变系数偏微分方程组一般解的构造

求解偏微分方程不仅有理论意义,而且有实用价值.本文利用共轭算子的性质给出构造偏微分方程组一般解的方法.     

Solution to a Class of Coupled Linear Partial Differential Equations

The solution to a coupled system of partial differential equationsinvolving a general linear time-independent operator L is presented.Examples of these equations include coupled diffusion equationsor coupled convection–dispersion equations. The solutionconsists of a convolution of the Green's function appropriatefor the operator L and a function independent of the operatorL. The method enables one to write software to calculate thesolution to a wide range of problems. The change of solutionupon changing the problem often only involves a substitutionof the Green's function. A specific example of physical significanceis given.     

On Chebyshev Solution of Parabolic Partial Differential Equations

Now at Mathemarics Department, Assiut University Egypt A method is presented to transform parabolic equations to asystem of ordinary differential equations for the solution atthe Chebyshev points. The system may be solved analyticallyor by numerical methods and the Chebyshev coefficients are computed.We have the exact solution of a perturbed problem.     

从偏导数恒等式变换到偏微分方程求解

从高等数学教材课后习题的偏导数恒等式变换求解,引导学生讨论一类偏微分方程的求解.在拓展课程内容、应用和常微分方程变量分离方法的基础上,巩固多元复合函数求导法则,常系数线性微分方程求解方法和傅里叶级数的相关理论与方法.     

关于某些非线性偏微分方程的准确解

以分层理论为基础，对偏微分方程给以一种全新的分类．     

非游荡算子半群标准及应用

根据非游荡算子半群的定义得到了非游荡算子半群的几个性质,给出了判定算子半群是非游荡半群的标准,应用给出的标准,在空间C([0,1],C)上讨论了偏微分方程au/at=γx(au/ax)+h(x)u,u(0,x)=f(x)的解半群的性质.     

逆谱问题中偏微分方程数值解的稳定性

讨论一类偏微分方程数值解的稳定性,这种方程源于Sturm-Liouville算子逆谱问题中变换算子法.证明这类偏微分方程差分格式解的存在性、唯一性、收敛性定理及稳定性定理成立.     

一类时间分数阶偏微分方程的解

考虑一类时间分数阶偏微分方程,该方程包含几种特殊情况:时间分数阶扩散方程、时间分数阶反应-扩散方程、时间分数阶对流-扩散方程以及它们各自相对应的整数阶偏微分方程． 通过Laplace-Fourier变换及其逆变换,该方程在空间全平面和半平面内的基本解可以求出,但其表达式则是通过适当的变形来求．另外,对于有限域上的初边值问题,则可由Sine(Cosine)-Laplace变换导出该方程的一种级数形式的解,并通过两个数值例子来说明该方法的有效性．     

在广义函数框架下求解偏微分方程的一种方法

本文是在广义函数的框架下,利用广义函数的一些特殊性质,讨论求解偏微分方程的一种方法。     

Singularities of Solution Surfaces for Quasilinear First-Order Partial Differential Equations

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.     

Regularization of Solution Operators for Quasilinear Hyperbolic-parabolic Partial Differential Equations

We list a hierarchy of hyperbolic-parabolic partial differential equations in terms of the regularization properties of their solution operators. This ranges from the most regularizing of the heat operator to the least, that of the hyperbolic conservation laws. We illustrate this with physical examples in gas dynamics and mechanics.     

Solution Bounds for Elliptic Partial Differential Equations via Feynman-Kac Representation

We transform suitable smooth functions into hard bounds for the solution to boundary value and obstacle problems for elliptic partial differential equations based on the probabilistic Feynman-Kac representation. Unlike standard approximate solutions, hard solution bounds are intended to limit the location of the solution, possibly to a large extent, and, thus, have the potential to be very useful information. Our approach requires two main steps. First, the violation of sufficient conditions is quantified for the test function to be a hard bounding function. After extracting those violation terms from the Feynman-Kac representation, it remains to deal with a boundary value problem with constant input data. Although the probabilistic Feynman-Kac representation is employed, the resulting numerical method is deterministic without the need for sophisticated probabilistic numerical methods, such as sample paths generation of reflected diffusion processes. Throughout this article, we provide numerical examples to illustrate the effectiveness of the proposed method.     

椭圆与抛物偏微分方程解的凸性

我们给出椭圆与抛物偏微分方程解或其水平集的凸性的一个文献综述.从三个经典例子开始,然后介绍凸性研究的常用方法,最后给出几个定量估计,其中注重与我个人研究有关的结果.     

随机偏微分方程——建模,分析与有效动力学

综述随机偏微分方程的基本概念、理论、方法与应用,内容包括Hilbert空间中的Wiener过程、Ito随机积分、随机偏微分方程的解及其有效动力学。还介绍了随机偏微分方程的粗糙轨道、正则结构以及在Kardar-ParisiZhang(KPZ)方程中的应用。还介绍了段金桥与王伟的著作《Effective Dynamics of Stochastic Partial Differential Equations(随机偏微分方程的有效动力学)》的基本内容。     

On the Explicit Solution of Certain First Order Systems of Partial Differential Equations

We consider two classes of systems of partial differential equations of first order. One consists of generalized Stokes-Beltrami equations $ Aw_z = w^*_z $ , $ \lambda Bw_{\bar z} -w^*_{\bar z} $ with square matrices A and B and a scalar factor u . The other may be written in matrix notation as $ v_{\bar z} = c{\bar v} $ where c denotes a square matrix. This system is known as a Pascali system. Both systems are in close connections to certain systems of second order for which the solutions can be represented using particular differential operators. On the basis of these relations we give the solutions of the first order systems explicitly.     

The Numerical Solution of Parabolic Partial Differential Equations Using the Method of Lanczos

Several types of parabolic equations are solved, subject tovarious boundary conditions. A polynomial solution is sought,which is the exact solution of a perturbed version of the originaldifferential equation. The method is developed for the heatequation and then extended to a wide range of problems.     

A Collocation Method for the Numerical Solution of Non-linear Parabolic Partial Differential Equations

Lagrange interpolation formulae are used to obtain a new algorithmfor the approximate polynomial solution of linear and non-linearparabolic equations. The algorithm is described for problemsin one space dimension, although it is applicable to problemsdefined in two or more dimensions. It is also shown how thealgorithm may be adapted to solve a moving boundary (Stefan)problem.     

Tensor Networks and Hierarchical Tensors for the Solution of High-Dimensional Partial Differential Equations

Hierarchical tensors can be regarded as a generalisation, preserving many crucial features, of the singular value decomposition to higher-order tensors. For a given tensor product space, a recursive decomposition of the set of coordinates into a dimension tree gives a hierarchy of nested subspaces and corresponding nested bases. The dimensions of these subspaces yield a notion of multilinear rank. This rank tuple, as well as quasi-optimal low-rank approximations by rank truncation, can be obtained by a hierarchical singular value decomposition. For fixed multilinear ranks, the storage and operation complexity of these hierarchical representations scale only linearly in the order of the tensor. As in the matrix case, the set of hierarchical tensors of a given multilinear rank is not a convex set, but forms an open smooth manifold. A number of techniques for the computation of hierarchical low-rank approximations have been developed, including local optimisation techniques on Riemannian manifolds as well as truncated iteration methods, which can be applied for solving high-dimensional partial differential equations. This article gives a survey of these developments. We also discuss applications to problems in uncertainty quantification, to the solution of the electronic Schrödinger equation in the strongly correlated regime, and to the computation of metastable states in molecular dynamics.