Second-order two-scale analysis and numerical algorithms for the hyperbolic–parabolic equations with rapidly oscillating coefficients

We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.

Regularity Results for a Nonlinear Elliptic-Parabolic System with Oscillating Coefficients

In this paper we study the initial boundary value problem for the system div(σ(u)∇φ)=0,u_(t)−∆u=σ(u)|∇φ|2.This problem is known as the thermistor problem which models the electrical heating of conductors.Our assumptions onσ(u)leave open the possibility that lim inf_(u→∞)σ(u)=0,while lim sup_(u→∞)σ(u)is large.This means thatσ(u)can oscillate wildly between 0 and a large positive number as u→∞.Thus our degeneracy is fundamentally different from the one that is present in porous medium type of equations.We obtain a weak solution(u,ϕ)with|∇φ|,|∇u|∈L∞by first establishing a uniform upper bound for eεu for some smallε.This leads to an inequality in∇φ,from which the regularity result follows.This approach enables us to avoid first proving the Holder continuity ofφin the space variables,which would have required that the elliptic coefficientσ(u)be an A2 weight.As it is known,the latter implies that lnσ(u)is“nearly bounded”.

The Diffusion Coefficient Using Sawtooth Oscillation in IR-T1 Tokamak

The investigation of runaway electrons is expanded by different methods. The aim of this study is to show sawtooth oscillations of hard x-ray emission and with the help of sawtooth oscillations to obtain radial diffusion coefficient and magnetic fluctuations. In the same way, the hard x-ray spectral evaluation is compared in several time intervals and it is shown that during discharge, the energy of the runaway electrons is less than 200？keV. Also, for typical plasmas, population of runaway electrons is measured at seven time intervals of 5ms and temporal evaluation of runaway electron mean energy. The sawtooth-like shape is observed in the hard x-ray range （10–1000？keV）. By the sawtooth oscillation method, the RE diffusion coefficient in radial transport in the IR-T1 plasma is Dr～0.5Dr～0.5m^2s^-1. The magnetic field fluctuation due to magnetic diffusion DmDm is given as br/Bt～10^-4.

Estimates for Green’s Functions of Elliptic Equations in Non-Divergence Form with Continuous Coefficients

We present a new method for the existence and pointwise estimates of a Green's function of non-divergence form elliptic operator with Dini mean oscillation coefficients.We also present a sharp comparison with the corresponding Green5s function for constant coefficients equations.

OSCILLATIONS OF SOLUTIONS OF NEUTRAL DIFFERENTIAL EQUATIONS WITH VARIABLE COEFFICIENTS AND DELAYS

Consider the neutral differential equations with variable coefficients and delays [x(t)-p(t)x(t-r(t))]'+ Qj(t)x(t-σj(t))=0. (1)We establish sufficient conditions for the oscillation of equation (1). Our condition is 'sharp' in the sense that when all the coefficients and delays of the equation are constants.Our conclusions improve and generalize some known results.

Transient Mould Friction Based on the Wavelet Theory

The mould friction is an important parameter reflecting the initial shell and mould lubrication conditions.The research of mould friction is very important to the optimization and developing of continuous casting.The measured mould friction under hydraulic oscillation mode was researched with wavelet analysis to reveal its time-frequency characteristics.Firstly,the mould friction signals under different production conditions were monitored and the mould friction was calculated.Then,mother wavelet function was selected from three wavelet functions which were chosen preliminary according to the characteristics of mould friction and wavelet theory.Through wavelet transformation,mould friction signal was projected onto the wavelet domain,and the time-frequency characteristics of mould friction under different production conditions were obtained and discussed.Mould friction under different production conditions such as different oscillation mode,casting speed fluctuation,increasing and decreasing stage of casting speed and breakout occurrence was reported in detail in the wavelet time-frequency maps.The characteristics of mould friction were reflected well through wavelet transformation which proved that wavelet analysis had a good feasibility for mould friction study and can further reveal the nature of mould friction.