\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 28, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/28\hfil Volterra integro-differential equations] {Numerical solutions for Volterra integro-differential forms of Lane-Emden equations of first and second kind using Legendre multi-wavelets} \author[P. K. Sahu, S. Saha Ray \hfil EJDE-2015/28\hfilneg] {Prakash Kumar Sahu, Santanu Saha Ray} \address{Prakash Kumar Sahu \newline National Institute of Technology Rourkela, Department of Mathematics, Rourkela, Odisha-769008, India} \email{prakash.2901@gmail.com} \address{Santanu Saha Ray \newline National Institute of Technology Rourkela, Department of Mathematics, Rourkela, Odisha-769008, India} \email{santanusaharay@yahoo.com} \thanks{Submitted June 23, 2014 Published January 29, 2015.} \subjclass[2000]{45D05, 45J05} \keywords{Legendre multi-wavelet; Volterra integral equation; \hfill\break\indent integro-differential equation; Lane-Emden equation} \begin{abstract} A numerical method based on Legendre multi-wavelets is applied for solving Lane-Emden equations which form Volterra integro-differential equations. The Lane-Emden equations are converted to Volterra integro-differential equations and then are solved by the Legendre multi-wavelet method. The properties of Legendre multi-wavelets are first presented. The properties of Legendre multi-wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved by any numerical method. Illustrative examples are discussed to show the validity and applicability of the present method. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In this article, we discuss a Lane-Emden equation of first kind \cite{3,1,2,4,5} of the form \begin{equation}\label{1.1} y''+\frac{\kappa}{x}y'+y^{m}=0,\quad y(0)=1,\quad y'(0)=0,\quad \kappa >1 \end{equation} and Lane-Emden equation of second kind \cite{6,8,7} of the form \begin{equation}\label{1.2} y''+\frac{\kappa}{x}y'+e^{y}=0,\quad y(0)=1,\quad y'(0)=0,\quad \kappa \geq 1 \end{equation} where $\kappa $ is the shape factor. Equation \eqref{1.1} is a basic equation in the theory of stellar structure \cite{Chandrasekhar:1967}. It is used in astrophysics for computing the structure of interiors of polytropic stars. This equation describes the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of thermodynamics \cite{2}. The Lane-Emden equation of the first kind appears also in other contexts such as radiative cooling, self-gravitating gas clouds, mean-field treatment of a phase transition in critical adsorption, and modeling of clusters of galaxies. Equation \eqref{1.2} is the Lane-Emden equation of the second kind that models the non-dimensional density distribution $y(x)$ in an isothermal gas sphere \cite{9}. In the study of stellar structures one considers the star as a gaseous sphere in thermodynamic and hydrostatic equilibrium for a certain equation of state \cite{10}. The well-known Lane-Emden equation has been used to model several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres, the theory of thermionic currents, and in the modeling of clusters of galaxies. A substantial amount of work has been done on these types of problems for various structures. The singular behavior that occurs at $x=0$ is the main difficulty of eqs. \eqref{1.1}--\eqref{1.2}. In this article, our main work is to establish Volterra integro-differential equation equivalent to the Lane-Emden equation of first and second kind. The newly established Volterra integro-differential equation will be solved by using the Legendre multi-wavelet method (LMWM). Legendre multi-wavelet method has been applied to solve the integral equations and integro-differential equations of different forms \cite{11,12,13,14,15}. The Legendre multi-wavelet method converts the Volterra integro-differential equation to a system of algebraic equations and that algebraic equations system again can be solved by any of the usual numerical methods. \section{Volterra integro-differential form of the Lane-Emden equation} \label{sec:2} Let us consider the Lane-Emden equation \begin{equation}\label{2.1} y''(x)+\frac{\kappa}{x}y'(x)+f(y)=0,\quad y(0)=\alpha ,\quad y'(0)=0,\quad \kappa \geq 1. \end{equation} Multiplying by $x^{\kappa}$ and integrating on $[0,x]$ we have \begin{equation}\label{2.4} y'(x)=-\int_{0}^{x}\big( \frac{t^{\kappa }}{x^{\kappa }}\big) f(y(t))dt \quad \kappa \geq 1, \quad y(0)=\alpha. \end{equation} Integrating again on $[0,x]$, \eqref{2.1} becomes \begin{equation}\label{2.2} y(x)=\alpha -\frac{1}{\kappa -1}\int_{0}^{x}t \Big( 1-\frac{t^{\kappa -1}}{x^{\kappa -1}}\Big) f(y(t))dt. \end{equation} \section{Properties of Legendre multi-wavelets} \label{sec:3} Wavelets constitute a family of functions constructed from dilation and translation of a single function called mother wavelet. When the dilation parameter $a$ and the translation parameter $b$ vary continuously, we have the following family of continuous wavelets as \begin{equation}\label{3.1} \Psi_{a,b}(x)=\vert a\vert^{-1/2}\Psi \big( \frac{x-b}{a}\big) ,\quad a,b\in \mathbb{R},\; a\neq 0 \end{equation} If we restrict the parameters $a$ and $b$ to discrete values as $ a=a_{0}^{-k} $, $ b=nb_{0}a_{0}^{-k} $, $ a_{0}>1 $, $ b_{0}>0 $ and $n$, and $k$ are positive integers, we have the following family of discrete wavelets: $$ \psi _{k,n}(x)=\vert a_{0}\vert^{-k/2}\psi (a_{0}^{k}x-nb_{0}), $$ where $\psi _{k,n}(x)$ forms a wavelet basis for $L^{2}(\mathbb {R})$. In particular, when $a_{0}=2$ and $b_{0}=1$, then $\psi _{k,n}(x)$ form an orthonormal basis. Legendre multi-wavelets $\psi _{n,m}(x)=\psi (k,n,m,x)$ have four arguments. $n=0,1,2,\dots ,2^{k}-1$, $k\in \mathbb{Z^{+}}$, where $m$ is the order of Legendre polynomials and $x$ is normalized time. These functions are defined on $[0,T)$ as (see \cite{16}) \begin{equation}\label{3.2} \psi _{n,m}(x)=\begin{cases} \sqrt{2m+1}\big( \frac{2^{k/2}}{\sqrt{T}}\big) P_{m}\big( \frac{2^{k}x}{T}-n\big) , & \frac{nT}{2^{k}}\leq x< \frac{(n+1)T}{2^{k}} \\ 0, & \text{otherwise}, \end{cases} \end{equation} where $m=0,1,\dots ,M-1$ and $n=0,1,2,\dots ,2^{k}-1$. The dilation parameter is $a=2^{-k}T$ and translation parameter is $b=n2^{-k}T$. Here $P_{m}(x)$ are the well-known shifted Legendre polynomials of order $m$, which are defined on the interval $[0,1]$, and can be determined with the aid of the following recurrence formulae \begin{gather*} P_{0}(x)=1,\quad P_{1}(x)=2x-1,\\ P_{m+1}(x)=\big(\frac{2m+1}{m+1}\big)(2x-1)P_{m}(x) -\big(\frac{m}{m+1}\big)P_{m-1}(x),\quad m=1,2,3,\dots \end{gather*} \section{Function approximation by Legendre multi-wavelets} \label{sec:4} A function $f(x)$ defined over $[0,T)$ can be expressed by the Legendre multi-wavelets as \begin{equation}\label{4.1} f(x)=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}c_{n,m}\psi _{n,m}(x) \end{equation} where $c_{n,m}=\langle f(x),\psi _{n,m}(x)\rangle$, in which $\langle \cdot ,\cdot \rangle$ denotes the inner product. If the infinite series in \eqref{4.1} is truncated, then \eqref{4.1} can be written as \begin{equation}\label{4.2} f(x) \cong \sum_{n=0}^{2^{k}-1}\sum_{m=0}^{M}c_{n,m}\psi _{n,m}(x)=C^{T}\Psi (x) \end{equation} where $C$ and $\Psi (x)$ are $(2^{k}(M+1)\times 1)$ matrices given by \begin{gather}\label{4.3} C=[c_{0,0},c_{0,1},\dots ,c_{0,M},c_{1,0},\dots ,c_{1,M},\dots , c_{2^{k}-1,0},\dots ,c_{2^{k}-1,M}]^{T},\\ \label{4.4} \Psi (x)=[\psi _{0,0}(x),\psi _{0,1}(x),\dots ,\psi_{0,M}(x),\dots , \psi _{2^{k}-1,0}(x),\dots ,\psi _{2^{k}-1,M}(x) ]^{T}. \end{gather} \section{Legendre multi-wavelet method for Volterra integro-differential equation form of Lane-Emden equation} \label{sec:5} Consider the Volterra integro-differential equation given in \eqref{2.4} which is the form of Lane-Emden equation defined in \eqref{2.1}. To apply the Legendre multi-wavelets, we first approximate the unknown function $y(x)$ as \begin{equation}\label{5.1} y(x)=C^{T}\Psi (x), \end{equation} where $C$ is defined similar to \eqref{4.3}. Integrating \eqref{2.4} and using the initial condition $y(0)=\alpha $, we have \begin{equation}\label{5.2} y(x)=\alpha -\int_{0}^{x}\Big[ \int_{0}^{z} \Big( \frac{t^{\kappa}}{z^{\kappa}}\Big)f(y(t))dt\Big] dz,\quad \kappa \geq 1 \end{equation} Then from \eqref{5.1} and \eqref{5.2}, we have \begin{equation} \label{5.3} \begin{aligned} C^{T}\Psi (x)&=\alpha -\int_{0}^{x} \Big[ \int_{0}^{z}\Big( \frac{t^{\kappa}}{z^{\kappa}}\Big)f(C^{T}\Psi (t))dt\Big] dz, \quad \kappa \geq 1 \\ &= \alpha -\int_{0}^{x}H(z)dz, \end{aligned} \end{equation} where \begin{equation*} H(z)=\int_{0}^{z}\Big(\frac{t^{\kappa}}{z^{\kappa}}\Big)f(C^{T}\Psi (t))dt. \end{equation*} Now we collocate \eqref{5.3} at $x_{i}=\frac{(2i-1)T}{2^{k+1}(M+1)}$, $i=1,2,\dots ,2^{k}(M+1)$ as \begin{equation}\label{5.4} C^{T}\Psi (x_{i})=\alpha -\int_{0}^{x_{i}}H(z)dz \end{equation} To use the Gaussian integration formula for \eqref{5.4}, we transfer the interval $[0,x_{i}]$ into the interval $[-1, 1]$ by means of the transformation $$ \tau =\frac{2}{x_{i}}z-1 $$ Equation \eqref{5.4} can be written as \begin{equation}\label{5.5} C^{T}\Psi (x_{i})=\alpha -\frac{x_{i}}{2}\int _{-1}^{1} H\big( \frac{x_{i}}{2}(\tau +1)\big)d\tau. \end{equation} Using the Gaussian integration formula, we obtain \begin{equation}\label{5.6} C^{T}\Psi (x_{i})\cong \alpha -\frac{x_{i}}{2}\sum_{j=1}^{s}w_{j} H\big( \frac{x_{i}}{2}(\tau _{j} +1)\big), \end{equation} where $\tau _{j}$ are $s$ zeros of Legendre polynomials $P_{s+1}$ and $w_{j}$ are the corresponding weights. The idea behind the above approximation is the exactness of the Gaussian integration formula for polynomials of degree not exceeding $2s+1$. Equation \eqref{5.6} gives a system of $2^{k}(M+1)$ nonlinear algebraic equations with same number of unknowns for coefficient matrix $C$. Solving this system numerically by Newton's method, we can get the values of unknowns for $C$ and hence we obtain the solution $y(x)=C^{T}\Psi (x)$. \section{Convergence analysis} \label{sec:6} \begin{theorem} \label{thm1} The series solution $y(x)=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}c_{n,m}\psi _{n,m}(x)$ defined in \eqref{4.1} using Legendre multi-wavelet method converges to $y(x)$. \end{theorem} \begin{proof} The set $\{\psi _{n,m}; n, m = 0, 1, \ldots\}$ is a complete orthonormal set in the Hilbert space $L^{2}(\mathbb{R})$. Let $y(x)=\sum _{m=0}^{M}C_{n,m}\psi _{n,m}(x)$ where $C_{n,m}=\langle y(x), \psi _{n,m}(x)\rangle$, for fixed $n$. Let us denote $\psi _{n,m}(x)=\psi (x)$ and let $\alpha _{j}=\langle y(x),\psi (x)\rangle$. Now we define the sequence of partial sum $\{ S_{n}\} $ of $\big( \alpha _{j}\psi (x_{j})\big)$. Let $\{ S_{n}\}$ and $\{ S_{m}\}$ be the partial sums with $n\geq m$. We have to prove $\{ S_{n}\}$ is a Cauchy sequence in the Hilbert space. Let $S_{n}=\sum _{j=1}^{n}\alpha _{j}\psi (x_{j})$. Now $$ \langle y(x),S_{n} \rangle = \langle y(x), \sum _{j=1}^{n}\alpha _{j}\psi (x_{j})\rangle = \sum _{j=1}^{n}\vert \alpha _{j}\vert ^{2}. $$ We claim that $$ \| S_{n}-S_{m}\|^{2}=\sum _{j=m+1}^{n}\vert \alpha _{j}\vert ^{2},\quad n>m. $$ Now $$ \| \sum _{j=m+1}^{n}\alpha _{j} \psi (x_{j})\|^{2} =\langle \sum _{j=m+1}^{n}\alpha _{j} \psi (x_{j}), \sum _{j=m+1}^{n}\alpha _{j} \psi (x_{j})\rangle =\sum _{j=m+1}^{n}\vert \alpha _{j}\vert ^{2}, $$ for $n>m$. Therefore, $$ \| \sum _{j=m+1}^{n}\alpha _{j} \psi (x_{j})\|^{2} =\sum _{j=m+1}^{n}\vert \alpha _{j}\vert ^{2}, \quad\text{for } n>m. $$ From Bessel's inequality, we have $\sum _{j=1}^{\infty }\vert \alpha _{j}\vert ^{2}$ is convergent and hence $$ \| \sum _{j=m+1}^{n}\alpha _{j} \psi (x_{j})\|^{2}\to 0\quad \text{as } n \to \infty. $$ So, $$ \| \sum _{j=m+1}^{n}\alpha _{j} \psi (x_{j})\| \to 0 $$ and $\{ S_{n}\}$ is a Cauchy sequence and it converges to $s$ (say). We assert that $y(x)=s$. In fact, \begin{align*} \langle s-y(x),\psi (x_{j})\rangle &= \langle s,\psi (x_{j})\rangle-\langle y(x), \psi (x_{j})\rangle\\ &= \langle \lim_{n\to \infty}S_{n}, \psi (x_{j})\rangle-\alpha _{j}\\ &= \alpha _{j}-\alpha _{j}. \end{align*} This implies $ \langle s-y(x), \psi (x_{j})\rangle=0$, which gives $y(x)=s$ and $\sum _{j=1}^{n}\alpha _{j}\psi(x_{j})$ converges to $y(x)$ as $n\to \infty$ and completes the proof. \end{proof} \section{Illustrative examples} \label{sec:7} \begin{example} \label{examp1}\rm Consider the generalized form of Lane-Emden equation of first kind $$ y''(x)+\frac{\kappa }{x}y'(x)+y^{m}(x)=0,\quad \kappa \geq 1,\quad y(0)=1,\quad y'(0)=0. $$ This equation is equivalent to the integro-differential equation $$ y'(x)=-\int_{0}^{x}\big( \frac{t^{\kappa }}{x^{\kappa }}\big) y^{m}(t)dt, \quad y(0)=1,\quad \kappa \geq 1. $$ The exact solutions of this problem for $\kappa =2$ and $m=0,1,5$ respectively are \begin{gather*} y(x)=1-\frac{1}{3!}x^{2}\\ y(x)=\frac{\sin x}{x}\\ y(x)=\big( 1+\frac{x^{2}}{3}\big)^{-1/2} \end{gather*} The approximate solutions obtained by Legendre multi-wavelet method $(M=7, k=1)$ for shape factor $\kappa =2$ and $m=0,1,5$ with their corresponding exact solutions and absolute errors have been shown in Tables \ref{table1}--\ref{table3} respectively. \end{example} \begin{table}[htb] \begin{center} \begin{tabular}{|l|l|l|l|} \hline $x$ & LMWM solution & Exact solution & Absolute error\\ \hline 0.2 & 0.993333 & 0.993333 & 2.66664E-12 \\ \hline 0.4 & 0.973333 & 0.973333 & 2.13333E-11 \\ \hline 0.6 & 0.940000 & 0.940000 & 7.20001E-11 \\ \hline 0.8 & 0.893333 & 0.893333 & 1.70667E-10 \\ \hline 1 & 0.833333 & 0.833333 & 3.33333E-10 \\ \hline \end{tabular} \end{center} \caption{Numerical solutions for Example \ref{examp1} when $\kappa =2$, $m=0$} \label{table1} \end{table} \begin{table}[htb] \begin{center} \begin{tabular}{|l|l|l|l|} \hline $x$ & LMWM solution & Exact solution & Absolute error\\ \hline 0.2 & 0.993347 & 0.993347 & 2.45593E-9 \\ \hline 0.4 & 0.973546 & 0.973546 & 5.46664E-10 \\ \hline 0.6 & 0.941071 & 0.941071 & 2.45289E-10 \\ \hline 0.8 & 0.896695 & 0.896695 & 1.94895E-10 \\ \hline 1 & 0.841471 & 0.841471 & 2.45936E-10 \\ \hline \end{tabular} \end{center} \caption{Numerical solutions for Example \ref{examp1} when $\kappa =2$, $m=1$} \label{table2} \end{table} \begin{table}[htb] \begin{center} \begin{tabular}{|l|l|l|l|} \hline $x$ & LMWM solution & Exact solution & Absolute error\\ \hline 0 & 1 & 1 & 2.66055E-9 \\ \hline 0.2 & 0.993399 & 0.993399 & 1.07934E-11 \\ \hline 0.4 & 0.974355 & 0.974355 & 1.17952E-11 \\ \hline 0.6 & 0.944911 & 0.944911 & 1.64531E-11 \\ \hline 0.8 & 0.907841 & 0.907841 & 2.17233E-11 \\ \hline \end{tabular} \end{center} \caption{Numerical solutions for Example \ref{examp1} when $\kappa =2$, $m=5$} \label{table3} \end{table} \begin{example} \label{examp2}\rm Consider the Lane-Emden equation of second kind $$ y''(x)+\frac{\kappa }{x}y'(x)+e^{y(x)}=0,\quad y(0)=y'(0)=0,\quad \kappa >1. $$ This equation is equivalent to $$ y'(x)=-\int_{0}^{x}\big( \frac{t^{\kappa }}{x^{\kappa }}\big) e^{y(t)}dt, \quad y(0)=1,\quad \kappa > 1. $$ The approximate solutions obtained by Legendre multi-wavelet method $(M=7, k=1)$ for shape factor $\kappa =2,3,4$ have been compared with the solutions obtained by a variational iteration method (VIM) \cite{5} cited in Table \ref{table4}. \end{example} \begin{table}[htb] \begin{center} \begin{tabular}{|l|ll|ll|ll|} \hline $x$& \multicolumn{2}{l|}{\quad\quad $\kappa =2$} &\multicolumn{2}{l|}{\quad \quad $\kappa =3$} &\multicolumn{2}{l|}{\quad \quad $\kappa =4$} \\ & LMWM& VIM& LMWM& VIM&LMWM & VIM\\ \hline 0 & -5.7433E-11 & 0 & -2.484E-11 & 0 & -1.2637E-11 & 0 \\ \hline 0.2 & -0.006653 & -0.006653 & -0.004992 & -0.004992 & -0.003994 & -0.003994 \\ \hline 0.4 & -0.026456 & -0.026456 & -0.019868 & -0.019868 & -0.015909 & -0.015909 \\ \hline 0.6 & -0.058944 & -0.058944 & -0.044337 &-0.044337 & -0.035544 & -0.035544 \\ \hline 0.8 & -0.103386 & -0.103386 & -0.077935 & -0.077935 & -0.062578 & -0.062578 \\ \hline \end{tabular} \end{center} \caption{Numerical solutions for Example \ref{examp2}} \label{table4} \end{table} \begin{example} \label{examp3}\rm Next, consider the Lane-Emden type equation given by $$ y''(x)+\frac{8 }{x}y'(x)+(18y(x)+4y(x)\ln(y(x))=0,\quad y(0)=1,\quad y'(0)=0 $$ The Volterra integro-differential form of this equation is given by $$ y'(x)+\int_{0}^{x}\frac{t^{8}}{x^{8}}(18y(t)+4y(t)\ln y(t))dt=0, \quad y(0)=1 $$ with exact solution $e^{-x^{2}}$ . The Legendre multi-wavelets solutions for $M=7, k=1$ along with their corresponding exact solutions and absolute errors have been shown in Table \ref{table5}. \end{example} \begin{table}[htb] \begin{center} \begin{tabular}{|l|l|l|l|} \hline $x$ & LMWM solution & Exact solution & Absolute error\\ \hline 0 & 1 & 1 & 3.95615E-8 \\ \hline 0.1 & 0.990050 & 0.990050 & 2.96242E-10 \\ \hline 0.2 & 0.960789 & 0.960789 & 3.82808E-10 \\ \hline 0.3 & 0.913931 & 0.913931 & 2.95619E-8 \\ \hline 0.4 & 0.852143 & 0.852143 & 4.68592E-7 \\ \hline 0.5 & 0.778797 & 0.778797 & 3.64064E-6 \\ \hline \end{tabular} \end{center} \caption{Numerical solutions for Example \ref{examp3}} \label{table5} \end{table} \begin{example} \label{examp4}\rm Consider the Lane-Emden type equation given by $$ y''(x)+\frac{1 }{x}y'(x)+(3y^{5}(x)-y^{3}(x))=0,\quad y(0)=1,\quad y'(0)=0 $$ The Volterra integro-differential form of this equation is given by $$ y'(x)+\int_{0}^{x}\frac{t}{x}(3y^{5}(t)-y^{3}(t))dt=0, \quad y(0)=1 $$ with exact solution $\frac{1}{\sqrt{1+x^{2}}}$. The Legendre multi-wavelets solutions for $M=7, k=1$ along with their corresponding exact solutions and absolute errors have been shown in Table \ref{table6}. \end{example} \begin{table}[htb] \begin{center} \begin{tabular}{|l|l|l|l|} \hline $x$ & LMWM solution & Exact solution & Absolute error\\ \hline 0 & 1 & 1 & 9.41731E-8 \\ \hline 0.2 & 0.980581 & 0.980581 & 8.91026E-10 \\ \hline 0.4 & 0.928477 & 0.928477 & 1.53517E-9 \\ \hline 0.6 & 0.857493 & 0.857493 & 1.16852E-9 \\ \hline 0.8 & 0.780869 & 0.780869 & 1.55470E-9 \\ \hline \end{tabular} \end{center} \caption{Numerical solutions for Example \ref{examp4}} \label{table6} \end{table} \begin{example} \label{examp5}\rm Consider the Lane-Emden type equation given by $$ y''(x)+\frac{2 }{x}y'(x)+4\Big( 2e^{y(x)}+e^{\frac{y(x)}{2}}\Big)=0,\quad y(0)=y'(0)=0. $$ The Volterra integro-differential form of this equation is given by $$ y'(x)+\int_{0}^{x}\frac{t^{2}}{x^{2}} \Big( 4\Big( 2e^{y(t)}+e^{\frac{y(t)}{2}}\Big) \Big) dt=0, \quad y(0)=0 $$ with exact solution $-2\ln (1+x^{2})$. The Legendre multi-wavelets solutions for $M=7, k=1$ along with their corresponding exact solutions and absolute errors have been shown in Table \ref{table7}. \end{example} \begin{table}[htb] \begin{center} \begin{tabular}{|l|l|l|l|} \hline $x$ & LMWM solution & Exact solution & Absolute error\\ \hline 0 & 1.1743E-7& 0 & 1.17430E-7 \\ \hline 0.2 & -0.078441& -0.078441 & 1.25003E-9 \\ \hline 0.4 & -0.296840& -0.296840 & 1.65908E-7 \\ \hline 0.6 & -0.614985& -0.614969 & 1.52712E-5 \\ \hline 0.8 & -0.989704& -0.989392 & 3.11348E-4 \\ \hline \end{tabular} \end{center} \caption{Numerical solutions for Example \ref{examp5}} \label{table7} \end{table} \begin{example} \label{examp6}\rm Consider the system of nonlinear Lane-Emden type equations given by \begin{gather*} y_{1}''(x)+\frac{8}{x}y_{1}'(x)+(18y_{1}(x)-4y_{1}(x)\ln y_{2}(x))=0\\ y_{2}''(x)+\frac{4}{x}y_{2}'(x)+(4y_{2}(x)\ln y_{1}(x)-10y_{2}(x))=0 \end{gather*} with initial conditions \begin{gather*} y_{1}(0)=1,\quad y_{1}'(0)=0, \\ y_{2}(0)=1,\quad y_{2}'(0)=0 \end{gather*} The system of nonlinear Volterra integro-differential form of the above system is given by \begin{gather*} y_{1}'(x)+\int_{0}^{x}\frac{t^{8}}{x^{8}}(18y_{1}(t)-4y_{1}(t)\ln y_{2}(t))dt=0,\\ y_{2}'(x)+\int_{0}^{x}\frac{t^{4}}{x^{4}}(4y_{2}(t)\ln y_{1}(t)-10y_{2}(t))dt=0, \end{gather*} with initial conditions $y_{1}(0)=1,\quad y_{2}(0)=1$. The corresponding exact solutions of this system are \[ y_{1}(x)=e^{-x^{2}},\quad y_{2}(x)=e^{x^{2}} \] The approximate solutions obtained by Legendre multi-wavelet method for $M=7, k=1$ along with their corresponding exact solutions and absolute errors have been shown in Table \ref{table8}. \end{example} \begin{table}[htb] \begin{center} \begin{tabular}{|l|ll|ll|ll|} \hline $x$ & \multicolumn{2}{l|}{LMWM solution}&\multicolumn{2}{l|}{Exact solution} & \multicolumn{2}{l|}{Absolute error}\\ &$y_{1}(x)$ &$y_{2}(x)$&$y_{1}(x)$&$y_{2}(x)$&$y_{1}(x)$&$y_{2}(x)$\\ \hline 0 & 1& 1 & 1& 1 & 7.15876E-8& 8.44232E-8 \\ \hline 0.1 & 0.99005& 1.01005 & 0.99005& 1.01005 & 5.61584E-10& 6.59049E-10 \\ \hline 0.2 & 0.960789& 1.04081 & 0.960789& 1.04081& 9.69923E-10& 3.34747E-10 \\ \hline 0.3 & 0.913931& 1.09417& 0.913931& 1.09417 & 3.5286E-8&4.47131E-8 \\ \hline 0.4 & 0.852144& 1.17351 & 0.852144& 1.17351 & 6.22823E-7& 8.00388E-7 \\ \hline 0.5 & 0.778805& 1.28402 & 0.778801& 1.28403 & 4.48153E-6& 7.03964E-6 \\ \hline \end{tabular} \end{center} \caption{Numerical solutions for Example \ref{examp6}} \label{table8} \end{table} \begin{example} \label{examp7}\rm To verify the accuracy of the presented method, we have considered a fractional order integro-differential equation \cite{Zhu-2012} as \begin{equation*} D^{\alpha}y(x)-\int_{0}^{1}xt[y(t)]^{2}dt=1-\frac{x}{4},\quad 0\leq x<1,\; 0<\alpha \leq 1, \end{equation*} with initial condition $y(0)=0$ and the exact solution $y(x)=x$ when $\alpha =1$. This problem has been solved by Chebyshev wavelet method (CWM) in \cite{Zhu-2012} for $\alpha=1$. The results obtained by the Chebyshev wavelet method \cite{Zhu-2012} have been compared with the results obtained by presented method and the root mean square errors (RMSE) of these two methods have been cited in Table \ref{table9}. \end{example} \begin{table}[htb] \begin{center} \begin{tabular}{|l|l|lll|} \hline Error & LMWM & & CWM \cite{Zhu-2012} & \\ & $k=3,M=2$& $k=3,M=2$& $k=4,M=2$&$k=5,M=2$\\ \hline RMSE & 3.92041E-10&2.9700E-7&1.8610E-8&1.1645E-9\\ \hline \end{tabular} \end{center} \caption{Root mean square errors for Example \ref{examp7}} \label{table9} \end{table} \begin{example} \label{examp8}\rm Again to verify the accuracy of the method here presented, we consider the nonlinear Volterra-Fredholm integro-differential equation (see \cite{Maleknejad-2011}) \begin{equation*} y'(x)+y(x)+\frac{1}{2}\int_{0}^{x}xy^{2}(t)dt -\frac{1}{4}\int_{0}^{1}ty^{3}(t)dt=g(x), \end{equation*} with $g(x)=2x+x^{2}+\frac{1}{10}x^{6}-\frac{1}{32}$ and initial condition $y(0)=0$. The exact solution of this problem is $x^{2}$. This problem has been solved by hybrid Legendre polynomials and Block-Pulse functions (HLPBPF) in \cite{Maleknejad-2011}. The results obtained using HLPBPF \cite{Maleknejad-2011} are compared with the results obtained by presented method and cited in Table \ref{table10}. The maximum absolute errors obtained by these two methods has been cited in Table \ref{table11}. \end{example} \begin{table}[htb] \begin{center} \begin{tabular}{|l|l|lll|l|} \hline $x$ & LMWM & & HLPBPF \cite{Maleknejad-2011}& & Exact \\ & $M=8,k=1$ & $M=8,n=2$ & $M=8,n=4$&$M=8,n=4$ & \\ \hline 0 & 0 & 0 & 0 & 0 & 0 \\ \hline 0.1 & 0.01 & 0.010917 & 0.010256& 0.010031 & 0.01 \\ \hline 0.2 & 0.04 & 0.041703 & 0.040487 & 0.040075 & 0.04 \\ \hline 0.3 & 0.09 & 0.092364 & 0.090698 & 0.090171 & 0.09 \\ \hline 0.4 & 0.16 & 0.162911 & 0.160866 & 0.160094 & 0.16 \\ \hline 0.5 & 0.25 & 0.253371 & 0.250997 &0.250228 & 0.25 \\ \hline 0.6 & 0.36 & 0.364244 & 0.361061 & 0.360502 & 0.36 \\ \hline 0.7 & 0.49 & 0.493830 & 0.490969 & 0.490583 & 0.49 \\ \hline 0.8 & 0.64 & 0.642375 & 0.640830 & 0.640374 & 0.64 \\ \hline 0.9 & 0.81 & 0.810337 & 0.810183 & 0.810047 & 0.81 \\ \hline \end{tabular} \end{center} \caption{Numerical solutions for Example \ref{examp8}} \label{table10} \end{table} \begin{table}[htb] \begin{center} \begin{tabular}{|l|l|lll|} \hline Error & LMWM & & HLPBPF \cite{Maleknejad-2011} & \\ & $M=8,k=1$& $M=8,n=2$& $M=8,n=4$&$M=8,n=8$\\ \hline Max. Abs. Err. & 1.85984E-9&4.244E-3&1.0610E-3&5.83E-4\\ \hline \end{tabular} \end{center} \caption{Maximum absolute errors for Example \ref{examp8}} \label{table11} \end{table} \subsection*{Conclusion} %\label{sec:8} Using the equivalence between the Lane-Emden equations of first and second kind and Volterra integro-differential equations a numerical method that overcomes the difficulty of the singular behavior at $x=0$ is established. The numerical method is reduced to solving a system of algebraic equations. Examples that demonstrate the validity and applicability of the present technique are included. These examples also exhibit the accuracy and efficiency of the proposed method. \subsection*{Acknowledgements} The authors would like to express their sincere thanks and gratitude to the anonymous reviewers for their kind suggestions for the betterment and improvement of the present paper. \begin{thebibliography}{00} \bibitem{14} J. Biazar, H. 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