The dynamics of a two-component fluid in the presence of capillary forces

In the present paper we study the qualitative behavior ast→∞ of the solution of the Cauchy problem for a system of equations describing a dynamics of a two-component viscous fluid. The model under consideration takes into account the mutual diffusion of the fluid components as well as their capillary interaction. We describe the ω-limit set of trajectories of the dynamical system generated by the problem. It is proved that the stationary solution of the problem, is a homogeneous stationary distribution of one of the components, is asymptotically stable. Any other stationary solution is not asymptotically stable and is even unstable if there are no close stationary solutions corresponding to a smaller energy level. Translated fromMatematicheskie Zametki, Vol. 62, No. 2, pp. 293–305, August, 1997. Translated by A. M. Chebotarev     

New numerical methods and some applied aspects of the p-regularity theory

The theory of p-regularity is applied to optimization problems and to singular ordinary differential equations (ODE). The special variant of the method of the modified Lagrangian function proposed by Yu.G. Evtushenko for constrained optimization problems with inequality constraints is justified on the basis of the 2-factor transformation. An implicit function theorem is given for the singular case. This theorem is used to show the existence of solutions to a boundary value problem for a nonlinear differential equation in the resonance case. New numerical methods are proposed including the p-factor method for solving ODEs with a small parameter.     

Spline Techniques for Solving Singularly-Perturbed Nonlinear Problems on Nonuniform Grids

A numerical method based on cubic splines with nonuniform grid is given for singularly-perturbed nonlinear two-point boundary-value problems. The original nonlinear equation is linearized using quasilinearization. Difference schemes are derived for the linear case using a variable-mesh cubic spline and are used to solve each linear equation obtained via quasilinearization. Second-order uniform convergence is achieved. Numerical examples are given in support of the theoretical results.     

INITIAL VALUE TECHNIQUES FOR THE HELMHOLTZ AND MAXWELL EQUATIONS

We study the initial value problem of the Helmholtz equation with spatially variable wave number. We show that it can be stabilized by suppressing the evanescent waves. The stabilized Helmholtz equation can be solved numerically by a marching scheme combined with FFT. The resulting algorithm has complexity n^2 log n on a n x n grid. We demonstrate the efficacy of the method by numerical examples with caustics. For the Maxwell equation the same treatment is possible after reducing it to a second order system. We show how the method can be used for inverse problems arising in acoustic tomography and microwave imaging.     

Numerical simulation of critical dependences for symmetric two-layered Josephson junctions

Partial critical dependences of the form current-magnetic field in a two-layered symmetric Josephson junction are modeled. A numerical experiment shows that, for the zero interaction coefficient between the layers of the junction, jumps of the critical currents corresponding to different distributions of the magnetic fluxes in the layers may appear on the critical curves. This fact allows a mathematical interpretation of the results of some recent experimental results for two-layered junctions as a consequence of discontinuities of partial critical curves.     

DIRICHLET BOUNDARY VALUE PROBLEMS FOR SECOND-ORDER QUASI-LINEAR DIFFERENTIAL EQUATIONS WITH CHANGING SIGN NONLINEARITIES

This paper is concerned with the existence of positive solutions of two-point Dirichlet singular and nonsingular boundary problems for second-order quasi-linear differential equations with changing sign nonlinearities.     

Dynamic behavior of discrete-time multiagent systems with general communication structures

In this paper, we discuss the dynamic behavior of networks of dynamic agents with general communication topologies. We first analyze the basic case: systems with communication topologies that have spanning trees, i.e., the systems that solve consensus problems. We establish an algebraic condition to characterize each agent's contributions to the final state. And we also study the influence of time-delays on each agent's contributions. Then, we investigate the general case: systems with weakly connected topologies. By using matrix theory, we prove that the states of internal agents will converge to a convex combination of boundary agents in the absence or presence of communication time-delays, and we also show that the coefficients of the convex combination are independent of time-delays even if the delays are time-varying. These results have broad applications in other areas, e.g., study of swarm behavior, formation control of vehicles, etc.     

Maximum-norm error analysis of a non-monotone FEM for a singularly perturbed reaction-diffusion problem

A non-monotone FEM discretization of a singularly perturbed one-dimensional reaction-diffusion problem whose solution exhibits strong layers is considered. The method is shown to be maximum-norm stable although it is not inverse monotone. Both a priori and a posteriori error bounds in the maximum norm are derived. The a priori result can be used to deduce uniform convergence of various layer-adapted meshes proposed in the literature. Numerical experiments complement the theoretical results. AMS subject classification (2000)  65L10, 65L50, 65L60     

关于非线性Schrodinger方程混合问题解的破裂和质量集中

本文讨论了一类非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程混合问题解的破裂现象，ｂｌｏｗ－ｕｐ时刻的估计和质量集中定理，并推广和改进了［ｌ～３］中的结果．     

Second-order necessary optimality conditions for domain optimization problems of the Neumann type

In this paper, we treat a domain optimization problem in which the boundary-value problem is a Neumann problem. In the case where the domain is in a three-dimensional Euclidean space, the first-order and the second-order necessary conditions which the optimal domain must satisfy are derived under a constraint which is the generalization of the requisite of constant volume.Portions of this paper were presented at the 13th IFIP Conference on System Modelling and Optimization, Tokyo, Japan, 1987.