Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type

Abstract

Lane–Emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models. In this paper, a pseudospectral technique is proposed to solve the Lane–Emden type equations on a semi-infinite domain. The method is based on rational Legendre functions and Gauss–Radau integration. The method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.

Introduction

Many science and engineering problems arise in unbounded domains. Different spectral methods have been proposed for solving problems on unbounded domains. The most common method is the use of polynomials that are orthogonal over unbounded domains, such as the Hermite spectral method and the Laguerre method [1], [2], [3], [4], [5], [6], [7].

Guo [8], [9], [10] proposed a method by mapping the original problem in an unbounded domain to a problem in a bounded domain and then using suitable Jacobi polynomials to approximate the resulting problems.

Another approach is replacing an infinite domain with $[-L\text{,}L]$ and a semi-infinite interval with $[0\text{,}L]$ by choosing L, sufficiently large. This method is named as domain truncation [11].

Another effective direct approach for solving such problems is based on rational approximations. Christov [12] and Boyd [13], [14] developed some spectral methods on unbounded intervals by using mutually orthogonal systems of rational functions. Boyd [14] defined a new spectral basis, named rational Chebyshev functions on the semi-infinite interval, by mapping them to the Chebyshev polynomials. Guo et al. [15] introduced a new set of rational Legendre functions which are mutually orthogonal in $L^2(0\text{,}+\infty )$. They applied a spectral scheme using the rational Legendre functions for solving the Korteweg-de Vries equation on the half line. Boyd et al. [16] applied pseudospectral methods on a semi-infinite interval and compared rational Chebyshev, Laguerre and mapped Fourier sine. Authors of [17] developed two pseudospectral methods based on Fourier series and rational Chebyshev function to solve the Nagumo equation.

Authors of [18], [19], [20] applied the spectral method to solve nonlinear ordinary differential equations on semi-infinite intervals. Their approach was based on rational Tau method. They obtained the operational matrices of derivative and product of rational Chebyshev and Legendre functions and then applied these matrices together with the Tau method [21], [22], [23] to reduce the solution of these problems to the solution of a system of algebraic equations.

In this paper, a pseudospectral technique based on rational Legendre functions is applied to solve nonlinear differential equations, i.e. Lane–Emden and white-dwarf on semi-infinite domain.

Many problems in mathematical physics and astrophysics which occur on semi-infinite interval, are related to the diffusion of heat perpendicular to the parallel planes and can be modeled by the heat equation

$$x^{-k}\frac{d}{\mathit{dx}}\left(x^k\frac{\mathit{dy}}{\mathit{dx}}\right)+f(x)g(y)=h(x)\text{,}x>0\text{,}k>0\text{,}$$

or equivalently

$$y^{″}+\frac{k}{x}{y}^{\prime }+f(x)g(y)=h(x)\text{,}x>0\text{,}k>0\text{,}$$

where y is the temperature. For the steady-state case and for $k=2\text{,}h(x)=0$, this equation is the generalized Emden–Fowler equation [24], [25], [26] given by

$$y^{″}+\frac{2}{x}{y}^{\prime }+f(x)g(y)=0\text{,}x>0\text{,}$$

subject to the conditions

$$y(0)=a\text{,}{y}^{\prime }(0)=b\text{,}$$

where $f(x)$ and $g(y)$ are given functions of x and y, respectively.

When $f(x)=1$, Eq. (3) reduces to the Lane–Emden equation which, with specified $g(y)$, was used to model several phenomena in mathematical physics and astrophysics such as the theory of stellar structure, the thermal behavior of a spherical cloud of gas, isothermal gas sphere and theory of thermionic currents.

Several authors have investigated this equation.

Bender et al. [27] proposed a perturbative technique for solving nonlinear differential equations such as Lane–Emden. It consists of replacing nonlinear terms in the Lagrangian such as $y^n$ by $y^{1+\delta }$ and then treating $\delta$ as a small parameter.

Shawagfeh [28] applied a nonperturbative approximate analytical solution for the Lane–Emden equation using the Adomian decomposition method. His solution was in the form of a power series. He used Padé approximation method [29] to accelerate the convergence of the power series.

Mandelzweig and Tabakin [30] used the quasilinearization approach to solve 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 the perturbation theories is not based on the existence of some kind of small parameters.

Wazwaz [31] employed the Adomian decomposition method [32] with an alternate framework designed to overcome the difficulty of the singular point. It was applied to the differential equations of Lane–Emden type. Further he used [33] the modified decomposition method for solving analytic treatment of nonlinear differential equations such as Lane–Emden equation. The modified method accelerates the rapid convergence of the series solution, dramatically reduces the size of work and provides the solution by using few iterations only without any need to the so-called Adomian polynomials.

Liao [34] provided an analytic algorithm for Lane–Emden type equations. This algorithm logically contains the well-known Adomian decomposition method. Different from all other analytical techniques, this algorithm itself provides us with a convenient way to adjust convergence regions even without Padé technique.

By the semi-inverse method, He [35] obtained a variational principle for the Lane–Emden equation, which gives much numerical convenience when applying finite element methods or Ritz method.

Parand and Razzaghi [20] presented a numerical technique based on a rational Legendre Tau method to solve higher ordinary differential equations such as Lane–Emden. In their work, the operational matrices of the derivative and product of rational Legendre functions together with the Tau method were utilized to reduce the solution of these physical problems to the solution of systems of algebraic equations.

Ramos [36], [37], [38] solved Lane–Emden equation through different methods. He presented linearization methods for singular initial value problems in second order ordinary differential equations such as Lane–Emden. These methods result in linear constant-coefficients ordinary differential equations which can be integrated analytically, thus they yield piecewise analytical solutions and globally smooth solutions [36]. Later, he developed piecewise-adaptive decomposition methods for the solution of nonlinear ordinary differential equations. Piecewise-decomposition methods provide series solutions in intervals which are subject to continuity conditions at the end points of each interval and their adaption is based on the use of either a fixed number of approximants and a variable step size, a variable number of approximants and a fixed step size or a variable number of approximants and a variable step size [37]. In [38], series solutions of the Lane–Emden equation based on either a Volterra integral equation formulation or the expansion of the dependent variable in the original ordinary differential equation are presented and compared with series solutions obtained by means of integral or differential equations based on a transformation of the dependent variables.

Yousefi [39] used integral operator and converted Lane–Emden equations to integral equations and then applied Legendre wavelet approximations. In his work properties of Legendre wavelet together with the Gaussian integration method were utilized to reduce the integral equations to the solution of algebraic equations. By his method, the equation was formulated on [0, 1].

Chowdhury and Hashim [40] obtained analytical solutions of the generalized Emden–Fowler type equations in the second order ordinary differential equations by homotopy-perturbation method (HPM). This method is a coupling of the perturbation method and the homotopy method. The main feature of the HPM [41] is that it deforms a difficult problem into a set of problems which are easier to solve. HPM yields solutions in convergent series forms with easily computable terms.

Aslanov [42] constructed a recurrence relation for the components of the approximate solution and investigated the convergence conditions for the Emden–Fowler type of equations. He improved the previous results on the convergence radius of the series solution.

Dehghan and Shakeri [43] investigated Lane–Emden equation using the variational iteration method and showed the efficiency and applicability of their procedure for solving this equation. Their technique does not require any discretization, linearization or small perturbations and therefore reduces the volume of computations.

Bataineh et al. [44] obtained analytical solutions of singular initial value problems (IVPs) of the Emden–Fowler type by the homotopy analysis method (HAM). Their solutions contained an auxiliary parameter which provided a convenient way of controlling the convergence region of the series solutions. It was shown that the solutions obtained by the Adomian decomposition method (ADM) and the homotopy-perturbation method (HPM) are only special cases of the HAM solutions.

As one more step in this direction, we use rational Legendre pseudospectral approach to solve Lane–Emden and white-dwarf equations which are nonlinear singular differential equations on semi-infinite interval. The main point of our analysis lies in the fact that there is no reconstruction of the problem on the finite domain. We show that our results have good agreement with exact results, which demonstrate the viability of the new technique. In this sense, this method has the potential to provide a wider applicability. On the other hand, the comparison of the results obtained by this method and the others shows that the new method provides more accurate solutions than those obtained by other methods.

The organization of the paper is as follows:

In Section 2, we explain the formulation of rational Legendre functions required for our subsequent development. In Section 3, after a short introduction to the essentials of Lane–Emden equation, we summarize the application of rational Legendre pseudospectral method for solving Lane–Emden and white-dwarf equations. Then, a comparison is made with the existing methods in the literature. Section 4 is devoted to conclusions.