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Using an effective numerical method for solving a class of Lane-Emden equations.

1. Introduction

Let us consider the following Lane-Emden problem:

u"(x) +n/xu'(x) + f(u) = g(x), x[member of][0, b], u (0) = [[lambda].sub.1], u'(0) = [[lambda].sub.2], (1)

where n [greater than or equal to] 0, g(x) is given bounded, continuous function, and f(u) is nonlinear function; in [1], we can see that the most popular form of f(u) is f(u) = [u.sup.m], where m is a constant parameter; this type of equation is the Lane-Emden equations of the first kind; in addition, f(u) can be the exponential functions f(u) = [e.sup.u]; this type of equations is called the Lane-Emden equations of the second kind; furthermore, the function f(u) can be logarithmic functions and trigonometric functions; all these types of equations are named after the astrophysicists Jonathan Lane and Robert Emden; they were the first to study these types of equations. Lane-Emden equations are widely used in various physical phenomena. Many scholars [2-5] devote their energies to this field, with the high development of computer technology; lots of numerical methods have been put forward to solve this type of equation, such as pseudospectral method, Haar wavelet method, and Adomian decomposition method (ADM) [6-11].

Reproducing kernel method (RKM) is an attractive method because of its accuracy, and it has already been applied to various fields. In this paper, we use the reproducing kernel method to solve (1) to show the efficiency and accuracy of this method.

2. The Reproducing Kernel Method

2.1. Practise Homogenization for Lane-Emden Equations. In order to use reproducing kernel method to solve (1), we need to practise homogenization for (1); previously, we multiplied (1) by x; we find that

(Lu) (x) = F(x, u), x[member of][0, b], u (0) = [[lambda].sub.1], u'(0) = [[lambda].sub.2], (2)

where Lu(x) = xu"(x) + nu'(x), F(x, u) = xg(x) - xf(u). Obviously, the solution of (2) is the solution of (1). So we only need to gain the solution of (2). The question (2) with nonhomogeneous boundary value conditions is equivalent to the problem of having a function v(x) satisfying

(Lv) (x) = [bar.F] (x, v), x[member of][0, b], v (0) = 0, v'(0) = 0, (3)

where [bar.F](x, v) = xg(x) - xf(v + [[lambda].sub.1] + [[lambda].sub.2]x) - [[lambda].sub.2]n.

2.2. Construct Reproducing Kernel Space. Aiming at the purpose of solving (3), we need to introduce the reproducing kernel space; previously, let us introduce the concept of the reproducing kernel space.

For each of x [member of] X, there is a function of two variables [K.sub.x](y) [member of] H, where H is Hilbert space and X is a set abstraction. If we can get

<u (y), [K.sub.x] (y)> = u(x), u(y) [member of] H, (4)

we say that H is the reproducing kernel Hilbert space and [K.sub.x](y) is the reproducing kernel of H.

We give a linear space [W.sup.3.sub.2][0, b] as follows:

[W.sup.3.sub.2][0, b] = {u | u,u', u"

is one-variable absolutely continuous function,

u'"[member of] [L.sup.2][0, b], u(0) = 0, u'(0) = 0}. (5)

According to [12, 13], we give the inner product as follows:

<u (y), v (y)> = u"(0) v"(0) + [[integral].sup.b.sub.0]u'"(y) v'"(y) dy. (6)

And according to [14], we can prove that [W.sup.3.sub.2][0, b] is a reproducing kernel space; its reproducing kernel R(x, y) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

In order to use reproducing kernel method to solve (3) and referring to [15, 16], we can get [[psi].sub.i](x) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where i = 1, 2, 3, ....

Then practise Gram-Schmidt orthonormalization for [{[[psi].sub.i](x)}.sup.[infinity].sub.i=1]; according to [17, 18] we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [[beta].sub.ik] are coefficients of Gram-Schmidt orthonormalization. If [{xi}.sup.[infinity].sub.i=1] are distinct points dense in [0, b] and [L.sup.-1] is existent, we get that

u (x) = [[infinity].summation over (i=1)][i.summation over (k=1)][[beta].sub.ik][bar.F] ([x.sub.k], v ([x.sub.k])) [[bar.[psi]].sub.i](x) + [[lambda].sub.1] + [[lambda].sub.2]x (10)

is the solution of (3). The proof of it refers to [19, 20]. If the equations are linear ones, [bar.F](x, v) = [bar.F](x), we can solve the problems directly. If they are nonlinear equations, we have to use iteration method to solve them, and the specific methodology refers to [21, 22].

2.3. The Approximate Solution. We denote the approximate solution of [u.sup.m](x) by

[u.sup.m] (x) =[m.summation over (i=1)][i.summation over (k=1)][[beta].sub.ik]F ([x.sub.k], v ([x.sub.k])) [[bar.[psi]].sub.i](x) + [[lambda].sub.1] + [[lambda].sub.2]x. (11)

According to the proof of [23] we can easily get that [parallel][u.sup.m](x) - u(x)[parallel] [right arrow] 0 and [u.sup.(k).sub.m](x) [right arrow] [u.sup.(k)](x), k = 0, 1, 2.

3. Numerical Experiment

Example 1. Let us talk about the well-known polytropic differential equation in [3]. Consider

u"+2/xu' + [u.sup.5](x) = 0, x [member of] [0, 1], u (0) = 1, u'(0) = 0, (12)

whose exact solution is given by [u.sub.T](x) = [(1 + [x.sup.2]/3).sup.-1/2]; using the reproducing kernel method, [x.sub.i] = ih, h = 1/N, i = 1, 2, ..., N, and N = 50. The numerical results are shown in Figure 1 and Table 1.

Example 2. Considering the following nonlinear equation:

u"(x) + 1/xu'(x) - [u.sup.3](x) + 3[u.sup.5](x) = 0, x [member of] [0, 1], u (0) = 1, u'(0) = 0, (13)

the exact solution is given by [u.sub.T](x) = 1/[square root of 1 + [x.sup.2]]; using the reproducing kernel method, [x.sub.i] = ih,h = 1/N, i = 1, 2, ..., N, and N = 50. The numerical results are shown in Figure 2 and Table 2.

Example 3. Consider a linear Lane-Emden equation:

u"(x) +1/xu'(x) + u(x) = g(x), x[member of][0, 1], u (0) = 0, u'(0) = 0, (14)

where g(x) = 4 - 9x + [x.sup.2]- [x.sup.3]; using the reproducing kernel method, [x.sub.i] = ih, h = 1/N, i = 1, 2, ..., N, and N = 50, 100, 150. The exact solution is given by [u.sub.T](x) = [x.sup.2]- [x.sup.3]. The numerical results are presented in Figure 3 and Table 3.

Example 4. Consider a Lane-Emden equation of the second kind in [1]. One has

u"(x) + n/xu'(x) + [e.sup.u](x)= 0, n [greater than or equal to] 0, u (0) = 0, u'(0) = 0, (15)

where n = 0; using the reproducing kernel method, [x.sub.i] = ih, h = 1/N, i = 1, 2, ..., N, and N = 50. The exact solution is given by [u.sub.T](x) = 2Ln(sech(x/[square root of 2)). The numerical results are presented in Figure 4 and Table 4.

Example 5. Consider a Lane-Emden equation in [3]. One has

u"(x) + 2/xu'(x) + 4(2[e.sup.u](x)+ [e.sup.u](x)/2) = 0, x [greater than or equal to] 0, u (0) = 0, u'(0) = 0.(16)

Using the reproducing kernel method, [x.sub.i] = ih, h = 1/N, i = 1, 2, ..., N, and N = 50. The exact solution is given by [u.sub.T](x) = -2Ln(1 + [x.sup.2]). The numerical results are presented in Figure 5 and Table 5.

4. Conclusions and Remarks

In this paper, reproducing kernel method has been used to solve some typical Lane-Emden examples; the computation implies that the solutions by the reproducing kernel method are very accurate. Moreover, the first and second derivatives of the solutions also have very high accuracy. From all of this, we can affirm that the reproducing kernel method is an efficient and accurate method. All computations are performed by the Mathematica 8.0 software package.

http://dx.doi.org/10.1155/2014/735831

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the reviewers for their valuable suggestions, which greatly improved the quality of the paper. This paper is supported by the Natural Science Foundation of China (no. 11361037), the Natural Science Foundation of Inner Mongolia (no. 2013MS0109), and Project Application Technology Research and Development Foundation of Inner Mongolia (no. 20120312).

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Yulan Wang, (1) Hao Yu, (1) Fugui Tan, (2) and Shuguang Li (1)

(1) Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China

(2) Jining Teachers College, Wulanchabu, Inner Mongolia 012000, China

Correspondence should be addressed to YulanWang; wylnei@163.com

Received 26 March 2014; Revised 12 May 2014; Accepted 22 May 2014; Published 18 June 2014

Academic Editor: Igor Leite Freire

TABLE 1: Numerical solutions for Example 1.

x      [u.sub.T](x)    [u.sub.50]      [absolute value of
                                         [u.sub.T](x) -
                                         [u.sub.50](x)]

0.0          1              1                  0
0.1      0.998337       0.998338     3.37916 x [10.sup.-7]
0.2      0.993399        0.9934      3.57116 x [10.sup.-7]
0.3      0.985329        0.98533      420119 x [10.sup.-7]
0.4      0.974355       0.974355     5.53188 x [10.sup.-7]
0.5      0.960769        0.96077     7.64858 x [10.sup.-7]
0.6      0.944911       0.944912     1.04891 x [10.sup.-6]
0.7      0.927146       0.927147     1.39484 x [10.sup.-6]
0.8      0.907841       0.907843     1.81934 x [10.sup.-6]
0.9      0.887357       0.887359      2.455 x [10.sup.-6]
1.0      0.866025       0.866029     3.77266 x [10.sup.-6]

x        [absolute value of       [absolute value of
          [u'.sub.T](x) -          [u".sub.T] (x) -
          [u'.sub.50](x)]          [u".sub.50](x)]

0.0              0                    0.00298864
0.1    1.12915 x [10.sup.-7]    3.93668 x [10.sup.-6]
0.2    3.49412 x [10.sup.-7]     5.2331 x [10.sup.-6]
0.3    9.49753 x [10.sup.-7]    8.31208 x [10.sup.-6]
0.4    1.72224 x [10.sup.-6]    1.10979 x [10.sup.-5]
0.5    2.49738 x [10.sup.-6]    1.31351 x [10.sup.-5]
0.6    3.16098 x [10.sup.-6]    1.37757 x [10.sup.-5]
0.7    3.77284 x [10.sup.-6]    1.06634 x [10.sup.-5]
0.8     4.9052 x [10.sup.-6]    3.95915 x [10.sup.-6]
0.9    8.51793 x [10.sup.-6]    5.08462 x [10.sup.-5]
1.0    1.98432 x [10.sup.-5]    1.76573 x [10.sup.-4]

TABLE 2: Numerical solutions for Example 2.

x      [u.sub.T](x)    [u.sub.50]      [absolute value of
                                         [u.sub.T](x) -
                                         [u.sub.50](x)]

0.0          1              1                  0
0.1      0.995037        0.99504     2.93318 x [10.sup.-6]
0.2      0.980581       0.980585     4.30454 x [10.sup.-6]
0.3      0.957826       0.957832     6.15161 x [10.sup.-6]
0.4      0.928477       0.928485     8.27303 x [10.sup.-6]
0.5      0.894427       0.894438     1.03314 x [10.sup.-6]
0.6      0.857493       0.857506     1.34827 x [10.sup.-5]
0.7      0.819232       0.819258     2.59653 x [10.sup.-5]
0.8      0.780869       0.780948     7.95391 x [10.sup.-5]
0.9      0.743294        0.74356     2.66219 x [10.sup.-4]
1.0      0.707107       0.707909     8.02319 x [10.sup.-4]

x        [absolute value of      [absolute value of
          [u'.sub.T](x) -          [u".sub.T](x) -
          [u'.sub.50](x)]          [u".sub.50](x)]

0.0              0                    0.0112026
0.1    1.29666 x [10.sup.-5]    1.64099 x [10.sup.-4]
0.2    1.58583 x [10.sup.-5]    1.26869 x [10.sup.-4]
0.3    2.06044 x [10.sup.-5]     1.3056 x [10.sup.-4]
0.4    2.10852 x [10.sup.-5]    1.15115 x [10.sup.-4]
0.5    2.12716 x [10.sup.-5]    1.93154 x [10.sup.-5]
0.6    5.30101 x [10.sup.-5]     7.5356 x [10.sup.-4]
0.7    2.43656 x [10.sup.-4]     3.6414 x [10.sup.-3]
0.8    9.70462 x [10.sup.-4]    1.22923 x [10.sup.-2]
0.9    3.10842 x [10.sup.-3]    3.31332 x [10.sup.-2]
1.0    8.31259 x [10.sup.-3]    7.52151 x [10.sup.-2]

TABLE 3: Numerical solutions for Example 3.

x          Exact        50-order RKM
         solution      approximation
       [u.sub.T](x)    [u.sub.50](x)

0.0          0               0
0.1        0.009         0.00886466
0.2        0.032         0.0318207
0.3        0.063         0.0627957
0.4        0.096         0.0957782
0.5        0.125          0.124764
0.6        0.144          0.143752
0.7        0.147          0.146741
0.8        0.128          0.12773
0.9        0.081         0.0807176
1.0          0          -0.000295352

x      100-order RKM     150-order RKM
       approximation     approximation
       [u.sub.100] M    [u.sub.150](x)

0.0          0                 0
0.1      0.00897728       0.00899228
0.2      0.0319718         0.0319906
0.3      0.0629686         0.0629896
0.4      0.0959661         0.0959888
0.5       0.124964         0.124988
0.6       0.143962         0.143987
0.7       0.14696          0.146986
0.8       0.127957         0.127985
0.9      0.0809539         0.0809837
1.0    -0.0000496372     -0.0000179019

TABLE 4: Numerical solutions for Example 4.

x      [u.sub.T](x)    [u.sub.50]      [absolute value of
                                         [u.sub.T](x) -
                                         [u.sub.50](x)]

0.0          0              0                  0
0.1     0.00499584     0.00497722    1.86175 x [10.sup.-5]
0.2      0.0199337      0.0198951    3.85998 x [10.sup.-5]
0.3      0.0446665      0.0446089    5.76023 x [10.sup.-5]
0.4      0.0789556      0.0788834    7.21411 x [10.sup.-5]
0.5      0.122479       0.122407     7.29154 x [10.sup.-5]
0.6      0.174846       0.174807     3.90139 x [10.sup.-5]
0.7      0.235604       0.235676     7.16611 x [10.sup.-5]
0.8      0.304261       0.304599     3.37896 x [10.sup.-4]
0.9      0.380294       0.381194     8.99901 x [10.sup.-4]
1.0      0.463163       0.465161     1.99826 x [10.sup.-3]

x        [absolute value of       [absolute value of
          [u'.sub.T](x) -          [u".sub.T](x) -
          [u'.sub.50](x)]          [u".sub.50](x)]

0.0              0                    0.0198013
0.1    2.00342 x [10.sup.-4]    2.32373 x [10.sup.-5]
0.2    1.97876 x [10.sup.-4]    1.09236 x [10.sup.-4]
0.3    1.76911 x [10.sup.-4]    4.26393 x [10.sup.-4]
0.4     9.9302 x [10.sup.-5]     1.3039 x [10.sup.-3]
0.5    1.17118 x [10.sup.-4]    3.30994 x [10.sup.-3]
0.6    6.28561 x [10.sup.-4]    7.38437 x [10.sup.-3]
0.7    1.71224 x [10.sup.-3]    1.50496 x [10.sup.-2]
0.8    3.83982 x [10.sup.-3]    2.87368 x [10.sup.-2]
0.9     7.7914 x [10.sup.-3]     5.2279 x [10.sup.-2]
1.0    1.48297 x [10.sup.-2]    9.16336 x [10.sup.-2]

TABLE 5: Numerical solutions for Example 5.

        Exact solution     50-order RKM    The approximation
         [u.sub.T](x)     approximation     solutions in [3]
                          [u.sub.50](x)       n(x) in [3]

0.0            0                0                  0
0.01     -0.0001999900     -0.00019999       -0.0001970587
0.1      -0.0199006617      -0.0199007       -0.0198967225
0.5      -0.4462871026      -0.446287        -0.4462840851
1.0      -1.3862943611       -1.38629        -1.3862934297
2.0      -3.2188758249       -3.21888        -3.2188763248
3.0      -4.6051701860       -4.60517        -4.6051709964
4.0      -5.6664266881       -5.66643        -5.6664274573
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Title Annotation:Research Article
Author:Wang, Yulan; Yu, Hao; Tan, Fugui; Li, Shuguang
Publication:Abstract and Applied Analysis
Article Type:Report
Date:Jan 1, 2014
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