A Representation of the Exact Solution of Generalized Lane-Emden Equations Using a New Analytical Method

and Applied Analysis 3


Introduction
Since the beginning of stellar astrophysics, the investigation of stellar structures has been a central problem. There have been continuous efforts to deduce the radial profiles of the pressure, density, and mass of a star, and one of the key results that came out of these efforts is the Lane-Emden equation, which describes the density profile of a gaseous star. Mathematically, the Lane-Emden equation is a second-order singular ordinary differential equation. In astrophysics, the Lane-Emden equation is essentially a Poisson equation for the gravitational potential of a self-gravitating, spherically symmetric polytropic fluid.
The Lane-Emden equation has been used to model several phenomena in mathematical physics, thermodynamics, fluid mechanics, and astrophysics, such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas spheres, and the theory of thermionic currents [9][10][11][12][13]. Lane-Emden-type equations were first published by Lane [14]; they were explored in more detail by Emden in 1870 [15], who considered the thermal behavior of a spherical cloud of gas that acts under the mutual attraction of its molecules and is subject to the classical laws of thermodynamics. The reader is kindly requested to peruse [9][10][11][12][13][14][15][16][17] to know more details about Lane-Emden-type equations, including their history, variations, and applications.
In the present paper, we introduce a simple new analytical method we call the residual-power-series (RPS) method [18] to discover series solutions to linear and nonlinear Lane-Emden equations. The RPS method is effective and easy to use to solve Lane-Emden equations without linearization, perturbation, or discretization. This method constructs an approximate analytical solution in the form of a polynomial. By using the concept of residual error, we obtain a series solution, which in practice tends to be a truncated series solution.
The RPS method has the following characteristics [18]: first, it obtains a Taylor expansion of the solution, and as a result, the exact solution is obtained whenever it is a polynomial. Moreover, the solutions and all of its derivatives are applicable for each arbitrary point in a given interval. Second, the RPS method has small computational requirements and high precision, and furthermore it requires less time.
In the present paper, the RPS method is used to obtain a symbolic approximate solution for generalized Lane-Emden 2 Abstract and Applied Analysis equations in the following form that are assumed to have a unique solution in the interval of integration: which is subject to both the initial conditions, and one of the following constraint-conditions cases: where , , ℎ are nonlinear analytic functions, , are analytic functions on [ 0 , 0 + ], ( ) is an unknown function of an independent variable that is to be determined, and 0 , , ∈ R with > 0. Throughout this paper, we assume that ( ) is an analytic function on the given interval.
In most cases, the Lane-Emden equation does not always have solutions that can be obtained using analytical methods. In fact, many of real physical and engineering phenomena that are encountered are almost impossible to solve by this technique; hence, these problems must be attacked by various approximate and numerical methods. Therefore, some authors have proposed numerical methods to approximate the solutions of a special case of (1) and (2). For example, the Adomian decomposition method has been applied to solve the Lane-Emden equation ( ) + ( / ) ( ) + ( ( )) + ( ) = 0 as described in [3]. In [1], the authors developed the optimal homotopy asymptotic method to solve the singular equation ( ) + ( / ) ( ) + ( ( )) + ℎ( ) = 0. Additionally, in [2], the authors provided the Hermite functions collocation method to further investigate the Lane-Emden equation ( ) + ( / ) ( ) + ( ) ( ( )) + ( ) = 0. Furthermore, the homotopy perturbation method is carried out in [4]  However, none of the previous studies propose a methodical way to solve (1) and (2). Moreover, the previous studies require more effort to achieve their results, and usually they are only suited for a special form of (1) and (2). However, the applications of other versions of series solutions to linear and nonlinear problems can be found in [20][21][22][23][24][25], and, to discern the numerical solvability of different categories of singular differential equations, one can consult [26].
The outline of the paper is as follows: in the next section, we present the formulation of the RPS method. Section 3 covers the convergence theorem. In Section 4, numerical examples are given to illustrate the capability of the proposed method. This paper ends in Section 5 with some concluding remarks.

The Formulation of the RPS Method
In this section, we employ the RPS method to find a series solution to the generalized Lane-Emden equation (1) that is subject to given initial conditions equation (2). First, we formulate and analyze the RPS method to solve such problems.

Convergence Theorem and Error Analysis
In this section, we study the convergence of the present method to capture the behavior of the solution. Afterwards, error functions are introduced to study the accuracy and efficiency of the method. Actually, continuous approximations to the solution will be obtained.
Taylor's theorem allows us to represent fairly general functions exactly in terms of polynomials with a known, specified, and bounded error. The next theorem will guarantee convergence to the exact analytic solution of (1) and (2). (1) and (2). Then the approximate solution obtained by the RPS method is in fact the Taylor expansion of ( ).

Theorem 1. Suppose that ( ) is the exact solution for
Proof. Assume that the approximate solution for (1) and (2) is as follows: In order to prove the theorem, it is enough to show that the coefficients in (11) take the form where ( ) is the exact solution for (1) and (2). Clearly, for = 0 and = 1 the initial conditions (2) give 0 = ( 0 ) and 1 = ( 0 ), respectively. Moreover, for = 2, differentiate both sides of (4) with respect to and substitute = 0 to obtain Indeed, from (11), one can writẽ By substituting (14) into (4), differentiating both sides of the resulting equation with respect to , and then setting = 0 , we obtain By comparing (13) and (15), it easy to see that 2 = (1/2!) ( 0 ). Hence, according to (14) the approximation for (1) and (2) is Furthermore, for = 3, differentiating both sides of (4) twice with respect to and then substituting = 0 yields the following result: By substituting (16) into (4), differentiating both sides of the resulting equation twice with respect to , and then setting = 0 , we obtain By comparing (17) and (18), we can conclude that 3 = (1/3!) ( 0 ). Thus, according to (16), we can write the approximation for (1) and (2) as By continuing the above procedure, we can easily prove (13) for = 4, 5, 6, . . .. Thus, the proof of the theorem is complete.

Corollary 2.
If ( ) is a polynomial, then the RPS method will obtain the exact solution.
It will be convenient to have a notation for the error in the approximation ( ) ≈ ( ). Accordingly, let Rem ( ) denote the difference between ( ) and its th Taylor polynomial, which is obtained from the RPS method; that is, let The functions Rem ( ) are called the th remainder of the RPS approximation of ( ). In fact, it often happens that the remainders Rem ( ) become smaller and approach zero as approaches infinity. The concept of "accuracy" refers to how closely a computed or measured value agrees with the true value. To show the accuracy of the present method, we report three types of error functions. The first one is the exact error, Ext, which is defined as follows: Abstract and Applied Analysis 5 Similarly, the consecutive error, which is denoted by Con, and the residual error, which is denoted by Res, are defined by respectively, where ∈ [ , ] and ( ) are the th-order approximation of ( ) that is obtained by the RPS method. An excellent account of the study of error analysis, which includes its definitions, varieties, applications, and method of derivations, can be found in [27].

Numerical Results and Discussion
The proposed method provides an analytical approximate solution in terms of an infinite power series. However, there is a practical need to evaluate this solution and to obtain numerical values from the infinite power series. The consequent series truncation and the corresponding practical procedure are realized to accomplish this task. The truncation transforms the otherwise analytical results into an exact solution, which is evaluated to a finite degree of accuracy.
In this section, we consider six examples to demonstrate the performance and efficiency of the present technique. Throughout this paper, all of the symbolic and numerical computations are performed using the Maple 13 software package.
As we mentioned earlier, if we select the first two terms of the approximations as 0 ( ) = 0 and 1 ( ) = 0, then the th-truncated series has the form To find the values of the coefficients , = 2, 3, 4, . . ., we employ our RPS algorithm. Therefore, we construct the residual function as follows: Consequently, the 4th-order approximation of the RPS solution for (23) and (6) according to this residual function is as follows: which agrees with Corollary 2. It easy to demonstrate that each of the coefficients for ≥ 5 in expansion (25)  ( ) . Thus, the analytic approximate solution to (23) and (6) is identical to the exact solution ( ) = 4 − 3 . Table 1 shows a comparison between the absolute errors of our method that were obtained from a 4th-order approximation, the optimal homotopy asymptotic method [1], and the Hermite functions collocation method [2]. From the table, it can be seen that the RPS method provides us with an accurate approximate solution to (23) and (6). In fact, the results reported in this table confirm the effectiveness and accuracy of our method.

Example 2. Consider the following nonlinear homogeneous Lane-Emden equation:
( ) + 8 ( ) + 9 ( ) + 2 ( ) ln ( ) = 0, which is subject to the initial conditions Assume that the initial guess approximation (which is the 1st approximation) has the form 1 ( ) = 1. Then, the 10th truncated series of the RPS solution of ( ) for (28) and (29) is as follows: Thus, the exact solution of (28) and (29) has a general form that coincides with the exact solution Let us carry out an error analysis of the RPS method for this example. Figure 1 shows the exact solution ( ) and the four iterated approximations ( ) for = 5, 10, 15, 20. This graph exhibits the convergence of the approximate solutions to the exact solution with respect to the order of the solutions. In Figure 2, we plot the exact error functions Ext ( ) when = 5, 10, 15, 20, 30, which approach the axis = 0 as the number of iterations increases. This graph shows that the exact errors become smaller as the order of the solutions increases, that is, as we progress through more iterations. These error indicators confirm the convergence of the RPS method with respect to the order of the solutions. From Figure 2, it is easy for the reader to compare the new result of the RPS method with the exact solution. Indeed, this graph shows that the current method has an appropriate convergence rate.
After we apply the RPS method to solve (32) and (33), we construct the residual function as follows: where ∑ =2 ( − 0 ) is the th-truncated series that approximates the solution ( ). As we mentioned earlier, if we select the first two terms of the approximations as 0 ( ) = 0 and 1 ( ) = 0 (which would imply that 1 = 2 = 0), then the first few terms of the approximations of the RPS solution for (32) and (33)    Furthermore, if we collect the above results, then the 10thtruncated series of the RPS solution for ( ) is given as Thus, the exact solution of (32) and (33) has the general form that coincides with the exact solution In most real-life situations, the Lane-Emden equation is too complicated to solve exactly, and, as a result, there is a practical need to approximate the solution. In the next two examples, the exact solution cannot be found analytically.
As we mentioned earlier, if we select the initial guess approximation as 1 ( ) = 1, then the Taylor series expansion of the solution for (38) and (39) is as follows: Consequently, the 10th-order approximation of the RPS solution for (38) and (39) is Our next goal is to show how the th value in the thtruncated series (3) affects the approximate solutions. In Table 2, the residual error has been calculated for various values of in [0, 2] to measure the extent of agreement between the th-order approximate RPS solutions when = 10, 15, 20, 25. As a result, Table 2 illustrates the rapid convergence of the RPS method by increasing the orders of approximation. To show the efficiency of the RPS method, numerical comparisons are also studied. Table 3 shows a comparison of ( ) that is obtained by the 10th-order approximation of the RPS method with those results that were obtained by the Adomian decomposition method [3], the Hermite functions collocation method [2], and the homotopy perturbation method [4]. Again, we find that our method has a similar degree of accuracy to these other methods.
8 Abstract and Applied Analysis Historically, this type of Lane-Emden equation was derived by Bonnor [28] in 1956 to describe what are now commonly known as Bonnor-Ebert [28,29] gas spheres. These gas spheres are isothermal gas spheres that have been embedded in a pressurized medium at the maximum possible mass that allows a hydrostatic equilibrium. The derivation is based on earlier work by Ebert [29], and hence, the equation is often referred to as the Lane-Emden equation of the second kind (which depends on an exponential nonlinearity). For a derivation of the Lane-Emden equation of the second kind, the reader is kindly requested to peruse [30][31][32][33].
In fact, this model appears in Richardson's theory of thermionic currents when the density and electric force of an electron gas in the neighborhood of a hot body in a thermal equilibrium [10] must be determined. For a thorough discussion of the formulation of (42) and (43) and the physical behavior of the emission of electricity from hot bodies, see [10,11]. It should be observed that this equation is nonlinear and has no analytic solution.
As we mentioned earlier, if we select the initial guess approximation as 2 ( ) = 0, then the Taylor series expansion of solutions to (42) and (43) is as follows: Consequently, the 10th-order approximation of the RPS solution for (42) Table 4 we compare our results to the results from the literature. Some of these results were obtained in [5] by constructing analytic approximations based on Euler transform series; others were obtained in [6] by using two acceleration methods to improve the convergence over the standard Taylor series results. In addition, the author in [7] applied the fractional approximation technique and the authors in [8] applied the Boubaker polynomials expansion scheme. In the above table, it can be seen that our results from the RPS method agree principally with the methods of [5,7]. In addition, we find that our results agree well with method [6]. However, for small , we find that the results of [8] agree well with the method of [7], and if is larger, we find that these results agree with method [6]. This conclusion is reasonable, as the fractional-approximationtechnique solution in [7] is of low order, and hence, it is valid for close to = 0. In contrast, the solution in [6] involves Padé approximation, which can improve the region of convergence.
In the next example, we show that the RPS method is capable of reproducing the exact solution to a new version of the Lane-Emden equation. Furthermore, we show that the consecutive error is a useful indicator in the iteration progresses, and moreover, this error can be used to study the structural analysis of the RPS method. 4.6. Example 6. Consider the following nonlinear nonhomogeneous singular initial-value problem: which is subject to the initial conditions where ( ) is chosen so that the exact solution is ( ) = sin( − 1) + cos( − 1). If we select the initial guess approximation as 2 ( ) = − (1/2)( − 1) 2 , then the 10th-truncated series of the RPS solution for (46) and (47) will be