Multipoint Problem for Nonisotropic Partial Differential Equations with Constant Coefficients

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x ₁,..., x _p for a nonisotropic (concerning differentiation with respect to t and x ₁,..., x _p) partial differential equation with constant complex coefficients. We establish conditions for the existence and uniqueness of a solution of this problem and prove metric theorems on lower bounds for small denominators appearing in the course of the construction of its solution.     

Multipoint Problem with Multiple Nodes for Partial Differential Equations

We establish conditions for the existence and uniqueness of a solution of a problem with multipoint conditions with respect to a selected variable t (in the case of multiple nodes) and periodic conditions with respect to x ₁,..., x _p for a nonisotropic partial differential equation with constant complex coefficients. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of this problem.     

Time-Variable Extension of the Solution of a Nonlocal Multipoint Problem for Partial Differential Equations with Constant Coefficients

A problem with nonlocal multipoint conditions for the nth-order partial differential equation with constant coefficients is considered. In the case where conditions of strict averaging of time intervals are specified, the existence of a solution of the problem in a cylinder that is the Cartesian product of a time interval and a p-dimensional spatial torus is discussed. It is found that under certain conditions of separability of the roots of the characteristic equation for almost all (in the sense of the Lebesgue measure) coefficients of the equation and parameters of the conditions, the solution of the problem cannot be extended in the time variable beyond the extreme points at which the conditions are given.     

A Problem with Nonlocal Conditions for Partial Differential Equations with Variable Coefficients

We establish conditions for the unique solvability of a problem for partial differential equations with coefficients dependent on variables t and x in a rectangular domain with nonlocal two-point conditions with respect to t and local boundary conditions with respect to x. We prove metric statements related to lower bounds of small denominators appearing in the course of solution of the problem.     

A Nonlocal Boundary-Value Problem for Differential Equations with Fractional Time Derivative with Variable Coefficients

Conditions for the existence and uniqueness of a classical solution of a nonlocal boundary-value problem for a differential equation with a regularized Riemann–Liouville fractional time derivative with variable coefficients are investigated.     

Partial Differential Equations That Are Hard to Classify

Semi-linear n×n systems of the form A∂u/∂x+B∂u/∂y=f can generally be solved, at least locally, provided data are imposed on non-characteristic curves. There are at most n characteristic curves and they are determined by the coefficient matrices on the left-hand sides of the equations. We consider cases where such problems become degenerate as a result of ambiguity associated with the definition of characteristic curves. In such cases, the existence of solutions requires restrictions on the data and solutions might not be unique.     

Dirichlet-Type Problems for Systems of Partial Differential Equations Unresolved with Respect to the Highest Time Derivative

We establish conditions for the correct solvability of problems for systems of partial differential equations unresolved with respect to the highest time derivative with Dirichlet-type conditions with respect to time and periodic conditions with respect to space variables. We prove metric theorems on lower bounds for small denominators appearing in the course of the solution of these problems.     

A Problem with Nonlocal Conditions for Partial Differential Equations Unsolved with Respect to the Leading Derivative

In the domain that is the product of a segment and a p-dimensional torus, we investigate the well-posedness of a problem with nonlocal boundary conditions for a partial differential equation unsolved with respect to the leading derivative with respect to a selected variable. We establish conditions for the the classical well-posedness of the problem and prove metric theorems on the lower bounds of small denominators appearing in the course of its solution.     

A Multipoint Problem for Pseudodifferential Equations

We investigate the well-posedness of a problem with multipoint conditions with respect to a chosen variable t and periodic conditions with respect to coordinates x ₁,...,x _p for equations unsolved with respect to the leading derivative with respect to t and containing pseudodifferential operators. We establish conditions for the unique solvability of this problem and prove metric assertions related to lower bounds for small denominators appearing in the course of its solution.     

小参数在高阶导数项的椭圆型方程第一边值问题的一致收敛差分格式

本文讨论了曲边区域上小参数ε在高阶导数项的椭圆型方程第一边值问题,从一致收敛的必要条件出发构造了特殊的差分格式,证明了差分方程问题解的一致收敛性,估计了收敛的阶数,并讨论了差分方程解的渐近性态.     

带VMO系数的拟线性抛物型方程斜微商问题

本文给出了拟线性抛物型方程Cu=ut-aij(x,t,u)Diju+a(x,t,u,Du)=0在正则斜微商边界条件Bu=biDiu=g下的强解u∈Wq2,1(QT)(q＞n+1)的存在、唯一性.这里我们假设系数aij是Caratheodory函数,且关于(x,t),aij∈VMO∩L∞;而函数a(x,t,u,Du)关于Du至多是二次增长.     

A Method for Obtaining Exact Solutions to Partial Differential Equations with Variable Coefficients

This paper describes techniques that can be used to transform PDEs with variable coefficients into equations with constant coefficients. The techniques are illustrated by calculating shear flows over quite general surfaces, by solving the signaling problem for diffusive processes in inhomogeneous materials, and by solving the signaling problem for acoustical waves when the sound speed varies with distance. The techniques may also be used to solve equations governing processes in inhomogeneous, anisotropic materials.     

Exact Solutions of Linear Partial Differential Equations with Variable Coefficients

Bäcklund transformations relating the solutions of linear PDE with variable coefficients to those of PDE with constant coefficients are found, generalizing the study of Varley and Seymour [2]. Auto-Bäcklund transformations are also determined. To facilitate the generation of new solutions via Bäcklund transformation, explicit solutions of both classes of the PDE just mentioned are found using invariance properties of these equations and other methods. Some of these solutions are new.     

The Link between Stochastic Differential Equations with Non-Markovian Coefficients and Backward Stochastic Partial Differential Equations

In this paper, we conjecture and prove the link between stochastic differential equations with non-Markovian coefficients and nonlinear parabolic backward stochastic partial differential equations, which is an extension of such kind of link in Markovian framework to non-Markovian framework.Different from Markovian framework, where the corresponding partial differential equation is deterministic, the backward stochastic partial differential equation here has a pair of adapted solutions, and thus the link has a much different form. Moreover, two examples are given to demonstrate the applications of the derived link.     

Averaging of Backward Stochastic Differential Equations and Homogenization of Partial Differential Equations with Periodic Coefficients

Abstract

We study the limit of the solutions of systems of semi-linear partial differential equations (PDEs) of second order of parabolic type, with rapidly oscillating periodic coefficients, a singular drift, and singular coefficients of the zero and second order terms. Our basic tool is the approach given by Pardoux [14 Pardoux , E. 1999 . Homogenization of linear and semilinear second order parabolic PDEs with periodic coefficients: a probabilistic approach . J. Funct. Anal. 167 : 498 – 520 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]]. In particular, we use the weak convergence of an associated backward stochastic differential equation (BSDE).     

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.     

The Conormal Derivative Problem for Elliptic Equations with BMO Coefficients on Reifenberg Flat Domains

We study the inhomogeneous conormal derivative problem for thedivergence form elliptic equation, assuming that the principalcoefficients belong to the BMO space with small BMO semi-normsand that the boundary is -Reifenberg flat. These conditionsfor the W^{1, p}-theory not only weaken the requirements on thecoefficients but also lead to a more general geometric conditionon the domain. In fact, the Reifenberg flatness is the minimalregularity condition for the W^{1, p}-theory. 2000 MathematicsSubject Classification 35R05 (primary), 35J15 (secondary).