The Lane-Emden-Fowler equation is a class of second-order nonlinear differential equations with applications in mathematical physics and engineering fields such as astrophysics, fluid mechanics, and thermodynamics. It can be used to describe phenomena including stellar structure and heat conduction.
Lane-Emden-Fowler 方程是一类二阶非线性微分方程，在数学物理和工程领域中有应用， 包括天体物理、流体力学和热力学等方向，可用于描述恒星结构和热传导等现象。
This project studies analytical and numerical approaches, including Kummer- Liouville transformations, Lie symmetries, Painleve analysis, and homotopy perturbation methods. The work also considers time-fractional variants where fractional derivatives introduce additional modeling flexibility and additional computational difficulty.
这个项目研究解析和数值方法，包括 Kummer-Liouville 变换、Lie 对称性、Painleve 分析和同伦扰动方法。 项目也考虑时间分数阶情形，其中分数阶导数带来更多建模灵活性，也带来额外的计算困难。
Keywords关键词
Mathematica, Lane-Emden-Fowler equations, numerical analysis, LH-HPM, fractional derivatives.
Methods方法
- Elzaki Transform Homotopy Perturbation Method.
- Laplace Transform Homotopy Perturbation Method.
- Fractional derivative extensions.
- Mathematica-based numerical computation and visualization.
Applications应用
- Astrophysics: stellar structure and evolution.
- Fluid mechanics: fluid motion and heat transfer.
- Thermodynamics: heat conduction and temperature distribution.
Technical challenges技术挑战
Some equations have computational loads that exceed direct Mathematica symbolic or numerical solving. The project therefore balances analytical simplification, numerical approximations, and visualization choices.
部分方程的计算量超出 Mathematica 直接符号或数值求解的承受范围，因此项目需要在解析化简、 数值近似和可视化方案之间做平衡。