On the Heat Convection Equations with Dissipation Term in Regions with Moving Boundaries

We consider an initial value problem for a system of equations describing the motion and the heat convection in a viscous and incompressible fluid which occupies a smooth region Ω_t⊂ℝ³ depending on time. In the equation for the distribution of temperature in the fluid we take into account not only the convective term but also the term responsible for the dissipation of energy. We prove local in time existence and uniqueness of solutions of the considered problem, and global in time existence for sufficiently small data. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

On the Structure of the Set of E-Functions Satisfying Linear Differential Equations of Second Order

We prove a special case of Siegel's conjecture concerning the representability of E-functions in the form of polynomials in hypergeometric functions. We prove several assertions (formulated earlier by A. B. Shidlovskii) about the transcendence and linear independence of values of E-functions.     

Integrable Structure Behind the WDVV Equations

An integrable structure behind the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations is identified with the reduction of the Riemann–Hilbert problem for the homogeneous loop group . The reduction requires the dressing matrices to be fixed points of an order-two loop group automorphism resulting in a subhierarchy of the hierarchy containing only odd-symmetry flows. The model has Virasoro symmetry; imposing Virasoro constraints ensures the homogeneity property of the Darboux–Egoroff structure. Dressing matrices of the reduced model provide solutions of the WDVV equations.     

论演化方程的结构

本文对演化方程进行了讨论，从而能够唯象地描述微观结构行为，如Ｍａｘｗｅｌｌ气体的部分可逆流动、恢复性、结构松弛以及从光散射和分子动力学得到的其它实验结果。讨论所得结果涉及Ｚａｒｅｍｂａ补充式（ａｎｓａｔｚ）的重新评价，可用来解决输运方程中具有显著非线性的问题。     

求解Hamilton-Jacobi方程的局部移动网格方法

对Hamilton-Jacobi方程设计了一个基于Runge-Kutta间断Galerkin方法的移动网格方法,并利用坏单元指示子进一步设计了一个局部移动网格方法.数值结果表明这两个方法相比均匀网格能提高数值解的质量.同时局部移动网格方法通过将网格移动局部化,在不影响精度的前提下节省了计算时间,提高了计算效率。     

Minkowski Equations in Nonlinear Magnetizable and Polarizable Moving Media

On the basis of the covariance of the Maxwell equations under Lorentz transformations, we generalize the Minkowski equations by deducing the material relations for nonlinear magnetizable and polarizable moving media.     

On the Theory of Mixed-Type Equations

We briefly review different well-posed boundary-value problems for mixed-type equations and their applications in transonic gas dynamics. We present barotropic relations for plane-parallel flow of a compressible gas that leads to mixed-type model equations on the hodograph plane. We also discuss the question on the existence of transonic solutions of profile flow problems, which is closely related to Dirichlet problems in mixed-type domains.__________Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 10, Suzdal Conference-4, 2003.     

On the Equivalence of Differential Equations

In this article we use the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Riccati equation and some polynomial differential equations. The results are applied to discussion of the qualitative behavior of periodic solutions of these complicated differential equations.     

轴向运动导电导磁梁的磁弹性振动方程

针对磁场环境中轴向运动导电导磁梁磁弹性耦合振动的理论建模问题进行研究.基于Timoshenko(铁木辛柯)梁理论并考虑几何非线性因素,给出轴向运动弹性梁在横向双向振动下的形变势能、动能计算式以及电磁力和机械力的虚功表达式.应用Hamilton(哈密顿)变分原理,推得磁场中轴向运动Timoshenko梁的非线性磁弹性耦合振动方程,并给出了简化形式的Euler-Bernoulli(欧拉 伯努利)梁磁弹性振动方程.根据电磁理论和相应的电磁本构关系,得到载流导电弹性梁所受电磁力的表达式,基于磁偶极子-电流环路模型给出铁磁弹性梁所受磁体力和磁体力偶的表述形式.通过算例,分析了轴向运动导电弹性梁的奇点分布及其稳定性问题.     

Volterra积分方程和泛函微分方程的振动性

本文建立了Ｖｏｌｔｅｒａ积分方程和一阶泛函微分方程解的振动准则．     

On Associativity Equations

We consider the associativity or Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations and discuss their solution class based on the existence of the residue formulas, which is most relevant for nonperturbative physics. We demonstrate that for this case, proving the associativity equations reduces to solving a system of linear algebraic equations. Particular examples of solutions related to Landau–Ginzburg topological theories, Seiberg–Witten theories, and the tau functions of semiclassical hierarchies are discussed in detail. We also discuss related questions including the covariance of associativity equations, their relation to dispersionless Hirota relations, and the auxiliary linear problem for the WDVV equations.     

Symmetry Structure of Integrable Nonevolutionary Equations

We study a class of evolutionary partial differential systems with two components related to second order (in time) nonevolutionary equations of odd order in spatial variable. We develop the formal diagonalization method in symbolic representation, which enables us to derive an explicit set of necessary conditions of existence of higher symmetries. Using these conditions we globally classify all such homogeneous integrable systems, i.e., systems which possess a hierarchy of infinitely many higher symmetries.     

On Remainder Equations

The remainder equations were introduced by S. O. Hansson. In this paper, we will give an exhaustive characterization of the set of solutions for remainder equations. Moreover, solutions to some unsolved problems proposed by Hansson are reported.     

On the Ultra Relativistic Euler Equations

We consider the relativistic Euler equations in isentropic fluids with the equation of state , which is the ultra-relativistic limit. We analyze the single shocks. We study the shock interaction, and give explicit example for the non-backward uniqueness. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Weak Solution of Moving Boundary Problems

The weak formulation of moving boundary problems with possiblyvanishing specific heat, that is governed by parabolic and/orelliptic differential equations, is developed. The uniquenessof the resulting weak solution is then proved. This approachis used to obtain numerical solutions to some physical examples,which arise in electrochemical machining processes, and in saturated/unsaturatedflow in porous media.     

On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane of the spectrum of the operator , generated by the closure in of the operation originally defined on smooth functions with values in a Hilbert space satisfying the Dirichlet conditions . Here and A is a model operator acting in . Criterial conditions on the parameter for the eigenfunctions of the operator to form a complete and minimal system as well as a Riesz basis in the Hilbert space H are given.