ON GREEN＇S FUNCTION FOR HYPERBOLIC-PARABOLIC SYSTEMS

We study the Green＇s function for a general hyperbolic-parabolic system, including the Navier-Stokes equations for compressible fluids and the equations for magnetohydrodynamics. More generally, we consider general systems under the basic Kawashima- Shizuta type of conditions. The first result is to make precise the secondary waves with subscale structure, revealing the nature of coupling of waves pertaining to different characteristic families. The second result is on the continuous differentiability of the Green＇s function with respect to a small parameter when the coefficients of the system are smooth functions of that parameter. The results significantly improve previous results obtained by the authors.     

Global Exact Boundary Controllability for Cubic Semi-linear Wave Equations and Klein-Gordon Equations

The authors prove the global exact boundary controllability for the cubic semi-linear wave equation in three space dimensions, subject to Dirichlet, Neumann, or any other kind of boundary controls which result in the well-posedness of the corresponding initial-boundary value problem. The exponential decay of energy is first established for the cubic semi-linear wave equation with some boundary condition by the multiplier method, which reduces the global exact boundary controllability problem to a local one. The proof is carried out in line with [2, 15]. Then a constructive method that has been developed in [13] is used to study the local problem. Especially when the region is star-complemented, it is obtained that the control function only need to be applied on a relatively open subset of the boundary. For the cubic Klein-Gordon equation, similar results of the global exact boundary controllability are proved by such an idea.     

NUMERICAL APPROXIMATION OF THE SMOLUCHOWSKI EQUATION USING RADIAL BASIS FUNCTIONS

The goal of this paper is to present a numerical method for the Smoluchowski equation,a drift-diffusion equation on the sphere,arising in the modelling of particle dynamics.The numerical method uses radial basis functions(RBF).This is a relatively new approach,which has recently mainly been used for geophysical applications.For a simplified model problem we compare the RBF approach with a spectral method,i.e.the standard approach used in related physical applications.This comparison as well as our other accuracy studies show that RBF methods are an attractive alternative for these kind of models.     

孤立波数值方法综述(Ⅰ)

This paper is devoted to introduce the history of numerical study of solitons and the main results for K. D. V. equation, R, L. W. equation, Klein-Gordon equation, Sine-Gordon equation and so on. The technique for constructing difference schemes based on conservation and characteristic is discussed. The two main methods for analysis of stability is given too. Then several finite element methods are discuaaed. The final part of this paper is for various. spectral methods.     

PLANE ELASTIC PROBLEMS OF DIFFERENT MEDIA WITH CRACKS ON THE INTERFACE

The equllibrium problem for the infinite elastic plane consisting of two different media with many cracks on the interface is discussed.It is transferred to a boundary value problem for analytic functions and then further reduced to a singular integral equation,the unique solvability and an effective method of solution for which are eatablished.A practical example in applications is illustrated,the solution of which is obtained in closed form.     

Solving Inhomogeneous Linear Partial Differential Equations

Lagrange＇s variation-of-constants method for solving linear inhomogeneous ordinary differential equations （ode＇s） is replaced by a method based on the Loewy decomposition of the corresponding homogeneous equation. It uses only properties of the equations and not of its solutions. As a consequence it has the advantage that it may be generalized for partial differential equations （pde＇s）. It is applied to equations of second order in two independent variables, and to a certain system of third-order pde＇s. Therewith all possible linear inhomogeneous pde＇s are covered that may occur when third-order linear homogeneous pde＇s in two independent variables are solved.     

PIECEWISE LINEAR NCP FUNCTION FOR QP FREE FEASIBLE METHOD

In this paper,a QP-free feasible method with piecewise NCP functions is proposed for nonlinear inequality constrained optimization problems.The new NCP functions are piece- wise linear-rational,regular pseudo-smooth and have nice properties.This method is based on the solutions of linear systems of equation reformulation of KKT optimality conditions,by using the piecewise NCP functions.This method is implementable and globally convergent without assuming the strict complementarity condition,the isolatedness of accumulation points.Fur- thermore,the gradients of active constraints are not requested to be linearly independent.The submatrix which may be obtained by quasi-Newton methods,is not requested to be uniformly positive definite.Preliminary numerical results indicate that this new QP-free method is quite promising.     

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.     

直接法的推广与广义Burgers方程的条件对称约化

A generalization of the direct method of Clarkson and Kruskal for finding similarity reductions of partial differential equations with arbitrary functions is found and discussed for the generalized Burgers equation. The corresponding reductions and the exact solutions due to the methods of the ordinary differential equations are then given by the methods. The results given here answer partially an open problem proposed by Clarkson, that is how to develop the direct method to seek symmetry reductions of nonlinear PDEs with arbitrary functions.     

Oscillation of Numerical Solution in the Runge-Kutta Methods for Equation x＇（t） = ax（t） ＋ aox（[t]）

The paper de ls with oscillation of Runge-Kutta methods for equation x＇（t） = ax（t） ＋ aox（[t]）. The conditions of oscillation for the numerical methods are presented by considering the characteristic equation of the corresponding discrete scheme. It is proved that any nodes have the same oscillatory property as the integer nodes. Furthermore, the conditions under which the oscillation of the analytic solution is inherited by the numerical solution are obtained. The relationships between stability and oscillation are considered. Finally, some numerical experiments are given.     

Existence of Positive Periodic Solution of Periodic Time-Dependent Predator-Prey System with Impulsive Effects

The general system of differential equations describing predator-prey dynamics with impulsive effects is modified by the assumption that the coefficients are periodic functions of time. By use of standard techniques of bifurcation theory, it is known that this system has a positive periodic solution provided the time average of the predator‘s net uninhibited death rate is in a suitable range.The bifurcation is from the periodic solution of the time-dependent logistic equation for the prey (which results in the absence of predator).     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

Two-direction refinable functions and two-direction wavelets with high approximation order and regularity

The concept of two-direction refinable functions and two-direction wavelets is introduced. We investigate the existence of distributional(or L~2-stable) solutions of the two-direction refinement equation: (?)(x)=(?)p_k~ (?)(mx-k) (?)p_k~-(?)(k-mx), where m≥2 is an integer.Based on the positive mask {p_k~ } and negative mask {p_k~-},the conditions that guarantee the above equation has compactly distributional solutions or L~2-stable solutions are established.Furthermore,the condition that the L~2-stable solution of the above equation can generate a two-direction MRA is given.The support interval of (?)(x) is discussed amply.The definition of orthogonal two-direction refinable function and orthogonal two-direction wavelets is presented,and the orthogonality criteria for two-direction refinable functions are established.An algorithm for construct- ing orthogonal two-direction refinable functions and their two-direction wavelets is presented.Another construction algorithm for two-direction L~2-refinable functions,which have nonnegative symbol masks and possess high approximation order and regularity,is presented.Finally,two construction examples are given.     

A Decomposition Theorem for the Solutions to the Interface Problems of Quasi-Linear Elliptic Equations

Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the form cr^a log^m rφ(θ), where the coefficients c depend on the H^1-norm of the solution, the C^(0,δ) -norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated.     

Asymptotic Analysis in a Gas-Solid Combustion Model with Pattern Formation

The authors consider a free interface problem which stems from a gas-solid model in combustion with pattern formation. A third-order, fully nonlinear, self-consistent equation for the flame front is derived. Asymptotic methods reveal that the interface approaches a solution to the Kuramoto-Sivashinsky equation. Numerical results which illustrate the dynamics are presented.     

A CLASS OF BOUNDARY PROBLEMS FOR SOME HYPERBOLIC EQUATIONS

A boundary problem for the Klein-Gordon equation in the strip O≤t≤T is considered with the boundary condition:the initial state at t=O and the final state at t=T.It is proven that the problem admits of an infinite number of solutions.The same result holds for a generic 2nd order hyperbolic equation in 2-variables.Using the result for the wave operator in 3-space dimensions we give a method to reconstruct functions whose integral on all unit spheres in R~3 is a given function.     

Surface Reconstruction of 3D Scattered Data with Radial Basis Functions

We use Radial Basis Functions （RBFs） to reconstruct smooth surfaces from 3D scattered data. An object＇s surface is defined implicitly as the zero set of an RBF fitted to the given surface data. We propose improvements on the methods of surface reconstruction with radial basis functions. A sparse approximation set of scattered data is constructed by reducing the number of interpolating points on the surface. We present an adaptive method for finding the off-surface normal points. The order of the equation decreases greatly as the number of the off-surface constraints reduces gradually. Experimental results are provided to illustrate that the proposed method is robust and may draw beautiful graphics.     

A Criterion of Existence of a Periodic Solution for n-th Order Systems of Autonomous Differential Equations (n\geq 3)

There have been many results^[1-7] on seeking a periodic solution for systems of autonomous differential equations by using the torus principl. The finest one amony them is still Liapunov’s method of rotating functions proposed by В. B. Nemytskii[3’4]. However, this method is limited in the case that the absolute values of derivatives of rotating functions are greater than some positive constants. This paper summarizes the methods proposed in[3-5] and gives a more careful criterion such that derivatives of rotating functions might be alternative functions. The author provides a sufficient condition of existence of a periodic solution for systems of autonomous differential equations by Using Liapunov’s method of rotating functions under more general circumstances.     

On a Linear Equation Arising in Isometric Embedding of Torus-like Surface

The solvability of a linear equation and the regularity of the solution are discussed. The equation is arising in a geometric problem which is concerned with the realization of Alexandroff＇s positive annul in R^3.     

Fourth-Order Splitting Methods for Time-Dependant Differential Equations

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes,e.g.,wave-propagation or heat-transfer,that are modeled by wave equations or heat equations.Here,we study both parabolic and hyperbolic equations.We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods,which are standard splitting methods of lower order,e.g.second-order.Our aim is to develop higher-order ADI methods,which are performed by Richardson extrapolation,Crank-Nicolson methods and higher-order LOD methods,based on locally higher-order methods.We discuss the new theoretical results of the stability and consistency of the ADI methods.The main idea is to apply a higher- order time discretization and combine it with the ADI methods.We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives.The higher-order methods are unconditionally stable.Some numerical experiments verify our results.