Partial Decomposition of a Domain Containing Thin Tubes for Solving the Heat Equation

An initial–boundary value problem for the heat equation in a three-dimensional domain containing thin cylindrical tubes is considered. The Neumann condition is set on the lateral boundaries of the tubes. The original three-dimensional problem is reduced to a hybrid-dimensional one in which the heat equation in the tubes is replaced by the one-dimensional heat equation in shorter cylinders (subtubes), and the three- and one-dimensional equations are matched on the bases of the subtubes. The difference between the solutions of the original and hybrid-dimensional problems is estimated using two geometric characteristics: the distance between the bases of the tubes and subtubes and the reciprocals of the minimal positive eigenvalues of the Neumann problem for the Laplace operator in the tube cross sections.     

Attractor of a Nonautonomous Hyperbolic Equation with a Small Parameter

Mathematical Notes -     

Positive Periodic Solution for a Nonautonomous Delay Differential Equation

In this paper, by using the Krasnoselskii fixed point theorem, we study the existence of one or multiple positive periodic solutions of a nonautonomous delay differential equation. We also give some examples to demonstrate our results.     

Increasing Powers in a Degenerate Parabolic Logistic Equation

The purpose of this paper is to study the asymptotic behavior of the positive solutions of the problem $$\partial _t u - \Delta u = au - b\left( x \right)u^p in \Omega \times \mathbb{R}^ + , u(0) = u_0 , \left. {u(t)} \right|_{\partial \Omega } = 0,$$ as p → + ∞, where Ω is a bounded domain, and b(x) is a nonnegative function. The authors deduce that the limiting configuration solves a parabolic obstacle problem, and afterwards fully describe its long time behavior.     

一类非自台非线性时滞差分方程正解的振动性

研究了一类非自治非线性时滞差分方程的正解关于平衡点l的振动性,所获结果改进和推广了文献[6]中的相关结论.     

一类非自治非线性时滞差分方程正解的振动性

研究了一类非自治非线性时滞差分方程△xn=rnxn1-xn-k/1-cxn-k的正确解关于平衡点1的振动性,所获结果改进和推广了献[6]中的相关结论。     

Sumsets Containing Powers of an Integer

Let n1, and let m be an integer with m2. We show that if a subset A of the interval [0,n] satisfies that 0A and |A|>1+n/2, then mA, the set of the sum of m (not necessarily distinct) elements in A, has a power of m. This result is best possible in the case that m is odd.     

三阶非自治微分方程的不稳定性定理

本文运用构造李雅普诺夫函数的方法，给出了一类三阶非自治微分方程零解不稳定的充分条件．     

On the Unique Solvability of a Nonautonomous Functional Differential Equation of Neutral Type

We present conditions for the existence and uniqueness of solutions of functional differential equations of the neutral type and their continuous dependence on the initial data.     

A Note on the Solution of a Stochastic Partial Differential Equation

Abstract

A physical model is described which justifies the appearance of a stochastic term in the two-dimensional Navier–Stokes equations. In this model, a linear oppositional control term accrues as well. The resulting stochastic partial differential equation is shown to have a unique stationary solution.     

一类非线性偏微分方程的数值解法

定义了再生核空间,在再生核空间中给出了一类带初、边值条件的非线性偏微分方程的数值解法,并给出了算法实例.     

Powers of Euler's Product and Related Identities

Ramanujan's partition congruences can be proved by first showing that the coefficients in the expansions of (q; q) ^r satisfy certain divisibility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r, the coefficients in the expansions of (q; q) ^r satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to above. We also obtain arithmetic relations for the coefficients in the expansions of (q; q) ^r (q ^t; q ^t) ^s , for t = 2, 3, 4 and various values of r and s. Our proofs are explicit and elementary, and make use of the Macdonald identities of ranks 1 and 2 (which include the Jacobi triple product, quintuple product and Winquist's identities). The paper concludes with a list of conjectures.     

Asymptotic Solution of the Multidimensional Burgers Equation Near A Singularity

We consider the Cauchy problem for the multidimensional Burgers equation with a small dissipation parameter and use the matching method to construct an asymptotic solution near the singularity determined by the vector field structure at the initial instant. The method that we use allows tracing the evolution of the solution with a hierarchy of differently scaled structures and giving a rigorous mathematical definition of the asymptotic solution in the leading approximation. We discuss the relation of the considered problem to different models in fundamental and applied physics.     

Conditional Symmetry and Exact Solutions of a Multidimensional Diffusion Equation

We investigate the conditional symmetry of a multidimensional nonlinear reaction–diffusion equation by its reduction to a radial equation. We construct exact solutions of this equation and infinite families of exact solutions for the corresponding one-dimensional diffusion equation.     

从一首古诗引入偏导数概念

将苏轼的诗《题西林壁》引入高等数学课堂教学,结合一元函数导数的几何意义,利用＂峰＂和＂岭＂的不同含义,从苏诗切入到多元函数偏导数概念的引入和偏导数的几何意义的讲解,以图改善教学手段,丰富授课方式并增加数学知识的人文气息.