Some properties of multi-degree-of-freedom potential systems and application to statistical equivalent non-linearization

This paper presents some properties of two restricted classes of multi-degree-of-freedom potential systems subjected to Gaussian white-noise excitations. Specifically, potential systems which exhibit damping terms with energy-dependent polynomial form are referred to. In this context, first systems with coupled stiffness terms and damping terms depending on the total energy are investigated. Then, systems with uncoupled stiffness terms and damping terms depending on the total energy in each degree-of-freedom are considered. For these two classes, it is found that algebraic relations among the stationary statistical moments of the energy functions can be derived by applying standard tools of Itô calculus. Further, it is noted that these relations are very useful within the framework of an equivalent statistical non-linearization technique to build approximate solutions for arbitrary non-linear systems.     

Sharp condition of global existence for second-order derivative nonlinear Schrödinger equations in two space dimensions

This paper discusses a class of second-order derivative nonlinear Schrödinger equations which are used to describe the upper-hybrid oscillation propagation. By establishing a variational problem, applying the potential well argument and the concavity method, we prove that there exists a sharp condition for global existence and blow-up of the solutions to the nonlinear Schrödinger equation. In addition, we also answer the question: how small are the initial data, the global solutions exist?     

Separable anisotropic differential operators and applications

The nonlocal boundary value problems for anisotropic partial differential-operator equations with a dependent coefficients are studied. The principal parts of the appropriate generated differential operators are nonself-adjoint. Several conditions for the maximal regularity and the fredholmness in Banach-valued Lp-spaces of these problems are given. These results permit us to establish that the inverse of corresponding differential operators belongs to Schatten q-class. Some spectral properties of the operators are investigated. In applications, the nonlocal BVP's for quasielliptic partial differential equations and for systems of quasielliptic equations on cylindrical domain are studied.     

The function of graphs and gestures in algorithmatization

The purpose of this paper is to present evidence supporting the conjecture that graphs and gestures may function in different capacities depending on whether they are used to develop an algorithm or whether they extend or apply a previously developed algorithm in a new context. I illustrate these ideas using an example from undergraduate differential equations in which students move through a sequence of Realistic Mathematics Education (RME)-inspired instructional materials to create the Euler method algorithm for approximating solutions to differential equations. The function of graphs and gestures in the creation and subsequent use of the Euler method algorithm is explored. If students’ primary goal was algorithmatizing ‘from scratch’, they used imagery of graphing and gesturing as a tool for reasoning. However if students’ primary goal was to make predictions in a new context, they used their previously developed Euler algorithm to reason and used graphs and gestures to clarify their ideas.     

A Nonlinear Drift - Diffusion System with Electric Convection Arising in Electrophoretic and Semiconductor Modeling

A multi-dimensional transient drift-diffusion model for (at most) three charged particles, consisting of the continuity equations for the concentrations of the species and the Poisson equation for the electric potential, is considered. The diffusion terms depend on the concentrations. Such a system arises in electrophoretic modeling of three species (neutrally, positively and negatively charged) and in semiconductor theory for two species (positively charged holes and negatively charged electrons). Diffusion terms of degenerate type are also possible in semiconductor modeling. For the initial boundary value problem with mixed Dirichlet - Neumann boundary conditions and general reaction rates, a global existence result is proved. Uniqueness of solutions follows in the Dirichlet boundary case if the diffusion terms are uniformly parabolic or if the initial and boundary densities are strictly positive. Finally, we prove that solutions exist which are positive uniformly in time and globally bounded if the reaction rates satisfy appropriate growth conditions.     

OSCILLATION OF A FORCED SECOND ORDER NONLINEAR EQUATION

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带耗散的一阶非线性偏微分方程组的Cauchy问题

本文采用整体迭代法证明了带小初值的一阶非线性耗散偏微分方程组的Cauchy问题的整体经典解的存在性及指数衰减性质。     

Cyclic reduction and FACR methods for piecewise hermite bicubic orthogonal spline collocation

Cyclic reduction and Fourier analysis-cyclic reduction (FACR) methods are presented for the solution of the linear systems which arise when orthogonal spline collocation with piecewise Hermite bicubics is applied to boundary value problems for certain separable partial differential equations on a rectangle. On anN×N uniform partition, the cyclic reduction and Fourier analysis-cyclic reduction methods requireO(N ²log₂ N) andO(N ²log₂log₂ N) arithmetic operations, respectively.     

A convection/reaction/switching system

We consider a spatially distributed hybrid system consisting of a convection/reaction system in which the reaction switches discontinuously in time between modes, independently at each spatial point on reaching “switching thresholds”. The model involves a novel formulation for evolution of the free boundary between the modal regions.     

Numerical analysis of time-dependent Boussinesq models

In this paper we analyse numerical models for time-dependent Boussinesq equations. These equations arise when so-called Boussinesq terms are introduced into the shallow water equations. We use the Boussinesq terms proposed by Katapodes and Dingemans. These terms generalize the constant depth terms given by Broer. The shallow water equations are discretized by using fourth-order finite difference formulae for the space derivatives and a fourth-order explicit time integrator. The effect on the stability and accuracy of various discrete Boussinesq terms is investigated. Numerical experiments are presented in the case of a fourth-order Runge-Kutta time integrator.