On the formal solution of initial value problem of Navier-Stokes equation

Ｉｎｒｅｆｅｒｅｎｃｅ [1 ] ,ｗｅｄｉｓｃｕｓｓｅｄｔｈｅｉｎｓｔａｂｉｌｉｔｙｏｆＮａｖｉｅｒ＿ＳｔｏｋｅｓｅｑｕａｔｉｏｎｏｎＲ3×Ｒ ,ａｎｄ ,ｉｎｔｈｅｉｌｌｕｓｔｒａｔｉｏｎｏｆｉｔ,ｗｅｐｒｅｓｅｎｔｅｄｓｏｍｅｅｘａｍｐｌｅｓｉｎｔｈｅｆｏｒｍｏｆｅｘａｃｔｓｏｌｕｔｉｏｎ .Ｔｈｅｂａｓｉｓｏｆｔｈｅｃｏｎｓｔｒｕｃｔｉｏｎｏｆｔｈｅｍｉｓｔｈｅ“ｅｑｕａｔｉｏｎｓｅｃｏｎｄａｉｒｅ”ｏｆｔｈｅｅｑｕａｔｉｏｎ .Ｉｎｆａｃｔ,ｔｈｅｓｅｅｘａｃｔｓｏｌｕｔｉｏｎｓａｒｅｓｏｍｅｌｉｍｉｔ…     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅰ)

The ill posed initial value problem of the Euler equations and the formal solvability of ill posed problem based on stratification theory are discussed. For some ill posed initial value problems, the existence conditions of formal solutions and the methods of how to construct a formal solution are given. Finally, an example is given to discuss the ill posedness of the initial value problem on hyper plane {t=0} in R⁴, and explain that the problem has more than one solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

New Explicit and Exact Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ ,ｔｈｅｓｔｕｄｙｋｅｒｎｅｌｏｆｍｏｄｅｒｎｓｃｉｅｎｃｅｉｓｃｈａｎｇｅｄｆｒｏｍｌｉｎｅａｒｔｏｎｏｎｌｉｎｅａｒｓｔｅｐｂｙｓｔｅｐ .Ｍａｎｙｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅｐｒｏｂｌｅｍｓｃａｎｓｉｍｐｌｙａｎｄｅｘａｃｔｌｙｂｅｄｅｓｃｒｉｂｅｄｂｙｕｓｉｎｇｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｏｆｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎ .Ｕｐｔｏｎｏｗ ,ｍａｎｙｉｍｐｏｒ…     

Global solution of the inverse problem for a class of nonlinear evolution equations of dispersive type

The inverse problem for a class of nonlinear evolution equations of dispersive type was reduced to Cauchy problem of nonlinear evolution equation in an abstract space. By means of the semigroup method and equipping equivalent norm technique, the existence and uniqueness theorem of global solution was obtained for this class of abstract evolution equations, and was applied to the inverse problem discussed here. The existence and uniqueness theorem of global solution was given for this class of nonlinear evolution equations of dispersive type. The results extend and generalize essentially the related results of the existence and uniqueness of local solution presented by YUAN Zhong-xin. Contributed by Chen Yu-shu Foundation item: the National Natural Science Foundation of China (Significance 199990510); the National Key Basic Research Special Foundation of China (G1998020316); Liuhui Center for Applied Mathematics, Nankai University & Tianjin University Biography: Chen Fang-qi (1963-)     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOWWATER EQUATIONS INCLUDING CURRENT AND SILTSEDIMENTATION （Ⅱ）——THE DISCRETE-TIMECASE ALONG CHARACTERISTICS

The mixed finite element(MFE) methods for a shallow water equation systemconsisting of water dynamics equations, silt transport equation, and the equation of bottomtopography change were derived. A fully discrete MFE scheme for the discrete-time alongcharacteristics is presented and error estimates are established. The existence andconvergence of MFE solution of the discrete current velocity, elevation of the bottomtopography, thickness of fluid column, and mass rate of sediment is demonstrated.     

ON THE WELL POSEDNESS OF INITIAL VALUE PROBLEM FOR EULER EQUATIONS OF INCOMPRESSIBLE INVISCID FLUID(Ⅱ)

The solvability of the Euler equations about incompressible inviscid fluid based on the stratification theory is discussed. And the conditions for the existence of formal solutions and the methods are presented for calculating all kinds of ill-posed initial value problems. Two examples are given as the evidences that the initial problems at the hyper surface does not exist any unique solution. Foundation item: the National Natural Science Foundation of China (19971054) Biography: Shen Zhen (1977−)     

MIXED FINITE ELEMENT METHODS FOR THE SHALLOW WATER EQUATIONS INCLUDING CURRENT AND SILT SEDIMENTATION （Ⅰ）——THE CONTINUOUS-TIME CASE

An initial-boundary value problem for shallow equation system consisting of water dynamics equations, silt transport equation, the equation of bottom topography change, and of some boundary and initial conditions is studied, the existence of its generalized solution and semidiscrete mixed finite element (MFE) solution was discussed, and the error estimates of the semidiscrete MFE solution was derived. The error estimates are optimal.     

On the stability of general Navier-Stokes type equation

Ｉｎｔｈｉｓｐａｐｅｒ,ｗｈａｔｗｅｄｉｓｃｕｓｂｅｌｏｎｇｓｔｏａｓｔａｂｉｌｉｔｙｐｒｏｂｌｅｍａｂｏｕｔｔｈｅｓｅｃｏｎｄｃｌａｓｇｅｎｅｒａｌＮａｖｉｅｒ＿Ｓｔｏｋｅｓｅｑｕａｔｉｏｎｓ．Ｉｔｔｏｕｃｈｅｓｓｏｍｅｉｍｐｏｒｔａｎｔｐｒｏｂｌ...     

2<Emphasis Type="Italic">T</Emphasis>-periodic solution for<Emphasis Type="Italic">m</Emphasis> order neutral type differential equations with time delays

Periodic solution of m order linear neutral equations with constant coefficient and time delays was studied. Existence and uniqueness of 2 T-periodic solutions for the equation were discussed by using the method of Fourier series. Some new necessary and sufficient conditions of existence and uniqueness of 2T-periodic solutions for the equation are obtained. The main result is used widely. It contains results in some correlation paper for its special case, improves and extends the main results in them. Existence of periodic solution for the equation in larger number of particular case can be checked by using the result, but cannot be checked in another paper. In other words, the main result in this paper is most generalized for (1), the better result cannot be found by using the same method. Foundation item: the Natural Science Foundation of Yunnan Education Committee (990002) Biography: ZHANG Bao-sheng (1962-)     

Similarity reductions for 2+1-dimensional variable coefficient generalized kadomtsev-petviashvili equation

ＩｎｔｒｏｄｕｃｔｉｏｎＤｕｒｉｎｇｔｈｅｓｔｕｄｙｏｆｓｈａｌｌｏｗｗａｔｅｒｗａｖｅｓ,ｐｅｏｐｌｅｏｂｔａｉｎｅｄｍａｎｙｗｅｌｌ＿ｋｎｏｗｎｃｏｍｐｌｅｔｅｌｙｉｎｔｅｇｒａｂｌｅｍｏｄｅｌｓ,ｓｕｃｈａｓＫｄＶｅｑｕａｔｉｏｎ,ｍＫｄＶｅｑｕａｔｉｏｎ,Ｂｏｕｓｓｉｎｅｓｑｅｑｕａｔｉｏｎ,Ｗｈｉｔｈａｍ＿Ｂｒｏｅｒ＿ＫａｕｐｅｑｕａｔｉｏｎａｎｄＫｈｏｋｈｌｏｖ＿Ｚａｂｏｌｏｔｓｋａｙａｅｑｕａｔｉｏｎｅｔｃ.[1～8].Ｆｏｒ2 1＿ｄｉｍｅｎｓｉｏｎａｌｖａｒｉａｂｌｅｃｏｅｆｆｉｃｉｅｎｔｇ…     

THE MULTI-SYMPLECTIC ALGORITHM FOR “GOOD” BOUSSINESQ EQUATION

The multi-symplectic formulations of the “Good” Boussinesq equation were considered. For the multi-symplectic formulation, a new fifteen-point difference scheme which is equivalent to the multi-symplectic Preissman integrator was derived. The numerical experiments show that the multi-symplectic scheme have excellent long-time numerical behavior. Foundation items: the Foundation for Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences; the Natural Science Foundation of Huaqiao University. Biography: ZENG Wen-ping (1940-), Professor (E-mail: qmz@1sec.cc.ac.cn)     

The solution of weak formulation for axisymmetric problem of orthotropic cantilever cylindrical shell

Weak formulation of equilibrium equations including boundary conditions of laminated cylindrical shell is presented, and thermal stresses mixed state equation for axisymmetric problem of closed cantilever cylindrical shell is established. A unified approach and weak solutions are obtained for closed laminated cantilever cylindrical shell of arbitrary thickness under thermal and mechanical loadings. The equation and boundary conditions proposed in this paper are weakened, the method of this paper would be easy to popularize in dynamics analysis of elasticity. Foundation item: the National Natural Science Foundation of China (19392305)     

The 3-layered explicit difference scheme for 2-D heat equation

A 3-layered explicit difference scheme for the numerical solution of 2-D heat equation is proposed. Firstly, a general symmetric difference scheme is constructed and its optimal error is obtained. Then two kinds of condition for choosing the parameters for optimal error and stable difference scheme are given. Finally, some numerical results are presented to show the advantage of the schemes Foundation items: the Science Foundation of Chinese Postdoctoral (2002031224); the Science Foundations of Southeast University (9209011148, 3007011043) Biography: Liu Ji-jun (1965-)     

The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｏｂｌｅｍｉｎｔｈｅＲｅｓｅａｒｃｈｏｆＴｏｒｏｉｄＴｈｉｓｐａｐｅｒｄｅａｌｓｗｉｔｈｔｈｅｅｘｉｓｔｅｎｃｅｏｆ2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｔｏｔｈｅｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｏｆｆｉｒｓｔ＿ｏｒｄｅｒｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｗｉｔｈａｄｅｖｉａｔｉｎｇａｒｇｕｍｅｎｔ ｘ(ｔ) =Ｂｘ(ｔ) Ｆ(ｘ(ｔ-τ) ) ｐ(ｔ) ,( 1 )ｗｈｅｒｅｘ(ｔ)∈Ｒ2 , ｘ(ｔ) =ｄｄｔｘ(ｔ) ,τ∈Ｒ ,Ｂ∈Ｒ2×2 ,Ｆ :Ｒ2 →Ｒ2 ｉｓｂｏｕｎｄｅｄａｎｄｐ∈Ｃ(…     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅲ)——RE-DISCUSS THE LINEAR THEORY OF NONLOCAL ELASTICITY

In this paper, it is proven that the balance equation of energy is the first integral of the balance equation of momentum in the linear theory of nonlocal elasticity. In other words, the balance equation of energy is not an independent one. It is also proven that the residual of nonlocal body force identically equals zero. This makes the transform formula of the nonlocal residual of energy much simpler. The linear nonlocal consitutive equations of elastic bodies are deduced in details, and a new formula to calculate the antisymmetric stress is given. Foundation item: the Natural Science Foundation of Jiangsu Province, China (BK97063)     

The high accuracy explicit difference scheme for solving parabolic equations 3-dimension

ＩｎｔｒｏｄｕｃｔｉｏｎＦｒｏｍｐｒａｃｔｉｃａｌｐｒｏｂｌｅｍ,ｗｅｃｏｕｌｄｃｏｎｃｌｕｄｅｍａｎｙｐｒｏｂｌｅｍｓａｂｏｕｔｓｏｌｖｉｎｇｐａｒａｂｏｌｉｃｐａｒｔｉａｌｄｉｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎ．Ｎｏｗｔｈｅｒｅａｒｅｍａｎｙｎｕ...     

A Simple Fast Method in Finding Particular Solutions of Some Nonlinear PDE

1　ＡＴｒｉａｌＦｕｎｃｔｉｏｎａｎｄａＲｏｕｔｉｎｅｔｏＦｉｎｄＡｎａｌｙｔｉｃａｌＳｏｌｕｔｉｏｎｏｆＴｗｏＴｙｐｅｓｏｆＮｏｎｌｉｎｅａｒＰＤＥ　　Ｗｅｔｒｅａｔｔｈｅｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎ ,ｗｈｉｃｈｉｓｆｏｒｍｅｄｂｙａｄｄｉｎｇｈｉｇｈｏｒｄｅｒｄｅｒｉｖａｔｉｖｅｔｅｒｍｓａｎｄｎｏｎｌｉｎｅａｒｔｅｒｍｓｔｏｔｈｅＢｕｒｇｅｒｓｅｑｕａｔｉｏｎ ｕ ｔ ｕ ｕ ｘ … ｕｐ ｕ ｘｑ α1 ｕ ｘ … αｎ ｎｕ ｘｎ =0 ,( 1)ｗｈｉｃｈｐ ,ｑ ,ｎａｎｄαｉ(ｉ =1,2…     

Poincaré-cartan integral invariants of Birkhoffian systems

Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincaré-Cartan’s type is constructed for Birkhoffian systems. Finally, one-dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincaré’s type is found.     

A CLASS OF ALTERNATING GROUP METHODOF BURGERS＇ EQUATION

Some new Saul‘ yev type asymmetric difference schemes for Burgers’ equation isgiven, by the use of the schemes, a kind of alternating group four points method for solvingnonlinear Burgers‘ equation is constructed here. The basic idea of the method is that thegrid points on the same time level is divided into a number of groups, the differenceequations of each group can be solved independently, hence the method with intrinsicparallelism can be used directly on parallel computer. The method is unconditionally stableby analysis of linearization procedure. The numerical experiments show that the method has good stability and accuracy.