On the Equivalence of Differential Equations

In this article we use the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Riccati equation and some polynomial differential equations. The results are applied to discussion of the qualitative behavior of periodic solutions of these complicated differential equations.     

Uniqueness of Quasi-Regular Solutions for a Bi-Parabolic Elliptic Bi-Hyperbolic Tricomi Problem

The Tricomi equation $ yu_{xx} + u_{yy} = 0 $ was established in 1923 by Tricomi who is the pioneer of parabolic elliptic and hyperbolic boundary value problems and related problems of variable type. In 1945 Frankl established a generalization of these problems for the well-known Chaplygin equation $ K(\,y)u_{xx} + u_{yy} = 0 $ subject to the Frankl condition 1 + 2( K / K ')' > 0, y <0. In 1953 and 1955 Protter generalized these problems even further by improving the above Frankl condition. In 1977 we generalized these results in R n ( n > 2). In 1986 Kracht and Kreyszig discussed the Tricomi equation and transition problems. In 1993 Semerdjieva considered the hyperbolic equation $ K_1 (\,y)u_{xx} + (K_2 {\rm (\,}y{\rm )}u_y )_y + ru = f $ for y<0. In this paper we establish uniqueness of quasi-regular solutions for the Tricomi problem concerning the more general mixed type partial differential equation $ K_1 (\,y)(M_2 {\rm (}x{\rm )}u_x )_x + M_1 (x)(K_2 {\rm (\,}y{\rm )}u_y )_y + ru = f $ which is parabolic on both lines x = 0; y = 0, elliptic in the first quadrant x > 0, y > 0 and hyperbolic in both quadrants x< 0, y > 0; x > 0, y< 0. In 1999 we proved existence of weak solutions for a particular Tricomi problem. These results are interesting in fluid mechanics.     

The Asymptotic Behavior of Solutions of Some Doubly Degenerate Nonlinear Parabolic Equations

Ｔｈｅ　Ａｓｙｍｐｔｏｔｉｃ　Ｂｅｈａｖｉｏｒ　ｏｆ　Ｓｏｌｕｔｉｏｎｓ　ｏｆ　Ｓｏｍｅ　Ｄｏｕｂｌｙ　Ｄｅｇｅｎｅｒａｔｅ　Ｎｏｎｌｉｎｅａｒ　Ｐａｒａｂｏｌｉｃ　ＥｑｕａｔｉｏｎｓＹａｎｇＪｉｎｓｈｕｎ（杨金顺）（ＤｅｐａｒｔｍｅｎｔｏｆＢａｓｉ...     

On the Vekua Pair in Spheroidal Geometry and its Role in Solving Boundary Value Problems

The Vekua pair forms a transformation between the kernel of the Laplace's and the kernel of the Helmholtz's operator. In fact, it provides an interior solution of the Helmholtz's equation once an interior harmonic function is given, and conversely, given an interior solution of the Helmhotz's equation an interior harmonic function is constructed. Consequently, it seems that the Vekua connection offers the perfect ground to obtain solutions of boundary value problems connected with Helmholtz operator. Vekua expressed his transformation in spherical coordinates. Nevertheless, when a change of coordinates is applied, the transformation assumes a much more complicated form, but it still remains a very useful technique for dealing with solutions of the equations of Laplace and Helmholtz. Here we extend the Vekua theory to a new integral transformation pair concerning solutions of the aforementioned operators in exterior domains. In addition, the form of the Vekua transformation is analyzed in spheroidal coordinates and its implication to boundary value problems is investigated.     

关于丢番图方程 χ^3一У^6=3pz^2的整数解

设p=5（mod 6）为素数．证明了丢番图方程χ^3一У^6=3pz^2。在p=5（mod 12）为素数时均无正整数解;在P=11（mod 12）为素数时均有无穷多组正整数解,并且还获得了该方程全部正整数解的通解公式,同时还给出了该方程的部分整数解．     

二阶常微分方程及Thomas－Fermi方程的边值问题

本文利用上、下解方法推广了Ｍ．Ｌｅｅｓ关于半线性边值问题的结果，给出了另一类边值问题解的存在唯一性的一个充分条件；并利用所得的结果证明了Ｔｈｏｍａｓ－Ｆｅｒｍｉ方程的边值问题解的存在唯一性，从而推广了Ｃ．ＤＬｕｎｉｎｇ的结果．     

ON ANALYTIC SOLUTION OF SCHRDINGER EQUATION AND KLEIN-GORDON EQUATION

1IntroductionConsiderSchrbdingerequationandKlein-GordonequationItiswellknownthatanalyticsolutionofnofllinearevolutionequationhasdrawnanincreasingattentionduringthepastyears,manyresearchers[1--4]havestudiedthesolitarysolutionofequation(1.1))(1.2)bythediffe…     

EXISTENCE AND THE NUMBER OF PERIODIC SOLUTIONS ABOUT ONE CLASS DIFFERENTIAL EQUATION

ＥＸＩＳＴＥＮＣＥＡＮＤＴＨＥＮＵＭＢＥＲＯＦＰＥＲＩＯＤＩＣＳＯＬＵＴＩＯＮＳＡＢＯＵＴＯＮＥＣＬＡＳＳＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮＦｅｎｇＺｈａｏｓｈｅｎｇ（冯兆生）（ＮｏｒｔｈｅｒｎＪｉａｏｔｏｎｇＵｎｉｖｅｒｓｉｔｙ，北方交通大学...     

Exact Solutions to a Finite-difference Model of a Nonlinear Reaction-advection Equation: Implications for Numerical Analysis

This article gives exact solutions to a finite-difference model of a nonlinear reaction-advection equation. We show that this partial difference equation and the corresponding stationary and spatially independent difference equations derived from this model give the best representation of the original partial differential equation. The relevance of this work to the elimination of chaotic behavior in numerical solutions of differential equations is discussed.     

Slowly Oscilating Solutions of Diferential Equations with Delays

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｙｏｆｓｌｏｗｌｙｏｓｃｉｌａｔｉｎｇ（Ｓ．Ｏ．）ｓｏｌｕｔｉｏｎｓｏｆｄｅｌａｙｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗａｓｉｎｉｔｉａｔｅｄｉｎｔｈｅ５０＇ｓｂｙｐｉｏｎｅｅｒｉｎｇｗｏｒｋｏｆＷｒｉ...     

A New Analysis of Volterra-Fredholm Boundary Integral Equations of Second Kind

In this paper, we study the existence, uniqueness and regularity of the solutions to Volterra-Fredholm boundary integral equations of second kind in a kind of boundary function spaces.     

Stability of Solutions for a Class of Quasilinear Degenerate Parabolic Equations

ＳｔａｂｉｌｉｔｙｏｆＳｏｌｕｔｉｏｎｓｆｏｒａＣｌａｓｓｏｆＱｕａｓｉｌｉｎｅａｒＤｅｇｅｎｅｒａｔｅＰａｒａｂｏｌｉｃＥｑｕａｔｉｏｎｓ￥ＺｈａｏＪｕｎｎｉｎｇ（赵俊宁）（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＪｉｌｉｎＵｎｉｖｅｒｓｉｔ...     

一类反应扩散方程解的爆破性

本文讨论反应扩散方程的定态解的存在性以及依赖时间的解的爆破性．     

约化波动方程的振荡性

本文给出了约化波动方程一切球对称解振荡的充分条件,本文的条件与[2]的条件互为独立,而且研究的方法也不同。     

THE ANALYSIS OF EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR SECOND ORDER LINEAR SINGULAR DIFFERENTIAL DIFFERENCE EQUATION WITH DELAY

1IntroductionaccallthefollowingequationssingularfunctionaldifferentialequationsdenotedbySFDE.WhereAisanxmconstantmatrix,dctA=0whenn=m,x(t)ER",tEThebasictheoryofFDEhasbeenwellestablished.ComparedwithFDE,theSFDEismuchmorecofllplicated,whichisshowninthispaper.Inregardtothesingularsystemswithoutdelay,therearemailyresults.ButtherearefewresultsonSingularsystemswithdelay,andtherearcmanyinterestingproblemstobestudiedonthisclassofsystems.Peoplehavediscussedthelinearordinarydifferentialequation…     

含不连续项的常微分积分方程的解

本文研究微分积分方程 u'=g(t,u)+integral from 0 to 1(k(t,s)f(s,u(s))ds),u(0)=x_0最小解、最大解的存在性.本文的特点是关于方程中函数g(t,x),f(t,x)没作任何连续性假定.     

Parabolic regularzation of a first order stochastic partial differential equation

In this paper we use regularization methods for proving the existence and uniqueness of smooth solutions of a first order semilinear stochastic partial differential equation. The regularizations are chosen in such a way so that the known theory of stochastich parabolic Ito equations can be applied. The existence of the generalized solutions and, if the the time parameter is the whole real axis, the existence of mean square bounded generalized solutions, is also considered     

On Singular Dirichlet Problems for Semilinear Elliptic Equation of Second Order Ⅱ

ＯｎＳｉｎｇｕｌａｒ　ＤｉｒｉｃｈｌｅｔＰｒｏｂｌｅｍｓｆｏｒＳｅｍｉｌｉｎｅａｒ　ＥｌｌｉｐｔｉｃＥｑｕａｔｉｏｎｏｆＳｅｃｏｎｄＯｒｄｅｒⅡＺｈａｎｇＺｈｉｊｕｎ（张志军）（ＤｅｐａｒｔｍｅｎｔｏｆＭｏｔｈｅｍａｔｉｃｓ，ＮｏｒｔｈｗｅｓｔＮｏｒ...     

Some Properties of Solutions for a Nonlinear Difusion System(II)

ＳｏｍｅＰｒｏｐｅｒｔｉｅｓｏｆＳｏｌｕｔｉｏｎｓｆｏｒａＮｏｎｌｉｎｅａｒＤｉｆｕｓｉｏｎＳｙｓｔｅｍ（Ｉ）＊）ＷｕＺｈｕｏｑｕｎ（伍卓群）ａｎｄＹｉｎＪｉｎｇｘｕｅ（尹景学）（ＤｅｐａｒｔｍｅｍｔｏｆＭａｔｈｍａｔｉｃｓ，ＪｉｌｉｎＵｎｉｖｅｒｓ...     

四阶非线性常微分方程非线性两点边值问题解的存在唯一性

本文利用Ｂｏｌｚａｎｏ定理，给出了四阶非线性常微微分方程具有非线性边界条件的两点边值问题（１）（２）２，（１）（２）３存在解与存在唯一解的一般性结果，并将所得结果应用于Ｌｉｐｓｃｈｉｚ方程，对Ｌｉｐｓｃｈｉｔｚ方程满足边界条件（２）２，（２）３的边值问题给出了存在解与存在唯一解的具体的充分条件。