非线性松弛方程的精确积分与可逆完备性
Exact Integration and Invertibility Completeness of the Nonlinear Relaxation Equation
- University of Chemistry and Technology, Prague（布拉格化学技术大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对多方气体渗透非线性松弛方程，通过分圆分解与微分代数准则，完整分类了所有显式闭式解，并扩展至 Lambert W 函数及对数积分情形。
AI中文摘要：
我们考察了模拟多方气体渗透的非线性初值问题 $\dot{y} = C(1 - y^\alpha)$。对于有理数 $\alpha$，我们通过分圆部分分式分解推导出精确原函数，并确立其实数三角表示。通过综合 Ritt 定理与 Rosenlicht 的微分代数残差准则，我们证明了精确阶跃响应 $y(t)$ 是初等函数当且仅当 $\alpha \in \{1, 2\}$。允许 Lambert $W$ 函数将闭式可解性严格扩展到 $\alpha \in \{-1, 1/2\}$。最后，渐近极限 $\alpha \to 0$ 收敛到由对数积分求解的奇异动力学。这些结果构成了该系统所允许的所有显式闭式解的完整分类。
英文摘要：
We examine the nonlinear initial value problem $\dot{y} = C(1 - y^α)$ modeling polytropic gas permeation. For rational $α$, we derive the exact primitive via cyclotomic partial fraction decomposition, establishing its real trigonometric representation. By synthesizing Ritt's theorem with Rosenlicht's differential algebraic residue criterion, we prove the exact step response $y(t)$ is an elementary function if and only if $α\in \{1, 2\}$. Admitting the Lambert $W$ function extends closed-form solvability strictly to $α\in \{-1, 1/2\}$. Finally, the asymptotic limit $α\to 0$ converges to a singular dynamics solved by the logarithmic integral. These results constitute a complete classification of all explicit closed-form solutions admitted by this system.
