位势

Initial boundary value problem of two dimensional unsteady diffusion equations is solved in this paper. By using time-dependent fundamental solution of two dimensional diffusion equations and the extension of double layer potential, we establish virtual boundary integral expression of diffusion equations. Then virtual boundary element method is used to implement the numerical computation.

本文在求解二维非定常扩散方程的初边值问题时，利用二维扩散方程与时间有关的基本解，基于双层位势的延拓，建立二维扩散方程的虚边界积分表达式，然后利用虚边界元法进行数值计算。

In this thesis, the virtual boundary integral equation is based on double layer potential with the virtual density to be determined on virtual boundary. Since this integral equation related to double layer potential only involves the computation of the normal derivative and second normal derivative of fundamental solution, the exponential integral function is not involved in it, so numerical computation for the exponential integral function is avoid.

本文则基于双层位势，引入虚拟矩密度函数来建立虚边界积分方程，并首先对时间变量进行解析积分，在虚、实边界上采用常单元和等额配点离散，该方程只涉及含基本解的法向导数及其二阶法向导数项的计算，对时间变量进行解析积分后，不会出现对指数积分函数的空间变量的积分计算。

Virtual boundary element; equivalent collation method; double layer potential; linear diffusion equation

基础科学，数学，数学分析虚边界元；等额配点法；双层位势；扩散方程

As known, using the porential theory Green's tormulae and the fundamental solutions wecan reduce the harmonic and biharmonic elliptic boundary value problems to integral equationson the boundary.

我们熟知，利用位势理论或由Green公式及基本解出发区域内调和及重调和边值问题可归化为边界上的积分方程。

The general forms of the conservation of momentum, t emperature and potential vorticity of coastal ocean are obtained in the xz pla ne for the nonlinear ocean circulation of Boussinesq fluid, and a elliptic type partial differential equations of second order are derived. Solution of the part ial differential equations are obtained under the conditions that the fluid move s along the topography.

在考虑了陆架地形后，在垂直海岸的xz剖面上，对Boussinesq流体的非线性海洋运动方程求得了总动量守恒、温度守恒和位势涡度守恒的普适形式，进而得到流函数所满足的椭圆型二阶偏微分方程，在给定流体沿地形运动的条件下，算出问题的解。

The contents are the following:In chapter two, the existence and multiplicity results for the following equation of p-Laplacian type are obtained.For the elliptic quasilinear hemivariational inequality involving the p-Laplacian operator,in order to use the mountain pass theorem proving the existence result, the authors usually need to use the uniform convexity of the Sobolev space to prove the energy function satisfies the PS condition. But for the p-Laplacian type equation mentioned above, this method is no use. To overcome this difficulty, the potential function is assumed to be convex, then I prove the existence result and by using the extension of the Ricceri theorem, the multiplicity result for the problem is obtained.

在第二章我们首先考虑关于以下p-Laplacian型(p-Laplacian type)方程非平凡解及多解的存在性对于带有p-Laplacian算子的椭圆拟线性半边分不等式问题，为应用非光滑的山路引理证明解的存在性，在证明方程所对应的能量泛函满足非光滑的PS条件时，需利用Sobolev空间的一致凸性，但是对于具有更一般形式的算子的p-Laplacian型方程，不具备上述性质，在文中为克服这一困难，本人对位势泛函做了一致凸的假设，从而证明了解的存在性，并应用推广的Ricceri定理，证明了方程三个解的存在性。

Potential theory is used to transfer the exterior boundary value problem into the boundary integral equation of the first kind. A iterated Tikhonov regularization and a modified Tikhonov regularization are presented to solve the integral equation.

利用位势理论将外边值问题转化为第一类边界积分方程，再分别利用迭代的Tikhonov正则化方法和改进的Tikhonov正则化方法求解积分方程。

By using the method of the gauge transformation,this paper obtains the gauge transformation of Eigenvalue problems of energy dependent potentials and the Backlund transformation.

用规范变换的方法，导出了位势依赖于能量的特征值问题的规范变换及相应的贝克隆变换。

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。

After study the potential energy of the dc-SQUID, the Hamiltonian of the device is presented.

通过引入直流超导量子干涉器的相位势，给出系统的哈密顿量。

This paper discusses design and realizable methods of remote test output interface from logical design angle.

本文从逻辑设计的角度讨论遥测输出接口的设计及实现方法。

This also 星体投射plies to buildings, structures and geological features.

这也适用于建筑物和地质特征。