Grid approximation of singularly perturbed parabolic reaction-diffusion equations on large domains with respect to the space and time variables

In an unbounded (with respect to x and t) domain (and in domains that can be arbitrarily large), an initial-boundary value problem for singularly perturbed parabolic reaction-diffusion equations with the perturbation parameter ε² multiplying the higher order derivative is considered. The parameter ε takes arbitrary values in the half-open interval (0, 1]. To solve this problem, difference schemes on grids with an infinite number of nodes (formal difference schemes) are constructed that converge ε-uniformly in the entire unbounded domain. To construct these schemes, the classical grid approximations of the problem on the grids that are refined in the boundary layer are used. Schemes on grids with a finite number of nodes (constructive difference schemes) are also constructed for the problem under examination. These schemes converge for fixed values of ε in the prescribed bounded subdomains that can expand as the number of grid points increases. As ε → 0, the accuracy of the solution provided by such schemes generally deteriorates and the size of the subdomains decreases. Using the condensing grid method, constructive difference schemes that converge ε-uniformly are constructed. In these schemes, the approximation accuracy and the size of the prescribed subdomains (where the schemes are convergent) are independent of ε and the subdomains may expand as the number of nodes in the underlying grids increases.     

On the stability and domain of attraction of asymptotically nonsmooth stationary solutions to a singularly perturbed parabolic equation

A stationary solution to the singularly perturbed parabolic equation ?u _t + ε² u _xx ? f(u, x) = 0 with Neumann boundary conditions is considered. The limit of the solution as ε → 0 is a nonsmooth solution to the reduced equation f(u, x) = 0 that is composed of two intersecting roots of this equation. It is proved that the stationary solution is asymptotically stable, and its global domain of attraction is found.     

Singular Perturbations of Elliptic Problems on Domains with Small Holes

Singular perturbation techniques are used to study the solutions of nonlinear second order elliptic boundary value problems defined on arbitrary plane domains from which a finite number of small holes of radius ρ_i(ε) have been removed, in the limit ε → 0. Asymptotic outer and inner expansions are constructed to describe the behavior of solutions at simple bifurcation and limit points. Since bifurcation usually occurs a eigenvalues of a linearized problem, we study in detail the dependence of the eigenvalues and eigenfunctions on ε, for ε → 0. These results are applied to the vibration of a rectangular membrane with one or two circular holes. The asymptotic analysis predicts a remarkably large sensitivity of eigenvalues and limit points to the ε-domain perturbation considered in this paper.     

A new approach to variational problems with multiple scales

We introduce a new concept, the Young measure on micropatterns, to study singularly perturbed variational problems that lead to multiple small scales depending on a small parameter ε. This allows one to extract, in the limit ε → 0, the relevant information at the macroscopic scale as well as the coarsest microscopic scale (say ε^α) and to eliminate all finer scales. To achieve this we consider rescaled functions Rx (t) := x (s + ε^αt) viewed as maps of the macroscopic variable s ∈ Ω with values in a suitable function space. The limiting problem can then be formulated as a variational problem on the Young measures generated by R^εx. As an illustration, we study a one‐dimensional model that describes the competition between formation of microstructure and highest gradient regularization. We show that the unique minimizer of the limit problem is a Young measure supported on sawtooth functions with a given period. © 2001 John Wiley & Sons, Inc.     

Dichotomies and moving singularities

We consider a linear differential system ε ^σ Φ (t,ε)Y′ =A(t, ε)Y, with ε a small parameter and Φ(t, ε) a function which may vanish in the domain of definition. Under some conditions imposed on the eigenvalues of the matrixA(t, ε), there exists an invertible matrixH(t, ε) which is continuous on ([0,a] × [0, ε₀]). The transformationY=H(t, ε)Z takes then dimensional linear system into two differential systems of orderk andn?k respectively, withk. Thus the investigaton ofn dimensional systems encountered in singular perturbation as well as in stability theory is considerably simplified.     

Singular perturbation analysis of a certain volterra integral equation

An investigation is made of the asymptotic behavior of the solutionu(t;ε) to the Volterra integral equation $$\varepsilon u(t;\varepsilon ) = \pi ^{ - \tfrac{1}{2}} \int\limits_0^t {(t - s)^{ - \tfrac{1}{2}} [f(s) - u^n (s;\varepsilon )]} ds, t \geqslant 0, n \geqslant 1$$ , in the limit as ε→0. This investigation involves a singular perturbation analysis. For the linear problem (n=1) an infinite, uniformly valid asymptotic expansion ofu(t;ε) is obtained. For the nonlinear problem (n≥2), the leading two terms of a uniformly valid expansion are found     

A generalized Lane-Emden-Fowler equation

The nonlinear potential problem Δ ? = –r^2m 0, ^{n = m} is studied with r ? 0, n ε (½, ∞), m ε (?1, ∞) and n m > 0. With various additional conditions we have regular and singular solutions which can be studied explicitly by homology transformation. It is proved that if m ? n + 1 = 0 the solution with ?(0) = 1, ≠ (0) = 0 is radiusstable. Numerical analysis of a number of cases with m ? n + 1 = 0 yields the same conclusion.     

Homogenization of variational inequalities of Signorini type for the <Emphasis Type="Italic">p</Emphasis>-Laplacian in perforated domains when <Emphasis Type="Italic">p</Emphasis> ∈ (1, 2)

The asymptotic behavior, as ε → 0, of the solution uε to a variational inequality with nonlinear constraints for the p-Laplacian in an ε-periodically perforated domain when p ∈ (1, 2) is studied.     

Long time behavior of a singular perturbation of the viscous Cahn–Hilliard–Gurtin equation

We consider a singular perturbation of the generalized viscous Cahn–Hilliard equation based on constitutive equations introduced by Gurtin. This equation rules the order parameter ρ, which represents the density of atoms, and it is given on a n‐rectangle (n?3) with periodic boundary conditions. We prove the existence of a family of exponential attractors that is robust with respect to the perturbation parameter ε>0, as ε goes to 0. In a similar spirit, we analyze the stability of the global attractor. If n=1, 2, then we also construct a family of inertial manifolds that is continuous with respect to ε. These results improve and generalize the ones contained in some previous papers. Finally, we establish the convergence of any trajectory to a single equilibrium via a suitable version of the ?ojasiewicz–Simon inequality, provided that the potential is real analytic. Copyright © 2007 John Wiley & Sons, Ltd.     

The Limit of the Stefan Problem with Surface Tension and Kinetic Undercooling on the Free Boundary

In this paper we consider the Stefan problem with surface tension and kinetic undercooling effects, that is with the temperature u satisfying the condition u = -σK - εV_n on the interface Γ_t, σ, ε = const. ≥ 0 where K and V_n are the mean curvature and the normal velocity of Γ_t, respectively. In any of the following situations: (1) σ > 0 fixed, ε > 0, (2) σ = ε → 0; (3) σ → 0, ε = 0, we shall prove the convergence of the corresponding local (in time) classical solution of the Stefan problem.     

Solving singular two-point boundary value problem in reproducing kernel space

In this paper, we present a new method for solving singular two-point boundary value problem for certain ordinary differential equation having singular coefficients. Its exact solution is represented in the form of series in reproducing kernel space. In the mean time, the n  -term approximation _un(x)$u_n(x)$ to the exact solution u(x)$u(x)$ is obtained and is proved to converge to the exact solution. Some numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method are compared with the exact solution of each example and are found to be in good agreement with each other.     

Embracing the giant component

Consider a game in which edges of a graph are provided a pair at a time, and the player selects one edge from each pair, attempting to construct a graph with a component as large as possible. This game is in the spirit of recent papers on avoiding a giant component, but here we embrace it. We analyze this game in the offline and online setting, for arbitrary and random instances, which provides for interesting comparisons. For arbitrary instances, we find that the competitive ratio (the best possible solution value divided by best possible online solution value) is large. For “sparse” random instances the competitive ratio is also large, with high probability (whp); If the instance has ¼(1 + ε)n random edge pairs, with 0 < ε ≤ 0.003, then any online algorithm generates a component of size O((log n)^3/2) whp , while the optimal offline solution contains a component of size Ω(n) whp . For “dense” random instances, the average‐case competitive ratio is much smaller. If the instance has ½(1 ? ε)n random edge pairs, with 0 < ε ≤ 0.015, we give an online algorithm which finds a component of size Ω(n) whp . © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Singular Perturbations of Limit Points with Application to Tubular Chemical Reactors

Singular perturbation techniques are used to study solutions of certain nonlinear boundary-value problems defined on domains with a circular hole of radius ε, in the limit ε → 0. Asymptotic expansions are constructed to describe the behavior of solutions at and near simple and double limit points (cusps). In particular, the behavior of axisymmetric solutions in an annular domain at limit points is investigated. The results are applied to two model problems arising in chemical-reactor theory. The asymptotic analysis predicts a surprisingly large sensitivity of limit points to the ε-domain perturbation considered here.     

Blow Up of Classical Solutions to $\Box$ U=|u|1+α in Three Space Dimensions

We study the life span of classical solutions to ◻u = |u|^{1+α} in three space dimensions with initial data t = 0: u = εf(x), u, = εg(x), where f and g have compact support and are not both identically zero, ε is a small parameter. We obtain respectively upper and lower bounds of the same order of magnitude for the life span for sufficiently small ε in case 1 ≤ α ≤ \sqrt{2}. We also proved that the classical solution always blows up even when ε = 1 in the critical case α = \sqrt{2}.     

Numerical analysis for phase field simulations of moving interfaces with topological changes

Phase field methods are a widely accepted tool for the approximation of moving free interfaces in sharp interface problems. Topological changes in the solution, such as nucleation or vanishing of particles or merging or pinching of interfaces, lead to singularities in the free boundary. In the sharp interface model, these singularities cause both numerical and theoretical problems, whereas they are handled “automatically” in phase field simulations. Phase field models contain a length scale ε > 0 that vanishes in the sharp interface limit. Therefore, when ε → 0, practical numerical methods have to be robust in the sense that error estimates may only depend polynomially on ε^-1, not exponentially. We show that robust error control is possible past the occurrence of topological changes and without restrictive assumptions on the initial data. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singularly perturbed two-dimensional parabolic problem in the case of intersecting roots of the reduced equation

The singularly perturbed parabolic equation ?u _t + ε²Δu ? f(u, x, ε) = 0, x ∈ D ? ?², t > 0 with Robin conditions on the boundary of D is considered. The asymptotic stability as t → ∞ and the global domain of attraction are analyzed for the stationary solution whose limit as ε → 0 is a nonsmooth solution to the reduced equation f(u, x, 0) = 0 that consists of two intersecting roots of this equation.     

Triple Cohomology and the Galois Cohomology of Profinite Groups

Given a cotriple 𝔾 = (G, ε, δ) on a category X and a functor E:X ^Opp→A into an abelian category A, there exists the cohomology theory of Barr and Beck: Hⁿ(X, E) ε |A| (n ≥ 0, X ε |X|), ([1], p.249). Almost all the important cohomology theories in mathematics have been shown to be special instances of such a general theory (see [1], [2] and [3]). Usually E arises from an abelian group object Y in X in the following manner: it is the contravariant functor from X into the category Ab of abelian groups that associates to each object X in X the abelian group X(X, Y) of maps from X to Y. In such a situation we shall write Hⁿ(X, Y)_𝔾 instead of Hⁿ(X, E)_G. Barr and Beck [2] have shown that the Eilenberg-MacLane cohomology groups H?ⁿ(π, A), n ≥ 2, can be re-captured as follows. One considers the free group cotriple 𝔾′ on the category Gps of groups, which induces in a natural manner a cotriple 𝔾 on the category (Gps, π) of groups over a fixed group π.     

Priority queues with semi-exhaustive service

This paper studies a new type of multi-class priority queues with semi-exhaustive service and server vacations, which operates as follows: A single server continues serving messages in queuen until the number of messages decreases toone less than that found upon the server's last arrival at queuen, where 1nN. In succession, messages of the highest class present in the system, if any, will be served according to this semi-exhaustive service. Applying the delay cycle analysis and introducing a super-message composed of messages served in a busy period, we derive explicitly the Laplace-Stieltjes transforms (LSTs) and the first two moments of the message waiting time distributions for each class in the M/G/1-type priority queues with multiple and single vacations. We also derive a conversion relationship between the LSTs for waiting times in the multiple and single vacation models.     

Bounds on a generalized domination parameter

Abstract

If n is an integer, n ≥ 2 and u and v are vertices of a graph G, then u and v are said to be K_n-adjacent vertices of G if there is a subgraph of G, isomorphic to K_n , containing u and v. For n ≥ 2, a K_n- dominating set of G is a set D of vertices such that every vertex of G belongs to D or is K_n-adjacent to a vertex of D. The K_n-domination number γK_n (G) of G is the minimum cardinality among the K_n-dominating sets of vertices of G. It is shown that, for n ε {3,4}, if G is a graph of order p with no K_n-isolated vertex, then γK_n (G) ≤ p/n. We establish that this is a best possible upper bound. It is shown that the result is not true for n ≥ 5.     

Dynamics of Hodgkin–Huxley systems revisited

We consider the singularly perturbed Hodgkin–Huxley system subject to Neumann boundary conditions. We construct a family of exponential attractors {?_ε} which is continuous at ε = 0, ε being the parameter of perturbation. Moreover, this continuity result is obtained with respect to a metric independent of ε, compared with all previous results where the metric always depends on ε. In the latter case, one needs to consider more regular function spaces and more smoother absorbing sets. Our results show that we can construct and analyse the stability of exponential attractors in a natural phase-space as it is known for the global attractor. Also, a new proof of the upper semicontinuity of the global attractor 𝒜_ε at ε = 0 is given.