On solvable Lie algebras and integration method of ordinary differential equations

On the symmetry solutions of two-dimensional systems not solvable by standard symmetry analysis

A class of two-dimensional systems of second-order ordinary differential equations is identified in which a system requires fewer Lie point symmetries than required to solve it. The procedure distinguishes among those which are linearizable, complex-linearizable and solvable systems. We also present the underlying concept diagrammatically that provides an analogue in $\Re^{3}$ of the geometric linearizability criteria in $\Re^2$.

带延迟变元常微分方程积分的对称性方法

Symmetry Approach to Integration of Ordinary Differential Equations with Retarded Argument

本文综述了利用李群对称性研究带延迟变元常微分方程的方法，完成一阶和二阶单延迟方程的李群分类，并通过诺特型定理给出二阶延迟方程的首积分，最后提出若干开放问题。

Invariance of second order ordinary differential equations under two-dimensional affine subalgebras of EP Lie algebra

Using the only admissible rank-two realisations of the Lie algebra of the affine group in one dimension in terms of the Lie algebra of Lie symmetries of the Ermakov-Pinney (EP) equation, some classes of second order nonlinear ordinary differential equations solvable by reduction method are constructed. One class includes the standard EP equation as a special case. A new EP equation with a perturbed potential but admitting the same solution formula as EP itself arises. The solution of the dissipative EP equation is also discussed.

Quasi-exactly solvable relativistic soft-core Coulomb models

By considering a unified treatment, we present quasi exact polynomial solutions to both the Klein-Gordon and Dirac equations with the family of soft-core Coulomb potentials $V_q(r)=-Z/\left(r^q+β^q\right)^{1/q}$, $Z>0$, $β>0$, $q\geq 1$. We consider cases $q=1$ and $q=2$ and show that both cases are reducible to the same basic ordinary differential equation. A systematic and closed form solution to the basic equation is obtain using the Bethe ansatz method. For each case, the expressions for the energies and the allowed parameters are obtained analytically and the wavefunctions are derive in t…

对合分布积分中的$\n\nmathcal{C}^{\infty}$-结构

$\mathcal{C}^{\infty}$-structures in the integration of involutive distributions

本文研究对合分布积分问题，提出比可解结构更一般的cinf-结构，弱化对称性要求，推广了积分因子与李点对称的关系，引入对称化因子并明确其在求积可积性中的作用，辅以应用实例。

Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups

In this paper, we characterize the class of distributions on an homogeneous Lie group $\fN$ that can be extended via Poisson integration to a solvable one-dimensional extension $\fS$ of $\fN$. To do so, we introducte the $ß'$-convolution on $\fN$ and show that the set of distributions that are $ß'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions. Moreover, we show that the $ß'$-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behaviour. Finally, we show that such distributions satisfy …

Solution of Conformable Fractional Ordinary Differential Equations via Differential Transform Method

Recently, a new fractional derivative called the conformable fractional derivative is given which is based on the basic limit definition of the derivative in [1]. Then, the fractional versions of chain rules, exponential functions, Gronwall's inequality, integration by parts, Taylor power series expansions is developed in [2]. In this paper, we give conformable fractional differential transform method and its application to conformable fractional differential equations.

组合与分解广义格林算子与常微分边值问题

Composing and Factoring Generalized Green's Operators and Ordinary Boundary Problems

本文研究了具有过多边界条件的线性常微分边值问题的解算子，探讨了广义格林算子的乘积性质及其对应的边值问题，通过实例展示分解方法的应用。

不规则路径上的确定性积分与常微分方程

Deterministic Integral and Ordinary Differential Equations over Irregular Paths

本文将关于不规则路径的确定性积分定义为标准线积分的极限，描述了Hölder指数在(0,1]范围内积分存在的路径类，并利用该积分理论研究依赖时间与路径的常微分方程解的存在性、唯一性与连续性，补充了Hölder指数小于1/3时的Young-Lyons-Gubinelli积分理论。

刚性常微分方程的一种六阶段三阶加性方法

A six-stage third order additive method for stiff ordinary differential equations

本文构造了一种用于求解加性分裂自治刚性常微分方程组的六阶段三阶加性方法，该方法对隐式部分L稳定且允许使用任意雅可比矩阵近似，并通过局部误差和稳定性控制实现自动步长选择。

任意线性常微分方程的整合

Integration of an arbitrary linear ODE

本文提出multex积分算子，用于显式求解任意线性常微分方程，弥补了传统方法在处理变量系数方程时的不足。

通过Jacobi场实现一阶ODE的积分

Integration of first-order ODEs by Jacobi fields

本文引入一类非Lie点对称的新向量场实现一阶常微分方程积分，基于二维Riemann流形与可积性关系提供积分程序，并揭示积分因子与Schrödinger型方程的联系。

On the integrability in finite terms of first-order differential equations according to Maximovic

We give a definition of integration by quadratures of first-order ordinary differential equations, and recover a little known result by Maximovic which states that a first-order ordinary differential equation can be integrated by quadratures only if it arises from the linear equation by a diffeomorphic transformation of the dependent variable. In the appendix this result is applied to the linear second-order equation in a restricted setting. A brief outline of Maximovic's life is also included.

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。