Self-similar solutions of some model degenerate partial differential equations of the second, third and fourth order

标量四阶半线性常微分方程的条件线性化准则

Conditional Linearizability Criteria for Scalar Fourth Order Semi-Linear Ordinary Differential Equations

利用几何方法将二阶三次半线性常微分方程和三阶五次半线性常微分方程线性化，并通过对三阶方程的微分推广到四阶。由此得到了一类四阶半线性常微分方程的可线性化准则，这类准则在以前的文献中没有讨论过。结果表明，该过程可以推广到高阶。虽然高阶的结果比较复杂，但通过代数计算是可以得到的。标准的李的方法，目前开发的似乎并不适合给出的结果，甚至可以处理代数计算。

Meromorphic Solution of the Degenerate Third Painlevé Equation Vanishing at the Origin

We prove that there exists the unique odd meromorphic solution of dP3, $u(τ)$ such that $u(0)=0$, and study some of its properties, mainly: the coefficients of its Taylor expansion at the origin and asymptotic behaviour as $τ\to+\infty$.

Painlevé IV and a third-order viewpoint

We take a third-order approach to the fourth Painlevé equation and indicate the value of such an approach to other second-order ODEs in the Painlevé-Gambier list of 50.

四阶常微分方程的点变换线性化

Linearization of fourth-order ordinary differential equations by point transformations

本文给出了用点变换法解决四阶方程线性化问题的方法。我们表明，所有的四阶方程是线性化的点变换包含在一类方程是线性的三阶导数。我们提供了线性化测试，并描述了获得线性化变换以及线性化方程的过程。

退化第三类Painlevé方程解的渐近性的联系公式。I

Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painlevé Equation. I

本文用等单值变形方法研究了退化的第三类Painlevé方程$u^{\prime \prime} = \frac{(u^{\prime})^{2}}{u} - \frac{u^{\prime}}τ + \frac{1}τ(-8 εu^{2} + 2ab) + \frac{b^{2}}{u}$，其中$ε,b \in \mathbb{R}$和$a \in \mathbb{C}$，以及相应的τ函数.得到了一般正则解作为$τ\to \pm 0$和$\pm i0$解与一般正则解作为$τ\to \pm \infty$和$\pm i \infty$解的渐近性的联系公式。

退化第三PII方程：在正则奇点邻域内解的完全渐近分类

The Degenerate Third Painleve' Equation: Complete Asymptotic Classification of Solutions in the Neighbourhood of the Regular Singular Point

本文研究了退化第三PII方程在正则奇点邻域内解的渐近行为，通过monodromy数据分类并给出解的完全渐近展开。

退化第三类Painleve方程：II解的渐近性的联系公式

Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painleve' Equation: II

本文用等单点变形方法研究了退化的第三类Painleve方程$u"(t)=(u'(t))^2/u(t)-u'(t)/t+1/t(-8c u^2(t)+2ab)+b^2/u(t)$，其中$c=+/-1$，$b>0$，$a$是复参数.本文利用文献[1]第一部分中提出的2 × 2线性辅助问题的单值数据，导出了一般正则解和奇异解$u(t)$的渐近性，并将其参数化为$t -> +/-\infty$和$t -> +/-i\infty$.利用这些结果，三个真正的参数家庭的解决方案，有无穷序列的零和极点是渐近位于沿实轴和虚轴的区别：这些零和极点的渐近性也得到。

Asymptotic Integration of a Linear Fourth Order Differential Equation of Poincaré Type

This article deals with the asymptotic behavior of fourth order differential equation where the coefficients are perturbations of linear constant coefficient equation. We introduce a change of variable and deduce that the new variable satisfies a third order differential equation of Riccati type. We assume three hypothesis. The first is the following: all roots of the characteristic polynomial associated to the fourth order linear equation has distinct real part. The other two hypothesis are related with the behavior of the perturbation functions. Under this general hypothesis we obtain four m…

常微分方程加性刚性系统的三阶非齐次方法

A non-homogeneous method of third order for additive stiff systems of ordinary differential equations

针对加性分裂自治刚性常微分方程组，构造了一种三阶L-稳定方法，其第四阶段显式且允许任意雅可比矩阵近似，通过局部误差和稳定性控制实现自动步长选择，数值实验验证了算法的可靠性和效率。

形式单值参数 $a=\pm i/2$ 且在原点为零的退化第三 Painlevé 方程的单参数亚纯解

One-Parameter Meromorphic Solution of the Degenerate Third Painlevé Equation with Formal Monodromy Parameter $a=\pm i/2$ Vanishing at the Origin

本文证明退化第三 Painlevé 方程在形式单值参数 $a=\pm i/2$ 下存在一个在原点为零的单参数亚纯解，并研究了其 Taylor 系数的数论性质与无穷远处渐近行为。

形式单值参数消失时退化第三潘勒韦方程的代数体解

Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter

本文研究形式单值参数为零时退化第三潘勒韦方程的代数体解，建立 $τ\to0$ 与 $τ\to+\infty$ 的渐近连接结果，并详细分析具有 $cτ^{1/3}$ 渐近行为的最简单解及其相关积分。