# Numerical Solution of Singular Lane-Emden Equation

Authors: Hossein Aminikhah; Sakineh Moradian
Journal: ISRN Mathematical Physics
 (2013)

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## Abstract

A new approach for solving the nonlinear Lane-Emden type equations has been proposed. The method is based on Legendre wavelets approximations. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.


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## Body

## 1. Introduction

The Lane-Emden type equations are nonlinear ordinary differential equations on semi-infinite domain. They are categorized as singular initial value problems. These equations describe the temperature variation of a spherical gas cloud under the mutual attraction of its molecules and subject to the laws of classical thermodynamics. The polytrophic theory of stars essentially follows out of thermodynamic considerations that deal with the issue of energy transport, through the transfer of material between different levels of the star. These equations are one of the basic equations in the theory of stellar structure and have been the focus of many studies. The general form of the Lane-Emden equations is the following form:(1)y′′(x)+mxy′(x)+f(x,y)=g(x),0<x≤1,m≥0,
with the following initial conditions:
(2)y(0)=A,y′(0)=B,
where f(x,y) is a continuous real-value function and g(x)   is an analytical function. Equation (1) was 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 sphere, and theory of thermionic currents [1, 2]. The solution of the Lane-Emden equation, as well as those of a variety of nonlinear problems in quantum mechanics and astrophysics such as the scattering length calculations in the variable phase approach, is numerically challenging because of the singular point at the origin. Bender et al. [3] proposed a new perturbation technique based on an artificial parameter  δ; the method is often called δ-method. El-Gebeily and O’Regan [4] used the quasilinearization approach to solve the standard Lane-Emden equation. This method approximates the solution of a nonlinear differential equation by treating the nonlinear terms as a perturbation about the linear ones, and unlike perturbation theories, it is not based on the existence of some small parameters. Approximate solutions to the above problems were presented by Shawagfeh [5] and Wazwaz [6, 7] by applying the Adomian method which provides a convergent series solution. Nouh [8] accelerated the convergence of a power series solution of the Lane-Emden equation by using an Euler-Abel transformation and Padé approximation. Mandelzweig and Tabakin [9] applied Bellman and Kalaba’s quasilinearization method, and Ramos [10] used a piecewise linearization technique based on the piecewise linearization of the Lane-Emden equation. Bozkhov and Gilli Martins [11] and later Momoniat and Harley [12] applied the Lie Group method successfully to generalized Lane-Emden equations of the first kind. Exact solutions of generalized Lane-Emden solutions of the first kind are investigated by Goenner and Havas [13]. Liao [14] solved Lane-Emden type equations by applying a homotopy analysis method. He [15] obtained an approximate analytical solution of the Lane-Emden equation by applying a variational approach which uses a semi-inverse method. Ramos [16] presented a series approach to the Lane-Emden equation and gave the comparison with homotopy perturbation method. Özis and Yildirim [17, 18] gave the solutions of a class of singular second-order IVPs of Lane-Emden type by using homotopy perturbation and variational iteration method. Parand et al. [19–22] presented three numerical techniques to solve higher ordinary differential equations such as Lane-Emden. Their approach was based on the rational Chebyshev, rational Legendre Tau, and Hermite functions collocation methods. In this paper, the new approximate analytical method will be introduced for exact solution of Lane-Emden equation.This paper is arranged as follows.In Section2, the properties of Legendre wavelets and the way to construct the wavelet technique for this type of equation are described. In Section 3, the proposed method is applied to some types of Lane-Emden equations, and a comparison is made with the existing analytic or exact solutions that were reported in other published works in the literature. Finally, we give a brief conclusion in the last section.


## 2. Legendre Wavelets Applied to Singular IVPs of Lane-Emden Type Equation

### 2.1. Wavelets and Legendre Wavelets

Wavelets constitute a family of functions constructed from dilation and translation of single function called the mother wavelet. When the dilation parametera and the translation parameter b vary continuously, we have the following family of continuous wavelets:
(3)ψa,b(x)=|a|-1/2ψ(x-ba),a,b∈ℝ,a≠0.If we restrict the parametera and b  to discrete values as  a=a0-k,b=nb0a0-k,a0>1,b0>0, and  n,k  positive integers, we have the following family of discrete wavelets:
(4)ψk,n(x)=|a0|k/2ψ(a0kx-nb0),
where ψk,n(x) forms a wavelet basis for L2(ℝ). In particular, when a0=2  and  b0=1  then  ψk,n(x) forms an orthonormal basis.Legendre waveletsψn,m(x)=ψ(k,n,m,x)  have four arguments,  n=1,2,3,…,2k-1:k can assume any positive integer, m is the order for Legendre polynomials, and  they are defined on the interval [0,1) as follows:
(5)ψnm(x)={m+122k/2Pm(2kx-2n+1),n-12k-1≤x<n2k-1,0,otherwise,
where   m=0,1,…,M-1, n=1,2,3,…,2k-1. The coefficient m+(1/2) is for orthonormality. Here, Pm(x) are the well-known Legendre polynomials of order m which are defined on the interval [-1,1] and can be determined with the aid of the following recurrence formulae:
(6)P0(x)=1,P1(x)=x,Pm+1(x)=(2m+1m+1)xPm(x)-(mm+1)Pm-1(x),m=1,2,…


### 2.2. Function Approximation

A functionf(x) defined on the interval [0,1) may be expanded by Legendre wavelet as
(7)f(x)=∑n=1∞∑m=0∞cnmψnm(x),
where
(8)cnm=(f(x),ψnm(x)).In (8), (·,·) denotes the inner product.If the infinite series in (7) is truncated, then (7) can be written as
(9)f(x)≈∑n=12k-1∑m=0M-1cnmψnm(x)=CTψ(x),
where C and ψ(x) are 2k-1M×1 matrices given by(10)C=[c10,c11,…c1M-1,c20,c21,…c2M-1,…,c2k-10,…,c2k-1M-1]T,(11)ψ(x)=[ψ10(x),ψ11(x),…ψ1M-1(x),ψ20(x),ψ21(x),…ψ2M-1(x),…,ψ2k-10(x),…,ψ2k-1M-1(x)]T.The integration of the product of two Legendre wavelets vector functions is obtained as(12)∫01ψ(x)ψT(x)dx=I,
where  I  is an identity matrix.


### 2.3. Legendre Wavelets Operational Matrix of Integration

The integration of the vectorψ(x)defined in (11) can be obtained as
(13)∫0xψ(t)dt=Pψ(x),
where  P  is the 2k-1M×2k-1M operational matrix of integration given by [23] as
(14)P=12k[LFF⋯FOLF⋱⋮OOL⋱F⋮⋱⋱⋱FO⋯OOL],
whereL,F, and Oare M×M matrices given by(15)L=[1330⋯00-3303350⋯00-553⋱⋱⋱⋮⋮0⋱⋱⋱0⋮⋯⋱-2M-3(2M-3)2M-1⋱2M-3(2M-3)2M-10⋯⋯0-2M-1(2M-1)2M-30],F=[20⋯000⋯0⋮⋮⋱⋮00⋯0],O=[00⋯000⋯0⋮⋮⋱⋮00⋯0].The following property of the product of two Legendre wavelet vector functions will also be used:(16)ψ(t)Tψ(t)C≈C^ψ(t),
where C is a vector given in (10) and C^ is a 2k-1M×2k-1M matrix, which is called the product operation of Legendre wavelet vector functions [23]. For  M=3 and  k=1, the matrix C^ is obtained:
(17)[c10c11c12c11c10+2c1252c115c122c115c10+25c127].


### 2.4. Solution of Lane-Emden Equations

We multiply both sides of (1) by x,
(18)xy′′(x)+my′(x)+xf(x,y)=xg(x),0<x≤1,m≥0,
in order to use Legendre wavelets to approximate y′′(x) as
(19)y′′(x)≈CTψ(x).Integrating (19) with respect to x twice from 0 to  x, we obtain
(20)y′(x)≈CTPψ(x)+y′(0)=CTPψ(x)+U1Tψ(x),(21)y(x)≈CTP2ψ(x)+xy′(0)+y(0)=CTP2ψ(x)+U0Tψ(x),
where coefficients U0 and U1 are known and can be obtained from the initial conditions, C and ψ(x) are defined similarly to (10) and (11), and P is 2k-1M×2k-1M operational matrix for integration, defined in (14).Also consider the following approximations:(22)xy′′(x)≈Y1Tψ(x),xf(x,y)≈x∑j=0n1j!((x-x0)∂∂x+(y-y0)∂∂y)jf(x0,y0)=Y2Tψ(x),xg(x)≈x∑j=0ng(n)(x0)n!(x-x0)n=GTψ(x),
where  Y1 and  Y2 are column vectors with the entries of the vectors  C and coefficients of G  known.Substitution of approximations (20) and (22) into (18) will be resulted to
(23)Y1Tψ(x)+m(CTP+U1T)ψ(x)+Y2Tψ(x)=GTψ(x).Simplifyingψ(x)   in (23), a nonlinear system in terms of  C  will be obtained:
(24)Y1T+m(CTP+U1T)+Y2T=GT.The element of vector functionsC  can be computed by solving these systems.


## 2.1. Wavelets and Legendre Wavelets

Wavelets constitute a family of functions constructed from dilation and translation of single function called the mother wavelet. When the dilation parametera and the translation parameter b vary continuously, we have the following family of continuous wavelets:
(3)ψa,b(x)=|a|-1/2ψ(x-ba),a,b∈ℝ,a≠0.If we restrict the parametera and b  to discrete values as  a=a0-k,b=nb0a0-k,a0>1,b0>0, and  n,k  positive integers, we have the following family of discrete wavelets:
(4)ψk,n(x)=|a0|k/2ψ(a0kx-nb0),
where ψk,n(x) forms a wavelet basis for L2(ℝ). In particular, when a0=2  and  b0=1  then  ψk,n(x) forms an orthonormal basis.Legendre waveletsψn,m(x)=ψ(k,n,m,x)  have four arguments,  n=1,2,3,…,2k-1:k can assume any positive integer, m is the order for Legendre polynomials, and  they are defined on the interval [0,1) as follows:
(5)ψnm(x)={m+122k/2Pm(2kx-2n+1),n-12k-1≤x<n2k-1,0,otherwise,
where   m=0,1,…,M-1, n=1,2,3,…,2k-1. The coefficient m+(1/2) is for orthonormality. Here, Pm(x) are the well-known Legendre polynomials of order m which are defined on the interval [-1,1] and can be determined with the aid of the following recurrence formulae:
(6)P0(x)=1,P1(x)=x,Pm+1(x)=(2m+1m+1)xPm(x)-(mm+1)Pm-1(x),m=1,2,…


## 2.2. Function Approximation

A functionf(x) defined on the interval [0,1) may be expanded by Legendre wavelet as
(7)f(x)=∑n=1∞∑m=0∞cnmψnm(x),
where
(8)cnm=(f(x),ψnm(x)).In (8), (·,·) denotes the inner product.If the infinite series in (7) is truncated, then (7) can be written as
(9)f(x)≈∑n=12k-1∑m=0M-1cnmψnm(x)=CTψ(x),
where C and ψ(x) are 2k-1M×1 matrices given by(10)C=[c10,c11,…c1M-1,c20,c21,…c2M-1,…,c2k-10,…,c2k-1M-1]T,(11)ψ(x)=[ψ10(x),ψ11(x),…ψ1M-1(x),ψ20(x),ψ21(x),…ψ2M-1(x),…,ψ2k-10(x),…,ψ2k-1M-1(x)]T.The integration of the product of two Legendre wavelets vector functions is obtained as(12)∫01ψ(x)ψT(x)dx=I,
where  I  is an identity matrix.


## 2.3. Legendre Wavelets Operational Matrix of Integration

The integration of the vectorψ(x)defined in (11) can be obtained as
(13)∫0xψ(t)dt=Pψ(x),
where  P  is the 2k-1M×2k-1M operational matrix of integration given by [23] as
(14)P=12k[LFF⋯FOLF⋱⋮OOL⋱F⋮⋱⋱⋱FO⋯OOL],
whereL,F, and Oare M×M matrices given by(15)L=[1330⋯00-3303350⋯00-553⋱⋱⋱⋮⋮0⋱⋱⋱0⋮⋯⋱-2M-3(2M-3)2M-1⋱2M-3(2M-3)2M-10⋯⋯0-2M-1(2M-1)2M-30],F=[20⋯000⋯0⋮⋮⋱⋮00⋯0],O=[00⋯000⋯0⋮⋮⋱⋮00⋯0].The following property of the product of two Legendre wavelet vector functions will also be used:(16)ψ(t)Tψ(t)C≈C^ψ(t),
where C is a vector given in (10) and C^ is a 2k-1M×2k-1M matrix, which is called the product operation of Legendre wavelet vector functions [23]. For  M=3 and  k=1, the matrix C^ is obtained:
(17)[c10c11c12c11c10+2c1252c115c122c115c10+25c127].


## 2.4. Solution of Lane-Emden Equations

We multiply both sides of (1) by x,
(18)xy′′(x)+my′(x)+xf(x,y)=xg(x),0<x≤1,m≥0,
in order to use Legendre wavelets to approximate y′′(x) as
(19)y′′(x)≈CTψ(x).Integrating (19) with respect to x twice from 0 to  x, we obtain
(20)y′(x)≈CTPψ(x)+y′(0)=CTPψ(x)+U1Tψ(x),(21)y(x)≈CTP2ψ(x)+xy′(0)+y(0)=CTP2ψ(x)+U0Tψ(x),
where coefficients U0 and U1 are known and can be obtained from the initial conditions, C and ψ(x) are defined similarly to (10) and (11), and P is 2k-1M×2k-1M operational matrix for integration, defined in (14).Also consider the following approximations:(22)xy′′(x)≈Y1Tψ(x),xf(x,y)≈x∑j=0n1j!((x-x0)∂∂x+(y-y0)∂∂y)jf(x0,y0)=Y2Tψ(x),xg(x)≈x∑j=0ng(n)(x0)n!(x-x0)n=GTψ(x),
where  Y1 and  Y2 are column vectors with the entries of the vectors  C and coefficients of G  known.Substitution of approximations (20) and (22) into (18) will be resulted to
(23)Y1Tψ(x)+m(CTP+U1T)ψ(x)+Y2Tψ(x)=GTψ(x).Simplifyingψ(x)   in (23), a nonlinear system in terms of  C  will be obtained:
(24)Y1T+m(CTP+U1T)+Y2T=GT.The element of vector functionsC  can be computed by solving these systems.


## 3. Numerical Examples

In this section, some examples of Lane-Emden equation are considered and will be solved by introduced method.Example 1.
Consider the following nonlinear Lane-Emden equation:(25)y′′(x)+6xy′(x)+14y(x)=-4y(x)ln(y(x)),0<x≤1,
subject to the initial conditions(26)y(0)=1,y′(0)=0,
which has the following analytical solution:
(27)y(x)=e-x2.
We solve (25) by the method discussed in this paper with  k=1  and  M=7.
We multiply both sides of (25) by  x,
(28)xy′′(x)+6y′(x)+14xy(x)=-4xy(x)ln(y(x)),0<x≤1.Let us consider the following approximations:(29)y′′(x)≈CTψ(x)y′(x)≈CTPψ(x)+U1Tψ(x)y(x)≈CTP2ψ(x)+U0Tψ(x)xy′′(x)≈Y1Tψ(x)xy(x)≈Y2Tψ(x)xy(x)ln(y(x))≈Y3Tψ(x).Substitution into (28) and simplifying will be resulted to:
(30)Y1T+6CTP+14Y2T=-4Y3T.By solving the system (30), we have
(31)c1,0=-0.75742553415,c1,1=0.8884200666,c1,2=0.02494929272,c1,3=-0.09212822515,c1,4=0.002866418366,c1,5=0.003597228202,c1,6=-0.00002910926654.Therefore, the approximate solution of (25) will be obtained as follows:
(32)y(x)≈(CTP2+U0T)ψ(x)=0.2023447118x6-0.3173001850x5+0.7344456412x4-0.08725168885x3-0.984175344x2-0.001172073003x+1.000020858.Table1 shows some values of the solutions and absolute errors at some x, and plots of the exact and approximate solutions are shown in Figure 1.Table 1
Numerical results of Example1.
x
Exact solution
Legendre wavelets
Absolute error
0.0
1
1.000020858
0.000020858
0.1
0.9900498337
0.9900449375
0.0000048962
0.2
0.9607894392
0.9607962885
0.0000068493
0.3
0.9139311853
0.9139303816
8.037
×
1
0
-
7
0.4
0.8521437890
0.8521354021
0.0000083869
0.5
0.7788007831
0.7787879122
0.0000128709
0.6
0.6976763261
0.6976230812
0.0000532449
0.7
0.6126263942
0.6124194854
0.0002069088
0.8
0.5272924240
0.5266984761
0.0005939479
0.9
0.4448580662
0.4434381168
0.0014199494
1
0.3678794412
0.3648016892
0.0030777520Figure 1
The exact and LWM solution of Example1.Example 2.
Consider the following nonlinear Lane-Emden equation:(33)y′′(x)+2xy′(x)+yn(x)=0,0<x≤1,
subject to the initial conditions(34)y(0)=1,y′(0)=0,
where  n≥0 is constant. Substituting  n=0,1, and  5  into (33) leads to the exact solution
(35)y(x)=1-13!x2,y(x)=sin(x)x,y(x)=(1+x23)-1/2,
respectively.Forn=0, we solve (33)  by the Legendre wavelet method with k=1 and M=3. For this equation, we find
(36)c10=-16,c11=0,c13=0.Therefore, we have(37)y(x)≈(CTP2+U0T)ψ(x)=-x26+1,
which is the exact solution.Forn=1, we solve (33)  by the method discussed in this paper with  k=1 and  M=10. We have
(38)c10=-0.3011686789,c11=0.02752116388,c12=0.006713250014,c13=-0.0002154247382,c14=-0.00002541095207,c15=5.308742329×10-7,c16=4.155816200×10-5,c17=-8.1255913×10-10,c18=9.51835270×10-11,c19=-8.086072988×10-11.Therefore, the following solution will result:(39)y(x)≈(CTP2+U0T)ψ(x)=-1.367445662×10-7x9+0.000003078850787x8-4.245445355×10-7x7-0.0001980757263x6-1.650008216×10-7x5+0.008333382135x4-8.134931025×10-9x3-0.1666666660x2+4.268843531×10-11x+1.Table2 shows that the Legendre wavelet solution is very near to the exact solution. Figure 2(a) shows that Legendre wavelet solution coincides with the exact solution.Table 2
Numerical results of Example2 for n=1.
x
Exact solution
Legendre wavelets
Absolute error
0.0
1
1
0
0.1
0.9983341665
0.9983341665
0
0.2
0.9933466540
0.9933466540
0
0.3
0.9850673556
0.9850673556
0
0.4
0.9735458558
0.9735458558
0
0.5
0.9588510772
0.9588510772
0
0.6
0.9410707892
0.9410707890
2
×
1
0
-
10
0.7
0.0920310982
0.0920310989
2
×
1
0
-
10
0.8
0.8966951136
0.8966951137
1
×
1
0
-
10
0.9
0.8703632328
0.8703632329
1
×
1
0
-
10
1
0.8414709848
0.8414709848
0(a) The exact and LWM solution of Example2 for  n=1. (b) The exact and LWM solution of Example 2 for  n=5.
(a)
(b)For solving (33) by Legendre wavelets with  k=1,M=12, and  n=5,  we find
(40)c10=-0.2165063510,c11=0.08910161508,c12=0.009961765824,c13=-0.005784759597,c14=0.0001927423729,c15=0.0001784630080,c16=-0.00002233585057,c17=-0.000003109896876,c18=8.759973237×10-7,c19=5.92941957×10-9,c110=-2.264665721×10-8,c111=1.961578560×10-9.The approximate solution ofy(x) is as follows:
(41)y(x)≈(CTP2+U0T)ψ(x)=0.000007988889774x11+0.0004944909178x10-0.003163180797x9+0.006776529202x8-0.002268670185x7-0.01058534314x6-0.00028270820x5+0.04171831604x4-0.000005679982758x3-0.1666663304x2-8.584262110×10-9x+1.Table3 shows that the Legendre wavelet solution is very near to the exact solution. Figure 2(b) shows that Legendre wavelet solution coincides with the exact solution.Table 3
Numerical results of Example2  for n=5.
x
Exact solution
Legendre wavelets
Absolute error
0.0
1
1
0
0.1
0.9983374884
0.9983374884
0
0.2
0.9933992679
0.9933992677
2
×
1
0
-
10
0.3
0.9853292781
0.9853292781
0
0.4
0.9743547036
0.9743547036
0
0.5
0.9607689228
0.9607689228
0
0.6
0.9449111826
0.9449111825
1
×
1
0
-
10
0.7
0.9271455411
0.9271455408
3
×
1
0
-
10
0.8
0.9078412992
0.9078412989
3
×
1
0
-
10
0.9
0.8873565093
0.8873565094
1
×
1
0
-
10
1
0.8660254038
0.8660254038
0Example 3.
Consider the following nonlinear Lane-Emden equation:(42)y′′(x)+2xy′(x)-2(2x2+3)y=0,0<x≤1,
subject to the initial conditions(43)y(0)=1,y′(0)=0,
which has the following analytical solution:
(44)y(x)=ex2.
Solving (42) by Legendre wavelets method with  k=1  and  M=12, we have
(45)c10=5.436563650,c11=3.464101614,c12=1.517605100,c13=0.4100561906,c14=0.1035241990,c15=0.02034845798,c16=0.003837769593,c17=0.0006174959039,c18=0.0000958708425,c19=0.00001333846701,c1,10=0.000001773800247,c1,11=2.395987744×10-7.
Therefore, the solution of the Lane-Emden equation will be obtained as follows:(46)y(x)≈(CTP2+U0T)ψ(x)=0.02527877277x11-0.08422571788x10+0.1670609573x9-0.1386060456x8+0.1250499043x7+0.109479685x6+0.017184049x5+0.4967026721x4+0.000380562x3+0.9999763578x2+6.313645775×10-7x+0.9999999958.Table4 shows that the Legendre wavelet solution is very near to the exact solution. Figure 3 shows that Legendre wavelet solution coincides with the exact solution.Table 4
Numerical results of Example3.
x
Exact solution
Legendre wavelets
Absolute error
0.0
1
0.9999999958
4.2
×
1
0
-
9
0.1
1.010050167
1.010050166
1
×
1
0
-
9
0.2
1.040810774
1.040810774
0
0.3
1.094174284
1.094174282
2
×
1
0
-
9
0.4
1.173510871
1.173510871
0
0.5
1.284025417
1.284025415
2
×
1
0
-
9
0.6
1.433329415
1.433329414
1
×
1
0
-
9
0.7
1.632316220
1.632316219
1
×
1
0
-
9
0.8
1.896480879
1.896480878
1
×
1
0
-
9
0.9
2.247907987
2.247907986
1
×
1
0
-
9
1
2.718281828
2.718281824
4
×
1
0
-
9Figure 3
The exact and LWM solution of Example3.Example 4.
Consider the following nonlinear Lane-Emden equation:(47)y′′(x)+2xy′(x)+4(2ey+ey/2)=0,0<x≤1,
subject to the initial conditions
(48)y(0)=0,y′(0)=0,
which has the following analytical solution:
(49)y(x)=-2ln(1+x2).We solve (47) by the method discussed in this paper with  k=1  and  M=6. We multiply both sides of (47) by  x,
(50)xy′′(x)+2y′(x)+4x(2ey+ey/2)=0,0<x≤1.Let us consider the following approximations:(51)y′′(x)≈CTψ(x)y′(x)≈CTPψ(x)+U1Tψ(x)y(x)≈CTP2ψ(x)+U0Tψ(x)xy′′(x)≈Y1Tψ(x)x(2ey+ey/2)≈Y2Tψ(x).Substitution into (50) and simplifying will be resulted to
(52)Y1T+2CTP+4Y2T=0.By solving the system (52), we have
(53)c10=-1.789983487,c11=1.607415602,c12=0.1463990297,c13=-0.04963748601,c14=0.04188454297,c15=-0.007221576268.The approximate solution ofy(x) is as follows:
(54)y(x)=CTP2ψ(x)=-0.1160867335x5+0.3967806389x4+0.4849181405x3-2.139040298x2+0.01441559362x-0.0003173071437.Table5 shows some values of the solutions and absolute errors at some x,s, and plots of the exact and approximate solutions are shown in Figure 4.Table 5
Numerical results of Example4.
x
Exact solution
Legendre wavelets
Absolute error
0.0
0
-
0.0003173071437
0.0003173071437
0.1
-
0.01990066171
-
0.01974271542
0.00015794629
0.2
-
0.07844142630
-
0.07851875395
0.00007732765
0.3
-
0.1723553925
-
0.1724816336
0.0001262411
0.4
-
0.2968400102
-
0.2967939002
0.0000461100
0.5
-
0.4462871026
-
0.4460837377
0.0002033649
0.6
-
0.6149693994
-
0.6145842735
0.0003851259
0.7
-
0.7975522400
-
0.7962728813
0.0012793587
0.8
-
0.9893924836
-
0.9850104864
0.0043819972
0.9
-
1.186653691
-
1.174680868
0.011972823
1
-
1.386294361
-
1.359329965
0.026964396Figure 4
The exact and LWM solution of Example4.Example 5.
Consider the following nonlinear Lane-Emden equation:(55)y′′(x)+8xy′(x)+xy(x)=x5-x4+44x2-30x,0<x≤1,
subject to the initial conditions(56)y(0)=0,y′(0)=0,
which has the following analytical solution:
(57)y(x)=x4-x3.We solve (55) by the method discussed in this paper with k=1and M=6. This implies that
(58)c10=0.9999999996,c11=1.732050807,c12=0.8944271910,c13=7.079278793×10-10,c14=-7.669124566×10-10,c15=9.994056888×10-10.The approximate solution ofy(x) is as follows:
(59)y(x)≈CTP2ψ(x)=-2.36323956×10-10x5+1.000000001x4-1.000000003x3+2×10-9x2-3.946836763×10-10x+5.809942853×10-12.Table6 shows that the Legendre wavelet solution is very near to the exact solution. Figure 5 shows that Legendre wavelet solution coincides with the exact solution.Table 6
Numerical results of Example5.
x
Exact solution
Legendre wavelets
Absolute error
0.0
0
5.809942853
×
1
0
-
12
5.8
×
1
0
-
12
0.1
-
0.0009
-
0.0009000000166
1.6
×
1
0
-
11
0.2
-
0.0064
-
0.006400000015
1.5
×
1
0
-
11
0.3
-
0.0189
-
0.01890000000
0
0.4
-
0.0384
-
0.03839999999
1
×
1
0
-
11
0.5
-
0.0625
-
0.06250000004
4
×
1
0
-
11
0.6
-
0.0864
-
0.08640000001
1
×
1
0
-
11
0.7
-
0.1029
-
0.1029000001
1
×
1
0
-
10
0.8
-
0.1024
-
0.1024000002
2
×
1
0
-
10
0.9
-
0.0729
-
0.07290000033
3.3
×
1
0
-
10
1
0
-
3.888737334
×
1
0
-
10
3.9
×
1
0
-
10Figure 5
The exact and LWM solution of Example5.


## 4. Conclusion

In this research, we have presented the Legendre wavelet method for solving nonlinear singular Lane-Emden equation. The Legendre wavelets operational matrix of integration is used to solve Lane-Emden equation. The present method reduces Lane-Emden equation into a set of algebraic equations. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique, and the results have been compared with the exact solution.



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*Source: 507145-2013-08-22.xml*