A new analytic algorithm of Lane–Emden type equations
Introduction
Many problems in mathematical physics and astrophysics can be modelled by the so-called Lane–Emden type equation [1], [2]subject to the boundary conditionswhere 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)=(u2−C)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 functionssuch thatwhere 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]subject to the boundary conditionswhere m⩾0 is a constant.
By means of Adomian decomposition method Shawagfeh [3] and Wazwaz [4] obtainedwhereHowever, 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⩽k⩽m,
Acknowledgements
This work is supported by National Science Fund for Distinguished Young Scholars of China (approval no. 50125923).
References (41)
A new algorithm for solving differential equations of Lane–Emden type
Appl. Math. Comput.
(2001)Nonlinear stochastic differential equations
J. Math. Anal. Appl.
(1976)A convenient computational form for the An polynomials
J. Math. Anal. Appl.
(1984)- et al.
On the solution of algebraic equations by the decomposition method
Math. Anal. Appl.
(1985) A review of the decomposition method and some recent results for nonlinear equations
Comput. Math. Appl.
(1991)- et al.
Convergence of Adomian’s method applied to nonlinear equations
Math. Compact. Model.
(1994) A new approach to the cubic Schrodinger equation: An application of the decomposition technique
Appl. Math. Comput.
(1998)The decomposition method applied to systems of partial differential equations and to the reaction–diffusion Brusselator model
Appl. Math. Comput.
(2000)- et al.
Domain decomposition techniques for reaction–diffusion equations in two-dimensional regions with re-entrant corners
Appl. Math. Comput.
(2001) - et al.
On the order of convergence of Adomian method
Appl. Math. Comput.
(2002)
Solution of nonlinear equations by modified Adomian decomposition method
Appl. Math. Comput.
Analytical approximate solutions for nonlinear fractional differential equations
Appl. Math. Comput.
The decomposition method for ordinary differential equations with discontinuities
Appl. Math. Comput.
Exact solutions for variable coefficients fourth-order parabolic partial differential equations in higher-dimensional spaces
Appl. Math. Comput.
A kind of approximate solution technique which does not depend upon small parameters: a special example
Int. J. Non-Linear Mech.
A kind of approximate solution technique which does not depend upon small parameters (ii): an application in fluid mechanics
Int. J. Non-Linear Mech.
An explicit, totally analytic approximation of Blasius viscous flow problems
Int. J. Non-Linear Mech.
An analytic approximation of the drag coefficient for the viscous flow past a sphere
Int. J. Non-Linear Mech.
Introduction to Nonlinear Differential and Integral Equations
Introduction to the Study of Stellar Structure
Cited by (158)
Romanovski-Jacobi spectral schemes for high-order differential equations
2024, Applied Numerical MathematicsAnalysis of perturbation factors and fractional order derivatives for the novel singular model using the fractional Meyer wavelet neural networks
2023, Chaos, Solitons and Fractals: XAnalytical and Numerical solutions for fourth order Lane–Emden–Fowler equation
2022, Partial Differential Equations in Applied MathematicsSwarming optimization to analyze the fractional derivatives and perturbation factors for the novel singular model
2022, Chaos, Solitons and Fractals