A new analytic algorithm of Lane–Emden type equations

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Abstract

An reliable, ease-to-use analytic algorithm is provided for Lane–Emden type equation which models many phenomena in mathematical physics and astrophysics. This algorithm logically contains the well-known Adomian decomposition method. Different from all other analytic techniques, this algorithm itself provides us with a convenient way to adjust convergence regions even without Páde technique. Some applications are given to show its validity.

Introduction

Many problems in mathematical physics and astrophysics can be modelled by the so-called Lane–Emden type equation [1], [2]u′′(x)+2xu(x)+f(u)=0,x⩾0,subject to the boundary conditionsu(0)=a,u(0)=0,where the prime denotes the differentiation with respect to x, a is a constant, f(u) is a nonlinear function of u(x). For example, it models the thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics [1], [3], [4] when f(u)=um, the gravitational potential of the degenerate white-dwarf stars [2] when f(u)=(u2C)3/2, the isothermal gas spheres [1] when f(u)=exp(u) and so on.

The difficult element in the analysis of this type of equations is the singularity behavior occurring at x=0. The series solution can be found by perturbation techniques and Adomian decomposition method. However, the series solutions are often convergent in restricted regions so that some techniques such as Páde method has to be applied to enlarge the convergence regions [1], [3], [4].

Liao developed a kind of analytic technique for nonlinear problems, namely the homotopy analysis method [5]. Unlike perturbation techniques [6], [7], [8], [9], [10] and other nonperturbative methods such as the artificial small parameter method [11], the δ-expansion method [12], the decomposition method [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and so on, the homotopy analysis method itself provides us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary. Briefly speaking, the homotopy analysis method has the following advantages:

  • 1.

    it is valid even if a given nonlinear problem does not contain any small/large parameters at all;

  • 2.

    it itself can provide us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary;

  • 3.

    it can be employed to efficiently approximate a nonlinear problem by choosing different sets of base functions.


The homotopy analysis method has been successfully applied to many nonlinear problems such as viscous flows [32], [33], [34], [35] and heat transfer [36], nonlinear oscillations [37], [38], nonlinear water waves [39], Thomas–Fermi’s atom model [40] and so on, and some elegant analytic results are obtained. Especially, by means of the homotopy analysis method Liao [41] gave a drag formula for a sphere in a uniform stream, which agrees well with experimental results in a considerably larger region of Reynolds number than those of all reported analytic drag formulas. All of these successful applications of the homotopy analysis method verify its validity for nonlinear problems in science and engineering. In this paper the homotopy analysis method is further applied to propose a reliable analytic algorithm for solving the Lane–Emden type equation and some applications are given. Our analytic approximate solutions contain Shawagfeh’s [3] and Wazwaz’s [4] solution given by Adomian decomposition method and besides are convergent in considerably large regions even without Páde technique.

Section snippets

Rule of solution expression

Obviously the Lane–Emden type equation can be expressed by the set of power functionsS1={xm|m⩾0}such thatu(x)=∑k=0+∞akxk,where ak is coefficient to be determined. This provides us with the first Rule of Solution Expression of the Lane–Emden type equation.

However, the set (3) is not the unique one to approximate the solution of the Lane–Emden type equation. Due to (1) the solution u(x) decreases monotonously as x increases. So, it is possible that u(x) can be approximate by the set of base

Lane–Emden equation

The thermal behavior of a spherical cloud of gas acting under the mutual attraction of its molecules and subject to the classical laws of thermodynamics is modelled by the well-known Lane–Emden equation [1], [3], [4]u′′(x)+2xu(x)+um(x)=0,x⩾0,subject to the boundary conditionsu(0)=1,u(0)=0,where m⩾0 is a constant.

By means of Adomian decomposition method Shawagfeh [3] and Wazwaz [4] obtainedu(x)=1+∑n=1+∞Anx2n,whereA1=−16,A2=m120,A3=−m(8m−5)3×7!,⋯However, for m>2, (27) is not valid in the whole

Conclusions and discussions

In the frame of the homotopy analysis method an analytic algorithm is given for Lane–Emden type equation which can model many phenomena in mathematical physics and astrophysics. The analytic algorithm is reliable and ease-to-use. Its validity is verified by three examples.

First of all, our solutions , , contain the corresponding results given by Adomian decomposition method [4], thus our algorithm is more general than Adomian decomposition method. This is mainly because μm,k(−1)=1 for 0⩽km,

Acknowledgements

This work is supported by National Science Fund for Distinguished Young Scholars of China (approval no. 50125923).

References (41)

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