06 \\
% 9.0 & 810.0000002462 & 810.0000000000 & 2.46e-07 \\
% 10.0 & 1100.0000072430 & 1100.0000000000 & 7.24e-06 \\
%\noalign{\smallskip}\hline
%\end{tabular}
%\end{table}
 %For $p(x)=1,~q(y(x))=y(x),~z(x)=6+12x+x^2+x^3,~A=0,~B=0,~$ Eq.(\ref{lane-eq}) will be on of the Lane-Emden type equations that is \\
%\begin{equation}
%\label{Eq1}
%y''(x)+\frac{2}{x}y'(x)+y(x)=6+12x+x^2+x^3, x\geq 0,
%\end{equation}\\
%subject to the boundry conditions\\
%\begin{equation}
%y(0)=0,~~~ y'(0)=0,\nonumber
%\end{equation}
%which has the following analytical solution:\\
%\begin{equation}
%\label{Eq2}
%y(x)=x^2+x^3.
%\end{equation}\\
%This equation has been solved by\citep{Ramos1,Ozis2,Hashim,Parand7} with linearization, VIM, HPM and HFC methods respectively. We applied the ICSRBF method to solve eq.(\ref{Eq1}).\\
%we construct the residual function as follows:\\
%\begin{equation}
%Res(x)=y_n(x)+\frac{2}{x}I_\vartheta(y_n(x))+I_\vartheta^{(2)}(y_n(x))-6-12x-x^2-x^3,\nonumber
%\end{equation}
%As said before, to obtain the coefficients $\xi_is,Res(x)$ is equalized to zero at $N$ point by (\ref{formula}) with $\gamma=1.7$:\\
%\begin{equation}
%Res(x_j)=0,~~~ j=0,1,2,...,N.\nonumber
%\end{equation}
%By solving this set of equations,we have the approximating function $u(x)$. Table \ref{Table. 16.} shows the comparison of $u(x)$ obtained by method proposed in this paper with $(N=35,r_\omega=12,D_\Omega\in[0,..,15])$,and analytic solution (\ref{Eq2}).the resulting graph of Eq. (\ref{Eq1}) with present method in comparison to the analytic solution is shown in Fig. \ref{Fig. 17.}. The graph of error for this equation for N= 10, 15, 20, 25, 30, and 35 is shown in Fig. \ref{Fig. 18.}.\\

%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig17.jpg}
%\caption{\scriptsize Graph of equation of example 10 in comparing the present method and analytic solution.}
%\label{Fig. 17.}
%\end{figure}


%\begin{figure}
%\center\includegraphics[scale=0.5]{Fig18.jpg}
%\caption{\scriptsize The error  of example 10  for   $N= 10,~ 15,~ 20,~ 25,~ 30,$ and 35}
%\label{Fig. 18.}
%\end{figure}

\section{ Conclusion}
\label{sec:6}
 Lane-Emden equations occur in the theory of stellar structure and describe the temperature variation of a spherical gas cloud. White Dwarf equation appears in the gravitational potential of the degenerate White Dwarf stars. Lane-Emden type equations have been considered by many mathematicians as mentioned before\citep{Dehghan3}. The fundamental goal of this paper has been to construct an approximation to the solution of non-linear Lane-Emden type equation in a semi-infinite interval. CSRBFs are proposed to provide an effective but simple way to improve the convergence of the solution by the collocation method. The validity of the method is based on the assumption that it converges by increasing the number of collocation points. A comparison is made among the numerical solution of Horedt\citep{Horedt} and series solutions of Wazwaz\citep{Wazwaz1}, Singh et al.\citep{Singh}, Kaur et al.\citep{Kaur} and the current work. It has been shown that our present work provides an acceptable approach for the Lane-Emden type equations. Also it was confirmed by maximum error and $\|Res\|_2$ figures, this approach has an exponentially convergence rate. Also, in this paper we show that the ICSRBF method for solving ordinary differential equations is simple and it has high accuracy and reliable convergence. \\



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\bibitem[{Abbasbandy et~al.(2014)Abbasbandy, Azarnavid, Hashim, and
  Alsaedi}]{Abbas3}
Abbasbandy, S., Azarnavid, B., Hashim, I., Alsaedi, A., 2014. Approximation of
  backward heat conduction problem using gaussian radial basis functions.
  U.P.B. Sci. Bull., Series A 76, 67--76.

\bibitem[{Adibi and Rismani(2010)}]{Adibi}
Adibi, H., Rismani, A.~M., 2010. On using modified legenre-spectral method for
  solving singular ivps of lane-emden type. Comput. Math. Applic. 60,
  2126--2130.

\bibitem[{Agrawala and O'Regnab(2007)}]{Ravi}
Agrawala, P.~R., O'Regnab, D., 2007. Appl. Math. Lett. 20, 119812005.

\bibitem[{Aky\"{u}z-Da\c{s}c{\i}o\u{g}lu and \c{C}erdik
  Yaslan(2011)}]{Dascioglu}
Aky\"{u}z-Da\c{s}c{\i}o\u{g}lu, A., \c{C}erdik Yaslan, H., 2011. The solution
  of high-order nonlinear ordinary differential equations. Appl. Math. Comput.
  217, 5658--5666.

\bibitem[{Aslanov(2008)}]{Aslanov}
Aslanov, A., 2008. Determination of convergence intervals of the series
  solution of emden-fowler equations using polytropes and isothermal sphere.
  Phys. Lett. A 372, 3555--3561.

\bibitem[{Azarnavid et~al.(2015)Azarnavid, Parvaneh, and Abbasbandy}]{Abbas2}
Azarnavid, B., Parvaneh, F., Abbasbandy, S., 2015. Picard-reproducing kernel
  hilbert space method for solving generalized singular nonlinear lane-emden
  type equations. Math. Model. Anal. 20, 754--767.

\bibitem[{Bataineh et~al.(2009)Bataineh, Noorani, and Hashim}]{Bataineh}
Bataineh, A.~S., Noorani, M. S.~M., Hashim, I., 2009. Homotopy analysis method
  for singular {IVP}s of {E}mden-{F}owler type. Commun. Nonlinear. Sci. Numer.
  Simul. 14, 1121--1131.

\bibitem[{Bender et~al.(1989)Bender, Milton, Pinsky, and Simmons}]{Bender}
Bender, C.~M., Milton, K.~A., Pinsky, S.~S., Simmons, L.~M., 1989. A new
  perturbative approach to nonlinear problems. J. Math. Phys. 30, 1447--1455.

\bibitem[{Benko et~al.(2008)Benko, Biles, Robinson, and Sparker}]{Benko}
Benko, D., Biles, D.~C., Robinson, M.~P., Sparker, J.~S., 2008. Nystr\"{o}m
  method and singular second-order differentil equations. Comput. Math. Applic.
  56, 1975--1980.

\bibitem[{Bharwy and Alofi(2012)}]{Alofi}
Bharwy, A.~H., Alofi, A.~S., 2012. A jacobi-gauss collocation method for
  solving nonlinear lane-emden type equations. Commun. Nonl. Sci. Numer. Simul.
  17, 62--70.

\bibitem[{Boubaker and Van-Gorder(2012)}]{Boubaker}
Boubaker, K., Van-Gorder, R.~A., 2012. Application of the bpes to lane-emden
  equations governing polytropic and isothermal gas sphere. New. Astron. 17,
  565--569.

\bibitem[{Boyd(2011)}]{Boyd}
Boyd, J.~P., 2011. Chebyshev spectral methods and the lane-emden problem.
  Numer. Math. Theor. Meth. Appl. 4 (2), 142--157.

\bibitem[{Bu et~al.(2015)Bu, Ting, Wu., and Yang}]{DH10}
Bu, W., Ting, Y., Wu., Y., Yang, J., 2015. Finite difference/finite element
  method for two-dimensional space and time fractional bloch–torrey
  equations. J. Comput. Phys. 293, 264--279.

\bibitem[{Buhman(2000)}]{Buhman2}
Buhman, M.~D., 2000. Radial basis functions. Acta Numerica, 1--38.

\bibitem[{Buhman(2004)}]{Buhman3}
Buhman, M.~D., 2004. Radial basis functions: Theory and Imolemantations.
  Cambridge University Press, New York.

\bibitem[{Caruntu and Bota(2013)}]{bota}
Caruntu, B., Bota, C., 2013. Approximate polynomial solutions of the nonlinear
  lane-emden type equations arising in astrophysics using the squared reminder
  minimization method. Comput. Phys. Commun. 184, 1643--1648.

\bibitem[{Chandrasekhar(1967)}]{Chandrasekhar}
Chandrasekhar, S., 1967. Introduction to the study of stellar structure. Dover,
  New York.

\bibitem[{Cheng et~al.(2003)Cheng, Golberg, Kansa, and Zammito}]{Cheng}
Cheng, A. H.~D., Golberg, M.~A., Kansa, E.~J., Zammito, Q., 2003. Exponential
  convergence and h-c multiquadric collocation method for partial differential
  equations. Numer. Meth. Part. D. E. 19, 571--594.

\bibitem[{Choi and Kweon(2016)}]{DH11}
Choi, H.~J., Kweon, J.~R., 2016. A finite element method for singular solutions
  of the navier–stokes equations on a non-convex polygon. J. Comput. Appl.
  Math. 292, 342--362.

\bibitem[{Chowdhury and Hashim(2009)}]{Hashim}
Chowdhury, M. S.~H., Hashim, I., 2009. Solution of {E}mden-{F}owler equations
  by {H}omptopy {P}erturbation method. J. Nonlinear Anal. Ser. A Theor. Method
  10, 104--115.

\bibitem[{Davis(1962)}]{Davis}
Davis, H.~T., 1962. Introduction of Nonlinear Differentail and Integral
  Equations. Dover, New York.

\bibitem[{Dehghan and Saadatmandi(2007)}]{Dehghan1}
Dehghan, M., Saadatmandi, A., 2007. The numerical solution of a nonlinear
  system of second-order boundry value problems using sinc-collocation method.
  Math. Comput. Model. 46, 1434--1441.

\bibitem[{Dehghan and Shakeri(2008{\natexlab{a}})}]{Dehghan3}
Dehghan, M., Shakeri, F., 2008{\natexlab{a}}. Approximat solution of a
  differential equation arising in {A}strophysics using the {VIM}. New Astron.
  13, 53--59.

\bibitem[{Dehghan and Shakeri(2008{\natexlab{b}})}]{Dehghan2}
Dehghan, M., Shakeri, F., 2008{\natexlab{b}}. Use of he's homotopy perturbation
  method for solving a partial differential equation arising in modeling of
  flow in porous media. J. Porous Media 11, 765--778.

\bibitem[{Dehghan and Shokri(2008)}]{DH1}
Dehghan, M., Shokri, A., 2008. A numerical method for solution of the
  two-dimensional sine-gordon equation using the radial basis functions. Math.
  Comput. Simul. 79, 700--715.

\bibitem[{Dehghan and Shokri(2009{\natexlab{a}})}]{dehghan2009meshless}
Dehghan, M., Shokri, A., 2009{\natexlab{a}}. A meshless method for numerical
  solution of the one-dimensional wave equation with an integral condition
  using radial basis functions. Numerical Algorithms 52~(3), 461--477.

\bibitem[{Dehghan and Shokri(2009{\natexlab{b}})}]{DH6}
Dehghan, M., Shokri, A., 2009{\natexlab{b}}. A meshless method for numerical
  solution of the one-dimensional wave equation with an integral condition
  using radial basis functions. Numer. Algorithms 52, 461--477.

\bibitem[{Dehghan and Shokri(2009{\natexlab{c}})}]{DH2}
Dehghan, M., Shokri, A., 2009{\natexlab{c}}. Numerical solution of the
  nonlinear klein–gordon equation using radial basis functions. J. Comput.
  Appl. Math. 230, 400--410.

\bibitem[{Doha et~al.(2013)Doha, Abd-Elhameed, and Youssri}]{Doha}
Doha, E.~H., Abd-Elhameed, W.~M., Youssri, Y.~H., 2013. Second kind chebyshev
  operational matrix algorithm for solving differential equations of lane-emden
  type. New. Astro. 23-24, 113--117.

\bibitem[{Fasshauer(2007)}]{MeshfreeFasshhauer}
Fasshauer, G.~E., 2007. Meshfree Approximation Methods With Matlab. World
  Scientific Publishing Co. Pte. Ltd.

\bibitem[{G\"{u}rb\"{u}z and Sezer(2014)}]{Gurbuz}
G\"{u}rb\"{u}z, B., Sezer, M., 2014. Laguerre polynomial approach for solving
  lane-emden type functional differential equations. Appl. Math. Comput. 242,
  255--264.

\bibitem[{He(2003)}]{He}
He, J.~H., 2003. Variational approach to the lane-emden equation. Appl. Math.
  Comput. 143, 539--541.

\bibitem[{Horedt(2004)}]{Horedt}
Horedt, G.~P., 2004. Polytropes: applications in Astrophysics and related
  fileds. Kluwer Academic Publishers, Dordecht.

\bibitem[{Hosseini and Abbasbandy(2015)}]{Abbas1}
Hosseini, S.~G., Abbasbandy, S., 2015. Solution of lane-emden type equations by
  combination of the spectral method and adomian decomposition method. U.P.B.
  Sci. Bull., Series A 2015, 1--10.

\bibitem[{Iqbal and Javad(2011)}]{IQbal}
Iqbal, S., Javad, A., 2011. Application of optimal homotopy asymptotic method
  for the analytic solution of singular lane-emden type equation. Appl. Math.
  Comput. 217, 7753--7761.

\bibitem[{Islam and S.~Haqb(2009)}]{Islam}
Islam, S.~U., S.~Haqb, A.~A., 2009. A meshfree method for the numerical
  solution of the rlw equation. J. Comput. Appl. Math. 223, 997--1012.

\bibitem[{Kansa(1990{\natexlab{a}})}]{Kansa}
Kansa, E.~J., 1990{\natexlab{a}}. Multiquadrics-a scratted dat approximation
  scheme with applications to computational fluid-dynamic-i surface
  approximations ans partial derivative estimates. Comput. Math. Appl. 19,
  127--145.

\bibitem[{Kansa(1990{\natexlab{b}})}]{Kansa2}
Kansa, E.~J., 1990{\natexlab{b}}. Multiquadrics-a scratted dat approximation
  scheme with applications to computational fluid-dynamic-ii solutions to
  parabolic, hyperbolic and elliptic partial differential equations. Comput.
  Math. Appl. 19, 147--161.

\bibitem[{Kansa and Hon(2000)}]{KansaHon}
Kansa, E.~J., Hon, Y.~C., 2000. Circumventing the ill-conditioning problem with
  multiquadric radial basis functions applications to elliptic partial
  defferential equations. Comput. Math. Appl. 39, 123--137.

\bibitem[{Kaur et~al.(2013)Kaur, Mittal, and Mishra}]{Kaur}
Kaur, H., Mittal, R.~C., Mishra, V., 2013. Haar wavelet approximate solutions
  for the generalized lane-emden equations arising in astrophysics. Comput.
  Phys. Commun. 184, 2169--2177.

\bibitem[{Kazemi-Nasab et~al.(2015)Kazemi-Nasab, K{\i}l{\i}\c{c}man, Atabakan,
  and Leong}]{Kazemi}
Kazemi-Nasab, A., K{\i}l{\i}\c{c}man, A., Atabakan, Z.~P., Leong, W.~J., 2015.
  A numerical approach for solving singular nonlinear lane-emden type equations
  arising in astrophysics. New. Astro. 34, 178--186.

\bibitem[{Lee and Hon(2004)}]{Li}
Lee, J.~C., Hon, Y.~C., 2004. Domain decomposition for radial basis meshless
  method. Numer. Methods Partial Differential Eq. 20, 450--462.

\bibitem[{Liao(2003)}]{LiaO}
Liao, S., 2003. A new analytic algorithm of lane-emden type equation. Appl.
  Math. Comput. 142, 1--16.

\bibitem[{Mai-Duy(2005)}]{Duy2}
Mai-Duy, N., 2005. Solving high order ordinary differential equations with
  radial basis function networks. Int. J. Numer. Meth. Eng. 62, 824--852.

\bibitem[{Mai-Duy and Tran-Cong(2001{\natexlab{a}})}]{Duy1}
Mai-Duy, N., Tran-Cong, T., 2001{\natexlab{a}}. Numerical solution of
  differential equations using multiquadric radial basis function networks.
  Neural Netw 14, 185--199.

\bibitem[{Mai-Duy and Tran-Cong(2001{\natexlab{b}})}]{Duy3}
Mai-Duy, N., Tran-Cong, T., 2001{\natexlab{b}}. Numerical solution of
  navier-stocks equations using multiquadric radial basis function networks.
  Int. J. Numer. Meth. Fl. 37, 65--86.

\bibitem[{Mall and Chakraverty(2014)}]{Mall}
Mall, S., Chakraverty, S., 2014. Chebyshev neural network based model for
  solving lane-emden type equations. Appl. Math. Comput. 247, 100--114.

\bibitem[{Mendelzweig and Tabakin(2001)}]{Mendelzweig}
Mendelzweig, V.~B., Tabakin, F., 2001. Quasilinearization approach to nonlinear
  problems in physics with application to nonlinear odes. Comput. Phys. Commun.
  141, 268--281.

\bibitem[{Nazari-Golshan et~al.(2013)Nazari-Golshan, Nourazar, Ghafoori-Fard,
  Y{\i}ld{\i}r{\i}m, and Campo}]{Golshan}
Nazari-Golshan, A., Nourazar, S.~S., Ghafoori-Fard, H., Y{\i}ld{\i}r{\i}m, A.,
  Campo, A., 2013. A modified homotopy perturbation method couled with the
  fourier transform for nonlinear and singular lane-emden equations. Appl.
  Math. Lett. 26, 1018--1025.

\bibitem[{Noye and Dehghan(1999)}]{DH9}
Noye, B.~J., Dehghan, M., 1999. New explicit finite difference schemes for
  two-dimensional diffusion subject to specification of mass. Numer. Meth. Par.
  Diff. Eq. 15, 521--534.

\bibitem[{Pandey and Kumar(2012)}]{Pandey2}
Pandey, R.~K., Kumar, N., 2012. Solution of lane-emden type equations using
  bernstein operational matrix of differentation. New. Astro. 17, 303--308.

\bibitem[{Pandey et~al.(2012)Pandey, Kumar, Bhardwaj, and Dutta}]{Pandey1}
Pandey, R.~K., Kumar, N., Bhardwaj, A., Dutta, G., 2012. Solution of lane-emden
  type equations using legendre operational matrix of differentation. Appl.
  Math. Comput. 218, 7629--7637.

\bibitem[{Parand et~al.(2011{\natexlab{a}})Parand, Abbasbandy, Kazem, and
  Rad}]{Parand9}
Parand, K., Abbasbandy, S., Kazem, S., Rad, J., 2011{\natexlab{a}}. A novel
  application of radial basis functions for solving a model of first-order
  integro-ordinary differential equation. Commun. Nonlinear Sci. Numer. Simul.
  16, 4250--4258.

\bibitem[{Parand et~al.(2011{\natexlab{b}})Parand, Abbasbandy, Kazem, and
  Rezaei}]{Parand12}
Parand, K., Abbasbandy, S., Kazem, S., Rezaei, A., 2011{\natexlab{b}}. An
  improved numerical method for a class of astrophysics problems based on
  radial basis functions. Physica Scripta 83~(1), 015--011.

\bibitem[{Parand et~al.(2013{\natexlab{a}})Parand, Dehghan, and
  Pirkhedri}]{Parand14}
Parand, K., Dehghan, M., Pirkhedri, A., 2013{\natexlab{a}}. The
  sinc-collocation method for solving the thomas--fermi equation. Journal of
  Computational and Applied Mathematics 237~(1), 244--252.

\bibitem[{Parand et~al.(2010{\natexlab{a}})Parand, Dehghan, Rezaei, and
  Ghaderi}]{Parand7}
Parand, K., Dehghan, M., Rezaei, A.~R., Ghaderi, S.~M., 2010{\natexlab{a}}. An
  approximation algorthim for the solution of the nonlinear {L}ane-{E}mden type
  equation arising in {A}storphisics using {H}ermit functions {C}ollocation
  method. Comput. Phys. Commun. 181, 1096--1108.

\bibitem[{Parand et~al.(2010{\natexlab{b}})Parand, Dehghan, and
  Taghavi}]{Parand13}
Parand, K., Dehghan, M., Taghavi, A., 2010{\natexlab{b}}. Modified generalized
  laguerre function tau method for solving laminar viscous flow: The blasius
  equation. International Journal of Numerical Methods for Heat \& Fluid Flow
  20~(7), 728--743.

\bibitem[{Parand et~al.(2016)Parand, Hossayni, and Rad}]{Parand17}
Parand, K., Hossayni, S.~A., Rad, J., 2016. An operation matrix method based on
  bernstein polynomials for riccati differential equation and volterra
  population model. Applied Mathematical Modelling.

\bibitem[{Parand and Khaleqi(2016)}]{Parand15}
Parand, K., Khaleqi, S., 2016. Rational chebyshev of second kind collocation
  method for solving a class of astrophysics problems. The European Physical
  Journal - Plus.

\bibitem[{Parand et~al.(2013{\natexlab{b}})Parand, Nikarya, and Rad}]{Parand8}
Parand, K., Nikarya, M., Rad, J.~A., 2013{\natexlab{b}}. Solving non-linear
  lane-emden type equations using bessel orthogonal functions collocation
  method. Celest. Mech. Dyn. Astr. 116, 97--107.

\bibitem[{Parand and Rad(2013)}]{DH7}
Parand, K., Rad, J.~A., 2013. Kansa method for the solution of a parabolic
  equation with an unknown spacewise-dependent coefficient subject to an extra
  measurement. Comp. Phys. Commun. 184, 582--595.

\bibitem[{Parand and Razzaghi(2004{\natexlab{a}})}]{Parand2}
Parand, K., Razzaghi, M., 2004{\natexlab{a}}. Rational chebyshev tau method for
  solving higher-order ordinary differential equations. Int. J. Comput. Math.
  81 (1), 73--80.

\bibitem[{Parand and Razzaghi(2004{\natexlab{b}})}]{Parand1}
Parand, K., Razzaghi, M., 2004{\natexlab{b}}. Rational legendre approximation
  for solving some physical problems on semi-infinite intervals. Phys. Scr. 69,
  353--357.

\bibitem[{Rad et~al.(2012)Rad, Kazem., and Parand}]{DH4}
Rad, J.~A., Kazem., S., Parand, K., 2012. A numerical solution of the nonlinear
  controlled duffing oscillator by radial basis function. Comput. Math. Appl.
  64, 2049--2065.

\bibitem[{Rad et~al.(2014)Rad, Kazem, Shaban, Parand, and Yildirim}]{DH5}
Rad, J.~A., Kazem, S., Shaban, M., Parand, K., Yildirim, A., 2014. Numerical
  solution of fractional differential equations with a tau method based on
  legendre and bernstein polynomials. Math. Meth. Appl. Sci. 37, 329--342.

\bibitem[{Rad et~al.(2015{\natexlab{a}})Rad, Parand, and Abbasbandy}]{radi1}
Rad, J.~A., Parand, K., Abbasbandy, S., 2015{\natexlab{a}}. Local weak form
  meshless techniques based on the radial point interpolation (rpi) method and
  local boundary integral equation (lbie) method to evaluate european and
  american options. Communications in Nonlinear Science and Numerical
  Simulation 22~(1), 1178--1200.

\bibitem[{Rad et~al.(2015{\natexlab{b}})Rad, Parand, and Ballestra}]{radi2}
Rad, J.~A., Parand, K., Ballestra, L.~V., 2015{\natexlab{b}}. Pricing european
  and american options by radial basis point interpolation. Applied Mathematics
  and Computation 251, 363--377.

\bibitem[{Ramos(2003)}]{Ramos4}
Ramos, J.~I., 2003. Linearization methods in classical and quantum mechanics.
  Comput. Phys. Commun. 153 (2), 199--208.

\bibitem[{Ramos(2005)}]{Ramos1}
Ramos, J.~I., 2005. Linearization techniques for singular initial-value
  problems of ordinary differential equations. J. Appl. Math. Comput. 161,
  525--542.

\bibitem[{Ramos(2007)}]{Ramos2}
Ramos, J.~I., 2007. Piecewise-adaptive decomposition method. Chaos. Solit.
  Fract. 40 (4), 1623--1636.

\bibitem[{Ramos(2008)}]{Ramos3}
Ramos, J.~I., 2008. Series approach to the lane-emden equation and comparison
  with the homotopy perturbation method. Chaos. Solit. Fract. 38 (2), 400--408.

\bibitem[{Rashidi et~al.(2014)Rashidi, Adibi, Rad, and Parand}]{DH8}
Rashidi, K., Adibi, H., Rad, J.~A., Parand, K., 2014. Application of meshfree
  methods for solving the inverse one-dimensional stefan problem,. Eng. Anal.
  Bound. Elem. 40, 1--21.

\bibitem[{Rismani and Monfared(2012)}]{Rismani}
Rismani, A.~M., Monfared, H., 2012. Numerical solution of singular inps of
  lane-emden type using a modified legendre-spectral method. Appl. Math. Model.
  36, 4830--4836.

\bibitem[{Sara(2005)}]{Sara}
Sara, S.~A., 2005. Adaptive radial basis function method for time dependent
  partial differential equations. Appl. Numer. Math. 54, 79--94.

\bibitem[{Shawagfeh(1993)}]{Shawagfeh}
Shawagfeh, N.~T., 1993. Nonperturbative approximate solution of {L}ane-{E}mden
  equation. J. Math. Phys. 34, 4364--4369.

\bibitem[{Shen(2012)}]{Shen}
Shen, Q., 2012. A meshless scaling iterative algorithm based on campactly
  supported radial basis functions for the numerical solution of
  lane-emden-fowler equation. Numer. Methods Partial Differential Eq. 28,
  554--572.

\bibitem[{Shokri and Dehghan(2010)}]{DH3}
Shokri, A., Dehghan, M., 2010. A not-a-knot meshless method using radial basis
  functions and predictor–corrector scheme to the numerical solution of
  improved boussinesq equation. Comput. Phys. Commun. 181, 1990--2000.

\bibitem[{Singh et~al.(2009)Singh, Pandey, and Singh}]{Singh}
Singh, O.~P., Pandey, R.~K., Singh, V.~K., 2009. An analytic algorithm of
  {L}ane-{E}mden type equations arising in {A}stropysics using modified
  {H}omotopy analysis method. Comput. Phys. Commun. 180, 1116--1124.

\bibitem[{Vanani and Aminataei(2010)}]{Vanani}
Vanani, S.~K., Aminataei, A., 2010. On the numerical solution of differential
  equations of lane-emden type. Comput. Math. Applic. 59, 2815--2820.

\bibitem[{Wazwaz(2001)}]{Wazwaz1}
Wazwaz, A.~M., 2001. A new algorithm for solving differential equations of
  lane-emden type. Appl. Math. Comput. 118, 287--310.

\bibitem[{Wazwaz(2006)}]{Wazwaz2}
Wazwaz, A.~M., 2006. The modified decomposition method for analytic treatment
  of differential equations. Appl. Math. Comput. 173, 165--176.

\bibitem[{Wendland(1995)}]{wendland}
Wendland, H., 1995. Piecewise polynomial, positive definite and compactly
  supported radial functions of minimal degree. Adv. Comput. Math. 4, 389--396.

\bibitem[{Wendland(2005)}]{Wendland2}
Wendland, H., 2005. Scattered Data Approximation. Cambridge University Press,
  New York.

\bibitem[{Wong et~al.(2002)Wong, Hon, and Golberg}]{Wong}
Wong, S.~M., Hon, Y.~C., Golberg, M.~A., 2002. Compactly supported radial basis
  function for shallow water equations. Appl. Math. Comput. 127, 79--101.

\bibitem[{Y{\i}ld{\i}r{\i}m and \"{O}zi\c{s}(2007)}]{Ozis1}
Y{\i}ld{\i}r{\i}m, A., \"{O}zi\c{s}, T., 2007. Solutions of singular ivps of
  lane-emden type homotopy perturabation method. Phys. Lett. A 369, 70--76.

\bibitem[{Y{\i}ld{\i}r{\i}m and \"{O}zi\c{s}(2009)}]{Ozis2}
Y{\i}ld{\i}r{\i}m, A., \"{O}zi\c{s}, T., 2009. Solutions of singular {IVP}s of
  {L}ane-{E}mden type equations by the {VIM} method. J. Nonlinear Anal. Ser. A
  Theor. Method 70, 2480--2484.

\bibitem[{Yousefi(2006)}]{Yousefi}
Yousefi, S.~A., 2006. {L}egendre {W}avlate method for solving differential
  equations of {L}ane-{E}mden type. Appl. Math. Comput 181, 1417--1422.

\bibitem[{Y\"{u}zba\c{s}{\i}(2011)}]{Yuzbasi1}
Y\"{u}zba\c{s}{\i}, S., 2011. A numerical approach for solving the high-order
  linear singular differential-difference equations. Comput. Math. Applic. 62,
  2289--2303.

\bibitem[{Y\"{u}zba\c{s}{\i} and Sezer(2013)}]{Yuzbasi2}
Y\"{u}zba\c{s}{\i}, S., Sezer, M., 2013. An improved bessel collocation method
  with a residual error function to solve a class of lane-emden differential
  equations. Math. Comput. Model. 57, 1298--1311.

\end{thebibliography}
