Solutions of the Singular IVPs of Lane-Emden type equations by combining Laplace transformation and perturbation technique

1 Introduction

The Lane–Emden equation is Poisson’s equation for the gravitational potential of a self-gravitating, spherically symmetric polytrophic fluid which arises in many applications of mathematical physics and constitutes a good model for many systems in various fields. The Lane–Emden type equations, first published by Jonathan Homer Lane in 1870 [1], and further explored in detail by Emden [2], represents such phenomena and having significant applications, is a second-order ordinary differential equation with an arbitrary index, known as the polytrophic index, involved in one of its terms. The general form of the Lane-Emden equations is the following form:

with the following initial conditions

where m, x > 0 and f(x, y) is a continuous real-value function and g(x) is an analytical function.

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 modified perturbation technique based on an artificial parameter δ, the method is often called δ–method. Mandelzweig and Tabakin [4] used the quasi-linearization 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 is not based on the existence of some small parameters. Wazwaz [5] has given a general way to construct exact and series solutions to Lane-Emden equations by employing the Adomian decomposition method. Yıldırım and Özis [6] gave the solutions of a class of singular second-order IVPs of Lane–Emden type by using homotopy perturbation method. He [7] obtained an approximate analytical solution of the Lane–Emden equation by applying a variational iteration method. Parand et al. [8, 9, 10, 11]. A numerical solution of Lane-Emden equations is given based on the Legendre wavelets methods [12] 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 approximate analytical method will introduced for exact solution of Lane-Emden equation. Yüzbaşı et al. [13, 14, 15, 16] gave the solutions of a class of singular second-order IVPs of Lane–Emden type by using collocation method. Homotopy method is a powerful device for solving functional equations [17, 18, 19, 20, 21]. Based on homotopy, which is a basic concept in topology, general analytical method namely the homotopy perturbation method is established by He [22] in 1998 to obtain series solutions of differential equations. This method has been already used to solve various functional equations. In this method, the problem is transferred to an infinite number of sub-problems and then the solution is approximated by the sum of the solutions of the first several sub-problems. This simple method has been applied to solve linear and nonlinear equations of heat transfer [23, 24, 25], fluid mechanics [26] and integral equations [27]. This paper is arranged as follows:

In Section 2, the combination of Laplace transform, polynomial series and perturbation technique and the way to construct the modified approach for this type of equation are described. In Section 3 the modified algorithm is applied to some types of Lane–Emden equations. Finally we give a brief conclusion in section 4.

2 Analysis of the method

For solving the equation (1), based on NHPM [28] and LTNHPM [29] we construct a following equation

where p is an artificial parameter, u₀(x) = ∑n=0∞α_nP_n (x) and α₀, α₁, α₂, … are unknown coefficients and P₀(x), P₁(x), P₂(x), … are specific polynomial functions depending on the problem. Obviously, when p = 1, from (3) we have original equation (1).

By the perturbation technique, assumed that the function y(x) can be expressed by an infinite series, y(x) = ∑n=0∞pⁿy_n(x), and nonlinear term f(x, y) can be decomposed into an infinite series of polynomials given by

where H_n(v₀, v₁, …, v_n) are defined by

Now let us write the equation (3) in the following form

where ∑n=0∞g(n)(0)n!xn is the Taylor series of g(x). By applying Laplace transform on both sides of (6), we have

or

Comparing coefficients of terms with identical powers of p, leads to

Now, let us determine α₀, α₁, α₂, … so that y₁ = 0, then from (9) we have y_n = 0, n = 2, 3, …. Setting p = 1, results in the solution of equation (3) as the following:

Therefore, in this method, only the first He’s polynomials is calculated, and does not need to solve the differential equation in each iteration.

3 Illustrative Examples

In this section, we apply the modified method for solution of Lane-Emden equation.

4 Conclusion

The main goal of this work is to provide analytical solution of the linear and nonlinear Lane-Emden equation by combining the Laplace transform, polynomial series and perturbation technique. This method has been applied to five examples successfully and exact solutions of the equations are achieved. We predict that the modified algorithm will be a promising method for investigating exact analytic solutions to nonlinear ODEs. The computations associated with the examples were performed using Maple 13.