特征方程

This section presents two kinds of characteristic schemes, one is the natural use of the usual backward Euler characteristic difference scheme ( [60] ) onto the triangular mesh, the other replaces the backward Euler's method by modified Euler formula to approximate the characteristic differential equation, so that the discretization for characteristic curve can get up to second order accuracy.

本节给出了两种三角形网格上的特征差分格式，第一种是通常的向后Euler型特征差分格式（[60]）在三角形网格上的自然运用，第二种采用§3.3、§3.4的离散思想，使用改进的Euler公式代替向后Euler公式来近似特征微分方程，使其对特征线的离散达到了二阶精度。

Douglas and Russell first presented the idea of characteristics in[13],in whichthe diffusion term is included in the derivative of characteristic direction of the solu-tion and the backward difference quotient along the time axis was replaced by it alongcharacteristic direction.

Douglas，Russell[13]最先提出了特征法的思想，其主要特点是将模型方程的对流项隐含在解沿特征方向的导数之中，以沿特征方向的向后差商逼近取代通常方法中沿时间方向的向后差商逼近。

Further, with the help of Riccati equations, an infinite number of conservation laws for the solton hierarchy are deduced. For the sake of simplicity, taking the general TD hierarchy as an illustrative example, we prove that its 2×2 Lenard pair of operators forms a Hamiltonian pair. Thus the isospectral evolution TD hierarchy is the general Hamiltonian system and possesses the Bi-Hamiltonian structures and Multi-Hamiltonian structures. By using the method of derivation of functional under some constraint condition, a complete one-to-one correspondence between the Hamiltonian functions of the hierarchy and its conservation density functions can be built. These results can also be applied to the isospectral evolution soliton hierarchy of this paper. Finally, there's a gauge transformation between the spectral problem of this paper and the AKNS system. Moreover, the potentials in these spectral problems satisfy the general Miura transformation, the corresponding relationship between the two soliton hierarchies is also given.

进一步本文还通过特征函数的组合关系所满足的Riccati方程，得到了该等谱方程族的无穷多个守恒律；为简便起见，本文以广义TD族为例，由它的2×2 Lenard算子对的性质证明了此算子对为Hamilton算子对，这说明广义TD族是广义Hamilton系统且具有Bi-Hamilton结构和Multi-Hamilton结构；进而利用它的依赖于谱参数的一般守恒密度的积分在约束条件下求泛函导数的方法，得到了广义TD族的Hamilton函数与守恒密度之间的对应关系，这些性质对于由本文提出的2×2谱问题所导出的等谱孤子族仍成立；另外此谱问题与AKNS系统存在着规范变换，位势之间有广义Miura变换，而孤子方程之间也满足一定的等价关系。

The main characteristics include: the quiet Sun corona has a rapid response to the flare heating, within 2 second the corona temperature increases nearly one order of magnitude, but later such increase becomes very slowly; the downward velocity at early stage of impulsive phase is too small to be negnected; chromospheric evaporation made by conduction shows some intensely, the maximum evaporated velocity may be as large as 1000 km/s; the chromospheric condensation appears just when the chromospheric evaporation takes place and its place always coincide with that of transition region in the early stage; the maximum of chromospheric condensation occurs also in the early stage of impulsive phase, at which the density in condensation may be larger two order of magnitude than that of surround and the downward velocity may be as large as 100 km/s; after the maximum of chromospheric condensation, its strongth decreases gradually but its width becomes greater with time and have a tendency to transport into deep chromosphere; the front of chromospheric condensation heats atmosphere obviously and the temperature in the chromospheric condensation is higher than the original case; the downware velocity of transition region may be greater than 100 km/s at the early stage; when the evaporated material collide at the top of loop, the densityand the presure there increase rapidly, which result in the formation of a shock wave; and so on.

在改善色球辐射损失和精确分辨过渡区的基础上，通过联立求解连续性方程、动量方程和能量方程，构造了一维磁限制性耀斑环在热沉积作用下的大气动力学模型，获得了耀斑热模型下脉冲相早期较为细致的大气演化理论图象，其主要特征是：耀斑对宁静日冕的加热相当迅速，仅2秒钟日冕温度增加即达一个量级，之后日冕温度增加缓慢；脉冲相早期日冕向下速度很小，几乎可以忽略，热传导驱动色球蒸发，其过程较为激烈，蒸发极大速度可达到1000km/s；色球压缩区与色球蒸发几乎同时形成，且在早期其位置总是与过渡区位置重合；色球压缩极大发生在脉冲相早期，其时密度可高于周围约2个量级，向下速度可超过100km/s；色球压缩极大后，其强度逐渐减弱，但宽度明显增大，且渐渐脱离过渡区往色球传播；压缩区内物质温度要高于原先的大气温度；过渡区下降在脉冲相早期相当迅速，可超过100km/s；当蒸发物质在环顶高速相碰时，环顶的密度、压力迅速增大，且温度也有所增加，导致形成激波，其波面以每秒上百公里的速度自环顶向下传播；在脉冲相前期，日冕软x辐射对色球的加热作用可以忽略；等。

Curvature is the result of geometric structure in image; a new third order partial differential equation is proposed which transports curvature information along isophotes. The third order POE is morphologically invariant. Helmholtz vorticity equation which annihilates vorticity and third order POE which forward transports curvature information are coupled to be a new model.

曲率是图像几何特征作用的结果，本文提出沿等照度线方向传输曲率信息的三阶形态学不变偏微分方程，并将抑制涡量的无粘亥姆霍兹方程和正向传输曲率信息的三阶偏微分方程耦合。

In this paper, we study nonlinear perturbation analysis of mesoscale verticalcirculation and evolution equation by means of f plane, non-hydrostatic balance andfilter model. Outstanding nonlinear characteristic of mesoscale disturbance invertical direction, we obtain nonlinear symmetric disturbance equation anddisturbance amplitude s evolution equation by use of multi-scale and singularperturbation technique.

中文摘要：本文采用f平面、非静力平衡、滤声波模式，讨论了大气中尺度非线性动力学方程，通过尺度分析重点突出了扰动在垂直方向上的非线性特征，从而得到了对称扰动非线性方程，运用多重尺度方法和摄动分析讨论了对称扰动非线性重力惯性波的演化过程，得到非线性重力惯性波振幅演化服从广义的非线性Schrdinger方程。

From research,It is known that the flux of single-phase transformer would offset and exciting current of single-phase transformer would be distorted under DC current inrushing.

直流电流通过两个Y0连接的变压器中性点侵入电力系统，故针对三相三柱变压器直流入侵情况建立了直流入侵下的三相三柱变压器的电路模型和相应的方程，结合三柱磁路的麦克斯韦方程和非线性的B—H曲线方程，利用数字迭代法，在MATLAB环境下编程计算了三相三柱变压器直流入侵下的变压器励磁电流和谐波特征。

According to the plastic equilibration equation and the theory of characteristic line, the stress equations of slip line in axial symmetry deformation on compressible materials were derived on basis of simplified yield criteria in this paper.

本文在可压缩材料简化屈服准则的基础上，根据塑性平衡方程和特征线理论推证出了可压缩材料轴对称问题的滑移线应力方程，提出了应力方程数值差分解

And then the influence of two of the damage models, the Kachanov and the gradient-dependent constitutive equations, on the well-posed properties of the fundamental equations in continuum damage mechanics is studied according to the characteristic method of the quasi-linear partial differential equations under the case of one dimension elastic damage.

首先从非局部理论出发，推导了应变梯度损伤本构方程；然后利用一阶拟线性偏微分方程组的特征理论，在一维弹性损伤情况下分析了两种不同的本构模型，即Kachanov损伤本构方程与应变梯度损伤本构方程，对连续介质损伤力学基本方程适定性的影响。

Concerned in this paper are the eigenvalues and the corresponding eigenfunctions for the crosswise vibration equation of a bean with continuous load, which can be related to a differential integral equation.

本文研究了附加连续荷载梁的横向振动方程的特征值及特征函数的性质。在一定的假设之下，附加连续荷载梁的横向振动方程对应一个微分积分方程。

As she looked at Warrington's manly face, and dark, melancholy eyes, she had settled in her mind that he must have been the victim of an unhappy attachment.

每逢看到沃林顿那刚毅的脸，那乌黑、忧郁的眼睛，她便会相信，他一定作过不幸的爱情的受害者。

Maybe they'll disappear into a pothole.

也许他们将在壶穴里消失