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An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method
K. Parand, Mehdi Dehghan, A. R. Rezaei, S. M. Ghaderi
TL;DR
The paper addresses nonlinear, singular Lane-Emden initial value problems on a semi-infinite domain, where exact solutions are limited to selected indices. It applies Hermite function collocation with a domain mapping and evaluates the approach on special cases, reporting appropriate convergence and agreement comparisons.
Problem
For physically interesting m in [0,5], singular behavior at x = 0 complicates analysis, while exact solutions are known only for m = 0, 1 and 5.
Method
The Hermite functions collocation method solves singular Lane-Emden type initial value problems by mapping (0, +∞) to the domain of Hermite functions and reducing the problem to algebraic equations.
Results
The method produces appropriate convergence rates in the tested standard Lane-Emden, isothermal gas spheres, and other Lane-Emden type examples, with comparisons to exact or previously obtained results.
Takeaways & Limitations
Hermite functions provide an effective and simple collocation-based approximation for nonlinear Lane-Emden type equations on semi-infinite intervals.
Takeaways & Limitations
The formulation assumes a polytropic index related to the gas’s specific-heat ratio and uses an altered density expression involving central stellar density and a related dimensionless quantity.
Abstract
from arXiv · showhide
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.
1. Introduction
Lane-Emden equations model astrophysical and theoretical-physics phenomena as nonlinear ODEs on a semi-infinite domain, with a singularity at x = 0. Physically relevant indices lie in [0,5], but exact solutions are known only for m = 0, 1, and 5, motivating numerical methods.
- Existing approaches: Spectral approaches for unbounded domains include Hermite and Laguerre methods, domain truncation, rational approximations, and mappings to bounded intervals.These methods address infinite or semi-infinite domains using orthogonal polynomials or rational bases.
- Problem: Lane-Emden equations are nonlinear ordinary differential equations on a semi-infinite domain and singular initial value problems.They describe phenomena including stellar structure, spherical gas clouds, isothermal gas spheres, and thermionic currents.
- Physical formulation: The Lane-Emden equation follows from hydrostatic equilibrium, the Poisson equation, and a pressure-density relation with polytropic index m.The constants K and m enter the pressure-density relation, while m is related to the ratio of specific heats of the stellar gas.
- Physical formulation: The dimensionless formulation uses x and y, with the central conditions r = 0 → x = 0, ρ = λ → y(0) = 1.The boundary conditions arise from hydrostatic equilibrium and normalization of the introduced variables.
- Problem: For physically interesting m in [0,5], exact solutions are known only for m = 0, 1, and 5; other cases require numerical integration.The singularity at x = 0 is identified as the main analytical difficulty.
2. Methods have been proposed to solve Lane-Emden equations
Prior work addresses Lane-Emden equations through analytical, perturbative, decomposition, variational, wavelet, rational spectral, and linearization methods. These approaches are designed to handle nonlinearities and the singular point at x = 0.
- Analytical methods: Analytical treatments include δ-method perturbation, Adomian decomposition, homotopy analysis, homotopy-perturbation, variational iteration, and Lie symmetry methods.Several methods specifically target singular initial value problems or the singular point.
- Alternative methods: Other analytical approaches include Ritz’s method, quasilinearization, hybrid function approximations, and modified homotopy analysis.Quasilinearization treats nonlinear terms as perturbations about linear ones without requiring a small parameter.
- Decomposition methods: Adomian decomposition variants and related algorithms use power-series, alternate decomposition, or adjustable convergence-region frameworks for Lane-Emden equations.Padé approximants were used in one approach to accelerate power-series convergence.
- Numerical methods: Numerical spectral approaches include rational Chebyshev and rational Legendre Tau methods for Lane-Emden and higher-order ordinary differential equations.These methods use rational bases suited to semi-infinite intervals.
- Numerical methods: Linearization and piecewise-adaptive decomposition methods have also been developed for singular initial-value problems and Lane-Emden equations.The reported linearization methods produce linear constant-coefficient ODEs that can be integrated analytically.
3. Hermite functions collocation method
The method uses well-behaved Hermite functions, transforms them from the whole real line to the semi-infinite domain, and applies collocation with optional domain scaling. The resulting approximation is constructed in a finite-dimensional Hermite space.
- 3. Hermite functions collocation method: The HFC method directly solves singular Lane-Emden initial value problems using highly accurate Hermite-function collocation.The paper motivates collocation as a useful approach for obtaining accurate differential-equation solutions.
- 3.1. Properties of Hermite functions: Hermite functions are preferred over Hermite polynomials because they are well behaved at infinity and have decay properties.Hermite polynomials are described as unsuitable in practice because of wild asymptotic behavior at infinities.
- 3.2. Approximations by Hermite functions: The approximation space H_N is the span of the first N + 1 normalized Hermite functions, with an associated L2 orthogonal projection.The paper defines convergence-rate spaces and norms for analyzing Hermite-function approximations.
- 3.3. Hermite functions transform: A variable transformation maps the Lane-Emden interval (0, +∞) into the whole-line setting where Hermite-function properties are available.The transformed basis functions are mutually orthogonal under the transformed weight and provide the basis on the semi-infinite interval.
- 3.4. Domain scaling: Domain scaling rescales the underlying variable to potentially increase computational accuracy in whole-line spectral approximations.The paper applies domain scaling in several subsequent examples.
4. Applications
The applications solve homogeneous and non-homogeneous Lane-Emden-type equations with transformed Hermite-function collocation. Boundary conditions are incorporated into the operator, and collocation reduces the problem to nonlinear algebraic equations for the expansion coefficients.
- 4. Applications: The applications cover Lane-Emden-type equations with varied f(x), g(y), A, and B, including homogeneous and non-homogeneous cases.The general formulation includes prescribed functions and real constants.
- 4. Applications: The boundary conditions are enforced by multiplying the operator by x and adding terms involving A and B.This construction addresses the behavior of the Hermite functions and satisfies the two boundary conditions.
- 4. Applications: The residual is formed by substituting the transformed Hermite approximation into the Lane-Emden equation.The approximation is represented through the transformed Hermite-function basis.
- 4. Applications: The N + 1 coefficient equations are obtained by setting the residual to zero at N + 1 transformed Hermite-Gauss points.The points are transformed roots of Hermite H_N+1(x).
- 4. Applications: The resulting N + 1 nonlinear equations can be solved by a method such as Newton’s method for the unknown coefficients.Solving this system yields the approximating function.
4.1. The homogeneous Lane-Emden type equations
The paper applies Hermite function collocation to several homogeneous Lane-Emden type equations, including standard Lane-Emden, isothermal gas sphere, and other nonlinear forms. The method constructs residual equations at transformed Hermite-Gauss points and compares approximations with exact, analytic, or previously published solutions.
- Collocation procedure: HFC determines approximation coefficients by setting the residual to zero at N + 1 transformed Hermite-Gauss points, then solving the resulting algebraic system.The resulting coefficients define the approximating function.
- Standard Lane-Emden equation: For the standard Lane-Emden equation, the method’s first zeros and solution values are compared with exact values and Horedt’s results for several m values.Table 1 compares first zeros for m = 1.5, 2, 2.5, 3, and 4; Tables 2 and 3 compare y(x) for m = 3 and 4.
- Standard Lane-Emden equation: The reported Hermite coefficients and logarithmic coefficient plots indicate an appropriate convergence rate for the standard Lane-Emden equation.The coefficients are reported for m = 2, 3, and 4, while the resulting solutions are graphed for m = 1.5, 2, 2.5, 3, and 4.
- Isothermal gas sphere equation: For the isothermal gas sphere equation, HFC approximations using N = 30, l = 2, and k = 2 are compared with Wazwaz’s results, with coefficient plots indicating an appropriate convergence rate.The model represents isothermal gas spheres, where temperature remains constant.
- Other Lane-Emden type equations: The HFC method is also applied to sinh(y), sin(y), exponential, and logarithmic Lane-Emden type equations, with comparisons to published series, analytic, or numerical solutions.For the logarithmic case, the transformation y(x) = e^z(x) is used before applying HFC; coefficient plots are reported to show an appropriate convergence rate.
4.2. The non-homogeneous Lane-Emden type equations
The section applies the Hermite function collocation method to two non-homogeneous Lane-Emden type equations, reducing each problem to algebraic equations and comparing the approximations with analytical solutions.
- Results: For both examples, the computed y(x) values and graphs are compared with the corresponding analytical solutions.The first comparison uses n = 30, k = 2/3 and l = 2; the second refers to analytic solution Eq. (4.20).
- Method: The HFC method is applied to two non-homogeneous Lane-Emden type equations previously treated with linearization, HPM, HAM, VIM, and TSADM methods.
- Method: For each equation, the authors construct a residual function and impose its vanishing at N + 1 transformed Hermite-Gauss points to determine the coefficients.The collocation equations produce an approximating function after solution of the resulting algebraic system.
5. Conclusions
The conclusion presents Hermite functions within a collocation method as an effective approximation approach for nonlinear Lane-Emden type equations on semi-infinite intervals. It reports acceptable solutions and exponential convergence based on coefficient plots, while noting that convergence is assumed to improve with more collocation points.
- Scope: The paper targets nonlinear Lane-Emden type equations arising in phenomena including stellar structure, spherical gas clouds, isothermal gas spheres, and thermionic currents.
- Method: The proposed method uses Hermite functions to approximate solutions on a semi-infinite interval through collocation.Its validity is based on the assumption that convergence improves as the number of collocation points increases.
- Results: The authors report acceptable results through comparisons with exact, numerical, and series solutions from the literature.
- Results: The logarithmic figures of the absolute Hermite-function coefficients indicate an exponential convergence rate for the approach.