An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method (1008.2063v1)

Abstract: In this paper we propose a collocation method for solving some well-known classes of Lane-Emden type equations which are nonlinear ordinary differential equations on the semi-infinite domain. They are categorized as singular initial value problems. The proposed approach is based on a Hermite function collocation (HFC) method. To illustrate the reliability of the method, some special cases of the equations are solved as test examples. The new method reduces the solution of a problem to the solution of a system of algebraic equations. Hermite functions have prefect properties that make them useful to achieve this goal. We compare the present work with some well-known results and show that the new method is efficient and applicable.

• The paper introduces a Hermite function collocation method (HFC) to efficiently solve nonlinear Lane-Emden equations on semi-infinite domains by transforming them into algebraic systems.
• Numerical results confirm the HFC method's accuracy, closely matching known solutions for astrophysical models with various polytropic indices like m=3.
• This robust HFC method provides a valuable computational tool for approximating solutions to complex ordinary differential equations in astrophysics and potentially other scientific fields.

An Approximation Algorithm for Nonlinear Lane-Emden Equations Using Hermite Functions

The paper provides an in-depth examination of a method to solve nonlinear Lane-Emden type equations, which are crucial in modeling astrophysical phenomena such as stellar structure and isothermal gas spheres. The authors present an innovative approach utilizing a Hermite function collocation (HFC) method. This method is designed to address the challenges associated with solving nonlinear ordinary differential equations (ODEs) on semi-infinite domains.

Methodology

The core of the proposed method is the usage of Hermite functions, which are known for their excellent approximation properties. The Hermite functions allow a reduction of the Lane-Emden problem to solving a system of algebraic equations, thereby simplifying computational complexity. The collocation method leverages the orthogonality of Hermite functions, which are well-defined and exhibit exponential decay, to ensure accuracy and stability.

Numerical Results and Analysis

The paper provides a comparative analysis of the HFC method with existing approaches and known solutions, focusing on different values of the polytropic index $m$. For instance, for $m = 1.5, 2, 2.5, 3, 4$, the method's results closely match those of Horedt's exact numerical solutions, confirming its precision. Strong numerical results were displayed, notably:

• For $m = 3$, the first zero of the equation was computed with a minor error compared to Horedt’s values.
• The method effectively reproduces the analytically known solutions for cases like $m=0$ and $m=5$.

Additionally, the authors explored various non-linear forms of Lane-Emden type equations, using the HFC method to solve equations involving exponential and trigonometric function nonlinearities. The resulting solutions showed good agreement with those derived through alternative techniques such as Adomian Decomposition and Homotopy Analysis Methods.

Theoretical and Practical Implications

The authors propose that this method provides a robust computational tool for approximating solutions to Lane-Emden equations, which historically require cumbersome numerical or perturbative methods. While there is a focus on semi-infinite intervals, the basis of utilizing orthogonal Hermite functions opens pathways to tackling a broader class of differential equations that extend beyond astrophysical applications. Such applicability can be observed in other fields requiring modeling with ODEs defined over unbounded domains.

The paper suggests the potential of the Hermite function collocation method to improve solution convergence through increased collocation points, an attribute desired in numerically stiff scenarios.

Future Directions

The authors hint at extending the methodology to encompass more complex boundary conditions and potentially higher-dimensional problems. There is also an interest in further refining the algorithm to enhance computational efficiency, especially in systems with highly oscillatory or rapidly decaying solutions.

This paper contributes to the body of numerical methods in computational mathematics, offering a high-accuracy, practical technique for addressing one of the fundamental equation classes in astrophysical modeling. The Hermite function collocation method emphasizes the importance of spectral methods and orthogonal function sets in solving intricate ODE problems efficiently.