常微分方程解的存在唯一性定理教学研究

探讨了常微分方程初值问题解的存在唯一性定理教学策略.为便于教学和有利于学生理解并掌握其思想方法,对定理证明过程的表述作了命题化处理,给出了Picard逐步逼近法的应用实例,提出了教学讨论与知识拓展的一些有益内容.     

模糊随机微分方程解的存在唯一性

将实数空间上的随机微分方程推广到模糊数空间,即为模糊随机微分方程.本文用Picard迭代的方法证明了其解的存在唯一性定理,推广了现有文献的结果,并且给出Picard迭代近似解误差的估计式.     

常微分方程初值问题解的一个存在唯一性结果

提出并证明了常微分方程初值问题解的一个存在唯一性结果.     

中立型无限时滞带跳随机泛函微分方程解的存在唯一性

本文首先在Lipschiz条件和线性增长条件下,通过Picard迭代法研究了带跳的无限时滞中立型随机微分方程解的存在唯一性,接着对这这类方程的Picard迭代解与精确解的误差进行估计,最后讨论了解的矩估计。     

Banach空间中二阶常微分方程初值问题解的存在唯一性

用单调迭代方法给出了二阶常微分方程初值问题的解的存在唯一性的结果.     

Existence and Uniqueness of Solutions for Initial Value Problems of Second Order Ordinary Differential Equations in Banach Space

用单调迭代方法给出了二阶常微分方程初值问题的解的存在唯一性的结果。     

Banach空间中脉冲积分-微分方程的初值问题

给出了 Banach空间中一阶线性脉冲积分 -微分方程初值问题解的存在唯一性的一个新证法 ,改进了已有结果 .利用它讨论了一阶非线性脉冲积分 -微分方程初值问题的解 ,所得结果大大推广了已有的相关结果 .     

三活性聚合反应方程组

三活性聚合反应的研究归结出一个包含无穷多个微分方程的方程组的初值问题。本文证明了这个初值问题的解的存在与唯一性。     

分数阶R-L微分方程初值问题解的存在唯一性

研究了R-L导数定义下的分数维微分方程初值问题解的存在性及其唯一性,给出了方程的Peano存在定理和不等式定理,基于逐次逼近的方法,利用对分数阶R-L微夯方程构造的Tonelli序列和Ascoli引理证明分数阶R-L微分方程解的存在性,根据分数阶不等式定理证明了分数阶R-L微分方程解的唯一性.     

一维波动方程的反演问题

本文讨论一维声波方程的反演问题。首先利用双曲方程奇性传播特性,将此反演问题归结为一个非线性偏微分方程组的初值问题,并对其解作了估计,证明了解的稳定性。然后证明此初值问题Picard迭代的收敛性,从而得到了反演问题解的存在性。     

Integral representation of solutions to Fuchsian system and Heun's equation

We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.     

Backward Stochastic Differential Equations in the Plane

This paper is devoted to study backward stochastic differential equations in the plane driven by a Brownian sheet, where the value of the solution at the corner (s ₀,t ₀) is fixed. The existence and uniqueness of a solution is obtained by means of Picard's approximation scheme and a suitable two-parameter Gronwall's type lemma.     

Preconditioners for Incompressible Navier-Stokes Solvers

<正>In this paper we give an overview of the present state of fast solvers for the solution of the incompressible Navier-Stokes equations discretized by the finite element method and linearized by Newton or Picard's method.It is shown that block preconditioners form an excellent approach for the solution,however if the grids are not to fine preconditioning with a Saddle point ILU matrix(SILU) may be an attractive alternative. The applicability of all methods to stabilized elements is investigated.In case of the stand-alone Stokes equations special preconditioners increase the efficiency considerably.     

声速及损耗系数的同时反演

引言 本文讨论有损层状介质的反演问题.有损介质的反演问题的研究,即使是一维情形,也未成熟,且无一般提法.G.Kristensson和R.J.Krueger的一系列文章,利用一般的二次双曲型方程,对有限厚度的介质进行反演,用到的响应信息是透射和反射数据,不     

ITERATIVE METHODS FOR THE BOUNDARY VALUE PROBLEM OF A THIRD ORDER DIFFERENCE EQUATION

ＩＴＥＲＡＴＩＶＥＭＥＴＨＯＤＳＦＯＲＴＨＥＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＡＴＨＩＲＤＯＲＤＥＲＤＩＦＦＥＲＥＮＣＥＥＱＵＡＴＩＯＮＷａｎｇＰｅｉｇｕａｎｇ（王培光）（ＨｅｂｅｉＵｎｉｖｅｒｓｉｔｙ，河北大学，邮编：０７１００２）＆Ｌｕ...     

Boundary value problem of a third-order mixed type differential-difference equation

In this paper, the existence and uniqueness of the solution for the boundary value problem of a third order mixed differential-difference equation is given by using Schauder's fixed point theorem, and an approximate solution is given by using the Picard's iterative methods.     

On a theorem of Picard

We extend Picard's theorem on the existence of elliptic solutions of the second kind of linear homogeneous -order scalar ordinary differential equations with coefficients being elliptic functions (associated with a common period lattice) to linear homogeneous first-order systems. In particular, the qualitative Floquet-type structure of fundamental systems of solutions in terms of elliptic and exponential functions, polynomials, and Weierstrass zeta functions of the independent variable is determined. Connections with completely integrable systems are mentioned.

     

SOME REMARKS ABOUT THE AREA-PRESERVING CONVEX CURVE FLOW IN THE PLANE

Using Picard‘s theorem and the l.eray-Schauder fixed point theorem to reinvestigate the area-preserving convex curve flow in the plane which is considered as a coupled system and thus different from the setting handled by Gage.     

Curves, Domains and Picard's Theorem

Based on computing spherical lengths of polygonal curves andspherical areas of domains bounded by polygonal curves on thecomplex plane, this paper proves a property of holomorphic functionsand, as an application, gives a very brief proof of a famousinequality obtained by L. V. Ahlfors which easily implies Picard'stheorem. Taken together with the arguments given by Alhfors,therefore, this paper in fact gives an elementary, brief andgeometrical proof of Picard's theorem.     

Integrating factors, adjoint equations and Lagrangians

Integrating factors and adjoint equations are determined for linear and non-linear differential equations of an arbitrary order. The new concept of an adjoint equation is used for construction of a Lagrangian for an arbitrary differential equation and for any system of differential equations where the number of equations is equal to the number of dependent variables. The method is illustrated by considering several equations traditionally regarded as equations without Lagrangians. Noether's theorem is applied to the Maxwell equations.