Research Article | Open Access
Osman Rasit Isik, Mehmet Sezer, "Bernstein Series Solution of a Class of Lane-Emden Type Equations", Mathematical Problems in Engineering, vol. 2013, Article ID 423797, 9 pages, 2013. https://doi.org/10.1155/2013/423797
Bernstein Series Solution of a Class of Lane-Emden Type Equations
The purpose of this study is to present an approximate solution that depends on collocation points and Bernstein polynomials for a class of Lane-Emden type equations with mixed conditions. The method is given with some priori error estimate. Even the exact solution is unknown, an upper bound based on the regularity of the exact solution will be obtained. By using the residual correction procedure, the absolute error can be estimated. Also, one can specify the optimal truncation limit which gives a better result in any norm. Finally, the effectiveness of the method is illustrated by some numerical experiments. Numerical results are consistent with the theoretical results.
Lane-Emden type equation that is presented in (1) models many phenomena in mathematical physics and astrophysics [1, 2]. Consider It describes the equilibrium density distribution in self-gravitating sphere of polytrophic isothermal gas . On the other hand , it plays an important role in various fields such as stellar structure , radiative cooling, and modeling of clusters of galaxies. It is a nonlinear ordinary differential equation that has a singularity at the origin. In the neighborhood of , it has an analytic solution . It is labeled by the names of the astrophysicists Lane  and Robert Emden.
In this paper, a class of Lane-Emden equations  is considered in the type of with the mixed conditions where and are functions defined on and , , , and are real constants. We will find an approximate solution, namely, Bernstein series solution, of (2) as such that satisfies (2) on the collocation nodes . Here, , , are Bernstein polynomials.
1.1. Recent Works
Recently, a number of numerical methods are used for handling the Lane-Emden type problems based on perturbation techniques or series solutions. Adomian decomposition method [6, 7] which provides a convergent series solution has been used to solve (1) [8–10]. Wazwaz  gave an algorithm to overcome the difficulty of the singular point in using Adomian decomposition method .
The quasilinearization method [11–13] can be considered as an example for iteration methods. Its fast convergence, monotonicity, and numerical stability were analyzed by Krivec and Mandelzweig . They verified this method on scattering length calculations in the variable phase approach to quantum mechanics. They also showed that the iterations converge uniformly and quadratically to the exact solution. The method gives accurate and stable answers for any coupling strengths, including super singular potentials for which each term of the perturbation theory diverges.
The Legendre wavelet method was given by Yousefi  to solve Lane-Emden equation. This method was used to convert Lane-Emden equations to integral equations and was expanded the solution by Legendre wavelets with unknown coefficients.
Ramos  applied a piecewise linearization method to solve the Lane-Emden equation. This method provided piecewise linear ordinary differential equations that can be easily integrated. Furthermore, it has given accurate results for hypersingular potentials, for which perturbation methods diverge. Homotopy analysis method (HAM) and modified HAM have also been used [16, 17] to solve (1). Parand et al.  proposed a collocation method based on a Hermite function collocation (HFC) method for solving some classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. A matrix method was given by Yuzbasi for solving nonlinear Lane-Emden type equations. Moreover, Yuzbasi and Sezer  applied a matrix method that depends on Bessel polynomials to solve (2). They estimated the absolute errors by using the residual correction procedure. In this study, a similar method to  was constructed. In addition, error analysis of the matrix method was developed.
In 2012, Pandey and coworkers [19–22] studied five methods. First, Pandey et al.  gave a numerical method for solving linear and nonlinear Lane-Emden type equations using Legendre operational matrix of differentiation. Second, Pandey et al.  studied a numerical method to solve linear and nonlinear Lane-Emden type equations using Chebyshev wavelet operational matrix. Third, Kumar et al.  presented a method for linear and nonlinear Lane-Emden type equations using the Bernstein polynomial operational matrix of integration. Fourth, Pandey and Kumar  proposed a numerical method for solving Lane-Emden type equations arising in astrophysics using Bernstein polynomials. This method is similar to the method used in the present study. And finally, a shifted Jacobi-Gauss collocation spectral method was proposed by Bhrawy and Alofi  for solving the nonlinear Lane-Emden type equation.
This paper is organized as follows. In Section 2, some definitions and theorems are given. The method is presented in Section 3. First, a matrix form for each term in (2) is found. Substituting these matrix forms into (2) gives a matrix equation, fundamental matrix equation. Then, a linear system by using collocation points is obtained. For the error analysis, in Section 4, some theorems that give some upper bounds for the absolute errors are presented. One of them guarantees the convergence if the solution is polynomial. The second one gives an upper bound in the case of the exact solution being unknown under the regularity condition. The residual correction procedure to estimate the absolute errors is also given so that the optimal truncation limit can be specified. On the other hand, this procedure gives a new approximate solution. Some numerical examples are given to illustrate the method.
Bernstein polynomials of th-degree are defined by where is the maximum range of the interval over which the polynomials are defined to form a complete basis .
We substitute the relation into (5) and obtain the relation
Let us consider pairs . The problem is to find a polynomial , called interpolating polynomial, such that The points are called interpolation nodes. If , the problem is over- or underdetermined.
Theorem 1 (see ). Given distinct nodes and corresponding values , then there exists a unique polynomial such that for .
Theorem 2 (see ). Let be distinct nodes, and let be a point belonging to the domain of a given function . Assume that , where is the smallest interval containing the nodes and . Then, the interpolation error at the point is given by where .
Let us denote the interpolation polynomial of by . Lagrange characteristic polynomials are defined as Thus, can be written the following form, Lagrange form:
Hermite interpolation polynomial of on is defined as follows . Suppose that are given data, with , , and . If is selected as and interpolation nodes are distinct, there exist a unique polynomial such that of the form where are the Hermite characteristic polynomials defined by Letting for , they satisfied the following recursive formula: where If , the interpolation error is given as follows: where .
The interpolation error may be reduced by using the roots of Chebyshev polynomials
3. Fundamental Relations
Let be Bernstein series solution of (2). Let us find the matrix forms of and . can be written as where Therefore, can be written as On the other hand, can be written as [26–28] where For , the relation is obtained where Substituting (24) into (22) yields Putting (26) into (19) yields the matrix form for as By substituting (19) and (27) into (2), we obtain a matrix equation as By using the collocation points in (28), one obtains the fundamental matrix equation
On the other hand, the matrix forms for the conditions can be written as where Replacing the condition matrices (31) by any two rows of , we get the augmented matrix as . Let the collocation points be selected such that the rank of is . Therefore, the unknown matrix is obtained as
4. Error Analysis and Estimation of the Absolute Error
In this section, some upper bounds of the absolute error are given by using Lagrange and Hermite interpolation polynomials. Also, an estimation of the error based on residual correction is given.
Theorem 3 (see ). Let be a nonsingular matrix and a vector. If and are, respectively, the solutions of the systems and , one has
Let be the exact solution of (2) and the interpolation polynomial of it on the nodes . If , then we can write as , where If is the Bernstein series solution of (2), then it satisfies (2) on the nodes. So, and are the solutions of and , respectively, where
Theorem 4. Let and be the Bernstein series solution and the exact solution of (2), respectively, and the interpolation polynomial of . Let be the function and the matrix which are defined earlier.If , then
Proof. Adding and subtracting gives thefollowing by triangle inequality: Since , the first term on the right hand side is bounded by Theorem 2. For the second term, by using Theorem 3 and properties of norm with (22), we get
Corollary 5. If the exact solution of (2) is a polynomial, then the method gives the exact solution for .
Proof. Since the exact solution is polynomial, for , ; the right hand side of (37) is zero.
The following theorem can be used for the estimation of the absolute error when the exact solution is unknown. Hence, an upper bound depending on is obtained under the condition for . It is well-known that if , then is bounded on .
Theorem 6. Let and be Bernstein series solution and the exact solution of (2), respectively. Let the interpolation nodes contain and . Let and be the Hermite interpolation polynomial of on the nodes . Then, the error function is bounded by where and .
Proof. Adding and subtracting the polynomials with triangle inequality yields The first term on the right hand side can be bounded by (17) since.
If the exact solution is unknown, the following steps can be used to find an upper bund of the absolute error. First, we construct the differential equation of . If is the Hermite interpolation polynomial of on the nodes , then satisfies the following differential equation: with the conditions Since is a polynomial, the method gives the exact solution by Corollary 5 under the condition . Thus, is obtained by finding Bernstein series solution of (42) so that an upper bound of the error is obtained depending on .
The following procedure, residual correction (e.g., see, [30–32]), can be given for the estimation of the absolute error. Moreover, one can estimate the optimal giving minimal absolute error using this procedure. The procedure is basic.
First, adding and subtracting the term to (2) yields the following differential equation, which admits as an exact solution: with the conditions Let be Bernstein series solution to (45). If is sufficiently small, the absolute error can be estimated by . Hence, the optimal for the absolute error can be obtained measuring the error functions for different values in any norm.
Note that the approximate solution is a better approximation than in the norm for . Let us call the approximate solution as corrected Bernstein series solution.
5. Numerical Examples
In this section, some numerical examples are given to illustrate the method. Some examples are given with their error estimation by using Theorem 4. Moreover, for these examples, the -norms of the error function , the estimate error function , and the absolute error of the corrected Bernstein series solution given in Corollary 7 are calculated for some and . The optimal truncation limit is specified for each example. All calculations are done in Maple . Since is a singular point, the equidistant nodes are selected as .
Example 8. Consider the Lane-Emden equation Applying the method for on the equidistant nodes, Bernstein series solution is obtained as which is the exact solution .
Example 9. Let us consider the equation with the boundary conditions and . The exact solution of (49) is  For different values , the norms and the upper bounds of the absolute errors are obtained on the equidistant nodes by using Theorem 4. Also, estimations of the absolute errors for and the norms of the absolute errors for corrected Bernstein series solutions, , are calculated on the Chebyshev nodes. All results are given in Table 1. The absolute error function for and the estimation of the error function, , are plotted in Figure 1. As seen from Table 1, the optimal truncation limit is specified as , which gives us the best approximation from the set . Moreover, the expected upper bounds are consistent with the absolute errors. Adding to yields the better results in -norm for .
Example 10. Let us consider the Lane-Emden equation having as exact solution [14, 18, 33]. Assuming that the exact solution is unknown and , an upper bound depending on is obtained by Theorem 6. The errors for and are given in Tables 2 and 3, respectively. To obtain and , the equidistant nodes and the Chebyshev collocation nodes are used, respectively. Here, and are the Hermite interpolation polynomials on the sets and , respectively. and represent the values of and in -norms, respectively. By calculating and using Theorem 6, the upper bounds of the absolute errors on the equidistant nodes are given in Table 4 by comparison with the absolute error. As seen from the table, these upper bounds bound the absolute error on some reference points. In Table 5, a comparison between Bernstein series solutions for and the approximate solution obtained by the Hermite functions collocation (HFC) method  for , and is given. The results are as follows.
Example 11. Let us consider the equation with the boundary conditions and . The exact solution of (49) is [14, 33] For different values and , the norms of the absolute errors, the estimations of the absolute errors, and the corrected absolute errors are obtained on the equidistant nodes and given in Table 6. As seen from Table 6, for , corrected absolute errors are better than the absolute errors. Moreover, residual correction procedure estimates the absolute errors accurately.