Mathematical Methods and Models in the Natural to the Life Sciences
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Using an Effective Numerical Method for Solving a Class of LaneEmden Equations
Abstract
We use the reproducing kernel method to solve the wellknown classes of LaneEmdentype equations. These classes of equations have the form of LaneEmden problem. Comparing the results of the reproducing kernel method with the analytical solutions by means of some typical examples, we can affirm that the reproducing kernel method is an efficient and accurate method.
1. Introduction
Let us consider the following LaneEmden problem: where , is given bounded, continuous function, and is nonlinear function; in [1], we can see that the most popular form of is , where is a constant parameter; this type of equation is the LaneEmden equations of the first kind; in addition, can be the exponential functions ; this type of equations is called the LaneEmden equations of the second kind; furthermore, the function 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. LaneEmden 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 LaneEmden Equations
In order to use reproducing kernel method to solve (1), we need to practise homogenization for (1); previously, we multiplied (1) by ; we find that where . 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 satisfying where .
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 , there is a function of two variables , where is Hilbert space and is a set abstraction. If we can get we say that is the reproducing kernel Hilbert space and is the reproducing kernel of .
We give a linear space as follows: According to [12, 13], we give the inner product as follows: And according to [14], we can prove that is a reproducing kernel space; its reproducing kernel is In order to use reproducing kernel method to solve (3) and referring to [15, 16], we can get as follows: where .
Then practise GramSchmidt orthonormalization for ; according to [17, 18] we get where are coefficients of GramSchmidt orthonormalization.
If are distinct points dense in and is existent, we get that is the solution of (3). The proof of it refers to [19, 20]. If the equations are linear ones, , 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 by According to the proof of [23] we can easily get that and .
3. Numerical Experiment
Example 1. Let us talk about the wellknown polytropic differential equation in [3]. Consider whose exact solution is given by ; using the reproducing kernel method, , , , and . The numerical results are shown in Figure 1 and Table 1.

Example 2. Considering the following nonlinear equation: the exact solution is given by ; using the reproducing kernel method, , , , and . The numerical results are shown in Figure 2 and Table 2.

Example 3. Consider a linear LaneEmden equation: where ; using the reproducing kernel method, , , , and . The exact solution is given by . The numerical results are presented in Figure 3 and Table 3.

Example 4. Consider a LaneEmden equation of the second kind in [1]. One has where ; using the reproducing kernel method, , , , and . The exact solution is given by . The numerical results are presented in Figure 4 and Table 4.

Example 5. Consider a LaneEmden equation in [3]. One has Using the reproducing kernel method, , , , and . The exact solution is given by . 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 LaneEmden 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.
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).
References
 A.M. Wazwaz, R. Rach, and J.S. Duan, “Adomian decomposition method for solving the Volterra integral form of the LaneEmden equations with initial values and boundary conditions,” Applied Mathematics and Computation, vol. 219, no. 10, pp. 5004–5019, 2013. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. Dehghan, S. Aryanmehr, and M. R. Eslahchi, “A technique for the numerical solution of initialvalue problems based on a class of Birkhofftype interpolation method,” Journal of Computational and Applied Mathematics, vol. 244, pp. 125–139, 2013. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 K. Parand, M. Dehghan, A. R. Rezaei, and S. M. Ghaderi, “An approximation algorithm for the solution of the nonlinear LaneEmden type equations arising in astrophysics using Hermite functions collocation method,” Computer Physics Communications, vol. 181, no. 6, pp. 1096–1108, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. Harley and E. Momoniat, “Instability of invariant boundary conditions of a generalized LaneEmden equation of the secondkind,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 621–633, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 E. Momoniat and C. Harley, “An implicit series solution for a boundary value problem modelling a thermal explosion,” Mathematical and Computer Modelling, vol. 53, no. 12, pp. 249–260, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A.M. Wazwaz, “The combined Laplace transformAdomian decomposition method for handling nonlinear Volterra integrodifferential equations,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1304–1309, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J.S. Duan, R. Rach, A.M. Wazwaz, T. Chaolu, and Z. Wang, “A new modified Adomian decomposition method and its multistage form for solving nonlinear boundary value problems with Robin boundary conditions,” Applied Mathematical Modelling, vol. 37, no. 2021, pp. 8687–8708, 2013. View at: Publisher Site  Google Scholar
 J. S. Duan, R. Rach, and A. Wazwaz, “Solution of the model of beamtype micro and nanoscale electrostatic actuators by a new modified Adomian decomposition method for nonlinear boundary value problems,” International Journal of NonLinear Mechanics, vol. 49, pp. 159–169, 2013. View at: Publisher Site  Google Scholar
 M. M. AlSawalha, M. S. M. Noorani, and I. Hashim, “Numerical experiments on the hyperchaotic Chen system by the Adomian decomposition method,” International Journal of Computational Methods, vol. 5, no. 3, pp. 403–412, 2008. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. Ghosh, A. Roy, and D. Roy, “An adaptation of Adomian decomposition for numericanalytic integration of strongly nonlinear and chaotic oscillators,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 4–6, pp. 1133–1153, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. Y. Al Bayati, A. J. AL Sawoor, and M. A. Samarji, “A multistage Adomian decomposition method for solving the autonomous Van Der Pol system,” Australian Journal of Basic and Applied Sciences, vol. 3, no. 4, pp. 4397–4407, 2009. View at: Google Scholar
 Y. L. Wang, X. J. Cao, and X. Li, “A new method for solving singular fourthorder boundary value problems with mixed boundary conditions,” Applied Mathematics and Computation, vol. 217, no. 18, pp. 7385–7390, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Y. L. Wang, M. J. Du, F. Tan, Z. Li, and T. Nie, “Using reproducing kernel for solving a class of fractional partial differential equation with nonclassical conditions,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 5918–5925, 2013. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Li, Y. Wang, F. Tan, X. Wan, and T. Nie, “The solution of a class of singularly perturbed twopoint boundary value problems by the iterative reproducing kernel method,” Abstract and Applied Analysis, vol. 2012, Article ID 984057, 7 pages, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Y. L. Wang, S. Lu, F. Tan, M. Du, and H. Yu, “Solving a class of singular fifthorder boundary value problems using reproducing kernel Hilbert space method,” Abstract and Applied Analysis, vol. 2013, Article ID 925192, 6 pages, 2013. View at: Publisher Site  Google Scholar
 Z. Y. Li, Y. L. Wang, and F. Tan, “The solution of a class of thirdorder boundary value problems by the reproducing kernel method,” Abstract and Applied Analysis, vol. 2012, Article ID 195310, 11 pages, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X. Y. Li, B. Y. Wu, and R. T. Wang, “Reproducing kernel method for fractional riccati differential equations,” Abstract and Applied Analysis, vol. 2014, Article ID 970967, 6 pages, 2014. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 F. Z. Geng and S. P. Qian, “Solving singularly perturbed multipantograph delay equations based on the reproducing kernel method,” Abstract and Applied Analysis, vol. 2014, Article ID 794716, 6 pages, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 L.H. Yang, H.Y. Li, and J.R. Wang, “Solving a system of linear Volterra integral equations using the modified reproducing kernel method,” Abstract and Applied Analysis, vol. 2013, Article ID 196308, 5 pages, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 L. H. Yang and M. G. Cui, “New algorithm for a class of nonlinear integrodifferential equations in the reproducing kernel space,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 942–960, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X. Q. Lv and M. G. Cui, “An efficient computational method for linear fifthorder twopoint boundary value problems,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1551–1558, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 M. G. Cui and Z. Chen, “The exact solution of nonlinear agestructured population model,” Nonlinear Analysis: Real World Applications, vol. 8, no. 4, pp. 1096–1112, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 J. N. Niu, Y. Z. Lin, and M. G. Cui, “A novel approach to calculation of reproducing kernel on infinite interval and applications to boundary value problems,” Abstract and Applied Analysis, vol. 2013, Article ID 959346, 7 pages, 2013. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
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Copyright © 2014 Yulan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.