Solutions for a class of iterated singular equations

Some fundamental solutions of radial type for a class of iterated elliptic singular equations including the iterated Euler equation are given.     

Symbolic computation of solutions for a forced Burgers equation

In this paper we give exact solutions for a forced Burgers equation. We make use of the generalized Cole-Hopf transformation and the traveling wave method.     

Boundedness of a Derived Function of a Solution About a Class of Diffusion Variational Equations

In this paper the boundedness of a derived function of a solution about a class of diffusion variational equations is discussed. The application of it to related stochastic analysis problems is also illustrated. What should be emphasized is that the problem discussed and the ways proved in this paper are fundamentally new and the conclusion of this paper is fairly profound.     

Properties of stationary solutions of a generalized Tjon-Wu equation

A generalized version of the Tjon-Wu equation is considered. Solutions of this equation are functions with values in the space of probability measures on [0,∞). We prove that the stationary solution μ_∗ of the equation has the following property: Either μ_∗ is supported at one point or suppμ_∗=[0,∞). Moreover we show that in the second case the distribution function of μ_∗ is continuous. Some open questions are discussed.     

一类扩散型变分方程问题

本文讨论了扩散型方程的某些分析性质,得到了一些较为深刻的结论,并应用这些结论证明了一类扩散型变分方程解的存在性。同时,还指出这类变分方程在随机控制的研究中有着重要的理论意义。     

Existence of solutions with a single semicycle for a general second-order rational difference equation

By making use of inclusion theorem, we show in this paper the existence of solutions with a single semicycle for a general second-order rational difference equation. As a corollary, our results positively confirm Conjectures 4.8.3 and 5.4.6 in [M.R.S. Kulenovic, G. Ladas, Dynamics of Second-Order Rational Difference Equations, with Open Problems and Conjectures, Chapman and Hall/CRC, 2002].     

Local solutions to a model of piezoelectric materials

A local existence theorem is proved for a non‐linear coupled system modelling the electromechanical motion of a one‐dimensional piezoelectric body with domain switching. The system is composed by a heat equation describing the behaviour of the number of electric dipoles and by a wave equation governing the dynamic of the electric displacement. The main coupling in the system appears in the time‐dependent velocity of the waves depending on the number of electric dipoles. The proof of the result relies on a time decay estimate satisfied by the number of electric dipoles and an uniform estimate of the solution of the regularized wave equation. Copyright © 2004 John Wiley & Sons, Ltd.     

Analysis of a Diagonal Form of the Fast Multipole Algorithm for Scattering Theory

Discretisation of the integral equations of acoustic scattering yields a system of linear equations with full coefficient matrices. In recent years a number of fast algorithms for the solution of this system have been proposed. In this paper we present a complete analysis for a fast multipole method for the Helmholtz equation. A one-level diagonal form of the multipole method is applied to a hypersingular integral equation arising from 2d scattering theory. The error of the approximation is analysed and the results used to establish the complexity of the method.     

On reflection principles supported on a finite set

We investigate the existence of reflection formulas supported on a finite set. It is found that for solutions of the Laplace and Helmholtz equation there are no finitely supported reflection principles unless the support is a single point. This confirms that in order to construct a reflection formula that is not ‘point to point’, it is necessary to consider a continuous support. For solutions of the wave equation ^∂2u/∂x∂y=0, there exist finitely supported reflection principles that can be constructed explicitly. For solutions of the telegraph equation ^∂2u/∂x∂y+λ²u=0, we show that if a reflection principle is supported on less than five points then it is a point to point reflection principle.     

均匀手性介质中理想导体光栅的电磁散射

§1Introduction Phenomenaofopticalactivityinspecialmaterialshavebeenknownsincethe beginningoflastcentury.Thoughopticalactivityhasbeenconsideredinopticsandin quantummechanicsformanyyears,itsanalysiswithintheframeworkofclassical electromagneticfieldtheoryarosemuchlater.Recently,therehasbeenaconsiderable interestinthestudyofscatteringanddiffractionbychiralmedium.Ingeneral,the electromagneticfieldsinsidethechiralmediumaregovernedbyMaxwellequations togetherwithDrude-Born-Fedorovequationsinwhichth…     

Oscillation and Nonoscillation Criteria for a Second Order Linear Equation

New oscillation and nonoscillation criteria are established for the equation

一个组合方程的单孤子解和周期尖波解

构造一个组合方程的单孤子解和周期尖波解.应用格林函数的性质,以及求一个非线性偏微分方程(简称PDE)弱解的方法.求出了这个组合方程的单孤子解和周期尖波解,推广了前人的研究成果.     

Solving a class of nonautonomous difference equations by generalized invariants

By using generalized invariants, we describe a method for solving a class of higher‐order nonautonomous difference equations. Solvability of the equations of order two, three, and four are studied in detail. Our results explain and extend some problems in the literature. As far as we know, the case when the order is four is considered for the first time in this paper. For the equations of second order, we also give an explanation how they can be obtained in a natural way.     

On a separating solution of a recurrent equation

We consider the recurrent equation

Explicit solution of a Kolmogorov equation

Ever since the technique of the Kalman-Bucy filter was popularized, there has been an intense interest in finding new classes of finite-dimensional recursive filters. In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It has been the major tool in studying the Duncan-Mortensen-Zakai equation. Recently the second author has constructed general finite-dimensional filters which contain both Kalman-Bucy filters and Benes filter as special cases. In this paper we consider a filtering system with arbitrary nonlinear driftf(x) which satisfies some regularity assumption at infinity. This is a natural assumption in view of Theorem 10 of [DTWY] in a special case. Under the assumption on the observation h(x)=constant, we propose writing down the solution of the Duncan-Mortensen-Zakai equation explicitly.This research was supported by Army Grant DAAH-04-93G-0006.     

A note on a class of singular integro-differential equations

A simplified analysis is employed to handle a class of singular integro-differential equations for their solutions     

On a class of functional and differential functional equations

One class of functional and differential functional equations with deviating argument depending on the unknown function is considered in this study. The procedure of constructing and investigating solutions to such equations is also proposed.     

Efficient Implementation of a Spectral-Tau Method for a Class of Two-Point Boundary-Value and Initial-Boundary-Value Problems

A simple and efficient spectral method for solving the second, third order and fourth order elliptic equations with variable coefficients and nonlinear differential equations is presented. It is different from spectral-collocation method which leads to dense, ill-conditioned matrices. The spectral method in this paper solves for the coefficients of the solution in a Chebyshev series, leads to discrete systems with special structured matrices which can be factorized and solved efficiently. We also extend the method to boundary value problems in two space dimensions and solve 2-D separable equation with variable coefficients. As an application, we solve Cahn-Hilliard equation iteratively via first-order implicit time discretization scheme. Ample numerical results indicate that the proposed method is extremely accurate and efficient.