A three level linearized compact difference scheme for the Cahn-Hilliard equation

This article is devoted to the study of high order accuracy difference methods for the Cahn-Hilliard equation.A three level linearized compact difference scheme is derived.The unique solvability and unconditional convergence of the difference solution are proved.The convergence order is O(τ 2 + h 4 ) in the maximum norm.The mass conservation and the non-increase of the total energy are also verified.Some numerical examples are given to demonstrate the theoretical results.     

二维变系数反应扩散方程的紧交替方向差分格式

1 引言 在研究热传导过程、气体扩散现象和电磁场的传播等问题时,常常遇到抛物型偏微分方程。用有限差分方法研究这类问题的数值解法目前已经有了许多工作。对于二维、三维抛物方程的数值求解比较理想的方法是交替方向法。     

近似对称约化简述

(同伦)近似对称方法由摄动法与对称约化方法相结合产生, 用于微分方程级数解的构造. 对称约化方法应用于微分方程或者其同伦模型经扰动展开分解而成的无穷多近似子方程, 可以得出通式形式的无穷多约化解和相应的约化方程, 再通过求解约化方程进一步得出原方程的截断级数解. 截断级数解的存在性体现原方程的可解性, 通常取决于扰动项的阶数与最高阶导数项奇偶性是否一致.     

INTEGRATED BISEMIGROUP AND APPLICATIONS TO IMPULSIVE DIFFERENTIALEQUATIONS

1IntroductionRecellt1yauth0rintr0ducethec0nceptofintegratedbisemigroupwhichliavebeendeveloped~,.lorsoivmgtheabstractkineticequation[11'Thisthe0ryextendstheoperatorsbisendgroupthe0r}toabstractboundaryvaluepr0blmewith0peratorwhichd0notgenerateastrong1ycontiuousbisentigroup.Ontheotherhand,thistheoryextendsthetheoryofintegratedsedrigr0upals0.ItismorericherthantheintegratedseAngroup8[5-91.InthispaPerwewillusethistlieoryt0studythes0lvabilityofabstractimpul8ivcdifferentialequati0ns.Insection2.werec…     

Approximate method for solving the boundary value problem for a parabolic equation with inhomogeneous transmission conditions of nonideal contact type

A linear parabolic equation in a disconnected domain with inhomogeneous transmission conditions of the nonideal contact type is studied. A generalized formulation of the problem is considered. An analogue of the Galerkin method is proposed for solving the problem, and the stability of the method is investigated. This makes it possible to prove existence and uniqueness theorems for the equation under different assumptions on the data smoothness.     

THE SOLVABILITY CONDITIONS FOR AN INVERSE EIGEN-PAIR PROBLEM

This paper discusses problem IEP:Given n×m matrix X and m×m diagonal matrix A, find an n×n matrix A such that AX=XA.The new solvablily conditions for the problem IEP are obtained. The eigenvalue dislribulaion of the solutions for the problem IEP are described in detail.     

On the global solvability in Gevrey classes on the n-dimensional torus

Let P be a linear partial differential operator with coefficients in the Gevrey class G^s(Tⁿ), where Tⁿ is the n-dimensional torus and s?1. We prove a necessary condition for the s-global solvability of P on Tⁿ. We also apply this result to give a complete characterization for the s-global solvability for a class of formally self-adjoint operators with nonconstant coefficients.     

Boundary value problem for the Burgers system

We consider the boundary value problem $\begin{gathered} div(\rho V) = 0, \rho |\Gamma _1 = \rho 0, \hfill \\ \rho (V,\nabla V) = v\Delta V, V|\Gamma = V^0 \hfill \\ \end{gathered} $ for a vector functionV=(V ₁,V ₂) and a scalar function ρ>-0 in a rectangular domain Ω ? ?² with boundary Γ. Here $\Gamma _1 = \{ x \in \Gamma :(V^0 ,n)< 0\} , V_1^0 |_\Gamma > 0, v = const > 0.$ We prove that this problem is solvable in Hölder classes.     

On the equivalence between strong solvability and strict semimonotonicity for some systems involving Z-functions

This paper focuses on the relationship between the strong solvability of a certain system involving a Z-function, and the strict semimonotonicity of such a function. Our main result shows that, for a system defined by a continuous, superhomogeneous Z-function, the additional condition of strict semimonotonicity is both necessary and sufficient for strong solvability.     

Yang-Lee theory and the conductor-insulator transition in asymmetric log-potential lattice gases

A feature of a conducting phase at low density is that there is a singularity in the fugacity expansion of the pressure, whereas the same expansion in the insulating phase gives an analytic series. The Yang-Lee characterization of a phase transition thus implies that in the conducting phase the zeros of the grand partition function must pinch the real axis in the complex scaled fugacity () plane at =0, whereas in the insulating phase a neighborhood of =0 must be zero free. Exact and numerical calculations are presented which suggest that for two-component log-potential lattice gases in one dimension with dimensionless coupling, the zeros pinch the point =0 for<2, while for2 a neighborhood of =0 is zero free. The conductor-insulator transition therefore takes place at=2 independent of the density and other parameters in the model.