抽象半线性发展方程的正解及应用

本文讨论了有序Ｂａｎａｃｈ空间中的正算子半群的特征，把通常常微分方程及偏微分方程的上、下解方法引入到有序Ｂａｎａｃｈ空间中的半线性发展方程，获得了整体解与正解的存在性．     

Banach空间中超线性Hammerstein型积分方程的解及其应用

本文利用不动点指数理论研究Ｂａｎａｃｈ空间中超线性Ｈａｍｍｅｒｓｔｅｉｎ型积分方程正解及非零解的存在性，并应用于Ｂａｎａｃｈ空间中超线性常微分方程的Ｓｔｕｒｍ－Ｌｉｏｕｖｉｌｌｅ问题，最后，本文给出了一个非线性常微分方程无穷组存在正解的例子．     

Banach空间中非线性刚性Volterra泛函微分方程稳定性分析

获得了Banach空间中非线性刚性Volterra泛函微分方程理论解的一系列稳定性、收缩性及渐近稳定性结果,为非线性刚性常微分方程、延迟微分方程、积分微分方程及实际问题中遇到的其他各种类型的泛函微分方程的解的稳定性分析提供了统一的理论基础.     

一类非线性抛物型方程组解的整体存在及爆破

研究了一类带有非线性边界条件的非线性抛物型方程组解的整体存在及解在有限时刻爆破问题.通过构造方程组的上、下解.得到了解整体存在及解在有限时刻爆破的充分条件.对指数型反应项和边界流采用了常微分方程方法构造其上下解,而其它例如第一特征值等方法运用于该方程就比较困难.     

一致连续性与解的整体存在性

将高等数学中的一致连续性概念用于研究常微分方程解的整体存在性.基于微分方程解的整体存在性定理,可证明微分方程dx/dt=f(t,x)的任一解x=x(t)都在(-∞, ∞)上有定义,其中f(t,x)一致连续.实例说明其应用.     

发展型p-Laplace方程(组)解的整体有限性

本文主要在多维空间中讨论一类发展型p-Laplace方程及方程组的初边值问题.这类问题在非牛顿渗流方程的理论研究中有着重要的意义.作者通过上、下解的方法证明了发展型p-Laplace方程解的整体有限性;同时利用两个关于常微分方程的比较原理,给出了相应的方程组解的整体有限性结果.     

Banach空间半线性发展方程的周期解

把上、下解方法引入到有序Ｂａｎａｃｈ空间中的半线性发展方程周期解问题，利用正算子半群特征与单调迭代程序，获得了最大周期解与最小周期解的存在性．所得的结果概括和推广了常微分方程与偏微分方程中的有关结论     

扩散引起的爆破问题

众所周知,在某些条件下常微分方程的解有限时刻爆破,与之相应的带齐次Dirichlet边界条件的反应扩散方程的解整体存在.也就是说,扩散阻止了解有限时刻爆破.一个自然的问题常微分方程的解是否整体存在,而与之相应的带齐次Dirichlet边界条件的反应扩散方程的解是否有限时刻爆破?即扩散能否引起解有限时刻爆破?本文将通过一个简单的例子给此问题一个确切的答案.     

扩散引起的爆破问题

众所周知,在某些条件下常微分方程的解有限时刻爆破,与之相应的带齐次Ｄｉｒｉｃｈｌｅｔ边界条件的反应扩散方程的解整体存在也就是说,扩散阻止了解有限时刻爆破一个自然的问题;常微分方程的解是否整体存在,而与之相应的带齐次Ｄｉｒｉｃｈｌｅｔ边界条件的反应扩散方程的解是否有限时刻爆破？即扩散能否引起解有限时刻爆破？本文将通过一个简单的例子给此问题一个确切的答案     

Banach空间二阶非线性微分方程的弱Carathéodory解

本文定义了二阶微分方程的弱 Carathéodory解 ,在不涉及紧型条件的情形下 ,直接用迭代法证明了 Banach空间二阶非线性常微分方程两点边值问题存在唯一解 ,并给出逼近解迭代序列的误差估计 ,对周期边值问题得到类似的结果     

Complex Lie symmetries for scalar second-order ordinary differential equations

A scalar complex ordinary differential equation can be considered as two coupled real partial differential equations, along with the constraint of the Cauchy–Riemann equations, which constitute a system of four equations for two unknown real functions of two real variables. It is shown that the resulting system possesses those real Lie symmetries that are obtained by splitting each complex Lie symmetry of the given complex ordinary differential equation. Further, if we restrict the complex function to be of a single real variable, then the complex ordinary differential equation yields a coupled system of two ordinary differential equations and their invariance can be obtained in a non-trivial way from the invariance of the restricted complex differential equation. Also, the use of a complex Lie symmetry reduces the order of the complex ordinary differential equation (restricted complex ordinary differential equation) by one, which in turn yields a reduction in the order by one of the system of partial differential equations (system of ordinary differential equations). In this paper, for simplicity, we investigate the case of scalar second-order ordinary differential equations. As a consequence, we obtain an extension of the Lie table for second-order equations with two symmetries.     

How to solve the polynomial ordinary differential equations

This paper discussed how to solve the polynomial ordinary differential equations. At first, we construct the theory of the linear equations about the unknown one variable functions with constant coefficients. Secondly, we use this theory to convert the polynomial ordinary differential equations into the simultaneous first order linear ordinary differential equations with constant coefficients and quadratic equations. Thirdly, we work out the general solution of the polynomial ordinary differential equations which is no longer concerned with the differential. Finally, we discuss the necessary and sufficient condition of the existence of the solution.     

Dichotomies for generalized ordinary differential equations and applications

In this work we establish the theory of dichotomies for generalized ordinary differential equations, introducing the concepts of dichotomies for these equations, investigating their properties and proposing new results. We establish conditions for the existence of exponential dichotomies and bounded solutions. Using the correspondences between generalized ordinary differential equations and other equations, we translate our results to measure differential equations and impulsive differential equations. The fact that we work in the framework of generalized ordinary differential equations allows us to manage functions with many discontinuities and of unbounded variation.     

三阶线性常微分方程Sinc方程组的结构预处理方法

三阶线性常微分方程在天文学和流体力学等学科的研究中有着广泛的应用.本文介绍求解三阶线性常微分方程由Sinc方法离散所得到的线性方程组的结构预处理方法.首先, 我们利用Sinc方法对三阶线性常微分方程进行离散,证明了离散解以指数阶收敛到原问题的精确解.针对离散后线性方程组的系数矩阵的特殊结构, 提出了结构化的带状预处理子,并证明了预处理矩阵的特征值位于复平面上的一个矩形区域之内.然后, 我们引入新的变量将三阶线性常微分方程等价地转化为由两个二阶线性常微分方程构成的常微分方程组, 并利用Sinc方法对降阶后的常微分方程组进行离散.离散后线性方程组的系数矩阵是分块2×2的, 且每一块都是Toeplitz矩阵与对角矩阵的组合.为了利用Krylov子空间方法有效地求解离散后的线性方程组,我们给出了块对角预处理子, 并分析了预处理矩阵的性质.最后, 我们对降阶后二阶线性常微分方程组进行了一些比较研究.数值结果证实了Sinc方法能够有效地求解三阶线性常微分方程.     

无限滞后测度泛函微分方程的平均化

本文研究了无限滞后测度泛函微分方程的平均化.利用广义常微分方程的平均化方法,在无限滞后测度泛函微分方程可以转化为广义常微分方程的基础上,获得了这类方程的周期和非周期平均化定理,推广了一些相关的结果.     

Efficient hybrid method for solving special type of nonlinear partial differential equations

In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh–coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh–coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two‐ and three‐dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed.     

On One Class of Systems of Differential Equations and on Retarded Equations

We establish a connection between solutions to a broad class of large systems of ordinary differential equations and solutions to retarded differential equations. We prove that solving the Cauchy problem for systems of ordinary differential equations reduces to solving the initial value problem for a retarded differential equation as the number of equations increases unboundedly. In particular, the class of systems under consideration contains a system of differential equations which arises in modeling of multiphase synthesis.     

On the approximation of nonlinear conflict-controlled systems of neutral type

We consider approximations of systems of nonlinear neutral-type equations in Hale’s form by systems of high-order ordinary differential equations. A procedure is given for the mutual feedback tracking between the motion of the original neutral-type conflict-controlled system and the motion of the approximating system of ordinary differential equations. The proposed mutual tracking procedure makes it possible to use approximating systems of ordinary differential equations as finite-dimensional modeling guides for neutral-type systems.     

Ermakov–Pinney and Emden–Fowler Equations: New Solutions from Novel Bäcklund Transformations

We study the class of nonlinear ordinary differential equations y″ y = F(z, y²), where F is a smooth function. Various ordinary differential equations with a well-known importance for applications belong to this class of nonlinear ordinary differential equations. Indeed, the Emden–Fowler equation, the Ermakov–Pinney equation, and the generalized Ermakov equations are among them. We construct Bäcklund transformations and auto-Bäcklund transformations: starting from a trivial solution, these last transformations induce the construction of a ladder of new solutions admitted by the given differential equations. Notably, the highly nonlinear structure of this class of nonlinear ordinary differential equations implies that numerical methods are very difficult to apply.     

Solving homogeneous fractional differential equations of Euler type

We study linear homogeneous differential equations with three left Riemann-Liouville fractional derivatives; these equations are analogs of Euler ordinary differential equations. By using the direct and inverse Mellin transforms and residue theory, we obtain a complete system of linearly independent solutions. As a corollary, related results are proved for Euler ordinary differential equations.