Mixed Boundary Value Problems for some Linear and Non-linear Elliptic Difference Equations

We obtain a priori bounds on difference quotients for some mixedboundary value problems associated with uniformly elliptic differenceequations over rectangular regions. These results are appliedto mildly quasi-linear and mildly non-linear problems to establishexistence and uniqueness criteria. Our estimates are usefulin the numerical computation of these solutions.     

A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow

A new family of mixed hp-finite elements is presented for thediscretization of planar Stokes flow on meshes of curvilinear,quadrilateral elements. The elements involve continuous pressuresand are shown to be stable with an inf–sup constant boundedbelow independently of the mesh-size h and the spectral orderp. The spaces have balanced approximation properties—theorders of approximation in h and p are equal for both the velocityand the pressure. This is the first example of a uniformly stablemethod with continuous pressures for spectral element discretizationof Stokes equations, valid for geometrically refined meshesand curvilinear elements.     

Finite-order meromorphic solutions and the discrete Painleve equations

Let w(z) be an admissible finite-order meromorphic solutionof the second-order difference equation

where R(z, w(z)) is rational in w(z) withcoefficients that are meromorphic in z. Then either w(z) satisfiesa difference linear or Riccati equation or else the above equationcan be transformed to one of a list of canonical differenceequations. This list consists of all known difference Painlevéequations of the above form, together with their autonomousversions. This suggests that the existence of finite-order meromorphicsolutions is a good detector of integrable difference equations.     

A Constructive Look at Functions of Bounded Variation

Functions with bounded variation and with a (total) variationare examined within Bishop's constructive mathematics. It isshown that the property of having a variation is hereditarydownward on compact intervals, and hence that a real-valuedfunction f with a variation on a compact interval can be expressedas a difference of two increasing functions. Moreover, if fis sequentially continuous, then the corresponding variationfunction, and hence f itself, is uniformly continuous. 1991Mathematics Subject Classification 26A45.     

Some Uniformly Convergent Spline Difference Schemes for Singularly Parturbed Boundary Value Problems

In this paper some difference schemes for singularly perturbedtwo-point boundary value problems are derived using cubic splinesv(x) C¹[0,1]. One of them is the well-known Allen-Southwell-Il'inscheme. These schemes are first-order uniformly convergent.Numerical examples support the theoretical results.     

The Modulus of Smoothness and Discrete Data in a Square Domain

This paper relates to a function f on a two-dimensional squaredomain that is finite, infinite, or semi-infinite. It is shownthat, if the second difference of f with respect to a uniformsquare lattice of mesh-size 2^–n is uniformly of order2^–n with 0<<2, then the h-stepsize second differenceof f is uniformly of order h in any direction in the domain.This is deduced from a corresponding, more general weak-type(i.e. Marchaud-type) inequality.     

Asymptotically Almost Periodic Solutions of Inhomogeneous Cauchy Problems on The Half-Line

Let u be a bounded, uniformly continuous, mild solution of aninhomogeneous Cauchy problem on R₊: u'(t)+Au(t)+(t)(t0). Supposethat u has uniformly convergent means, (A)iR is countable, and is asymptotically almost periodic. Then u is asymptoticallyalmost periodic. Related results have been obtained by Ruessand V, and by Basit, using different methods. A direct proofis given of a Tauberian theorem of Batty, van Neerven and Räbiger,and applications to Volterra equations are discussed. 1991 MathematicsSubject Classification 34C28, 44A10, 47D03.     

Clunie theorems for difference and q-difference polynomials

The main purpose of this paper is to prove difference and q-differencecounterparts of the Clunie and Mohon’ko lemmas from theNevanlinna theory of differential polynomials. We also giveapplications to the value distribution theory of meromorphicsolutions of some complex difference equations.     

High-accuracy Tridiagonal Finite Difference Approximations for Non-linear Two-point Boundary Value Problems

We discuss the construction of finite difference approximationsfor the non-linear two-point boundary value problem: y" = f(x,y), y(a)=A, y(b)=B. In the case of linear differential equations,the resulting finite difference schemes lead to tridiagonallinear systems. Approximations of orders higher than four involvederivatives of f. While several approximations of a particularorder are possible, we obtain the "simplest" of these approximationsleading to two high-accuracy methods of orders six and eight.These two methods are described and their convergence is established;numerical results are given to illustrate the order of accuracyachieved.     

Uniform Embeddings of Free Products into Uniformly Convex Banach Spaces

Let A and B be countable discrete groups, and let = A * B betheir free product. We show that if A and B are uniformly embeddableinto a uniformly convex Banach space, then so is . 2000 MathematicsSubject Classification 46L89, 20F65.     

A Sixth-order Tridiagonal Finite Difference Method for General Non-linear Two-point Boundary Value Problems

We present a sixth-order finite difference method for the generalsecond-order non-linear differential equation Y"=f(x, y, y')subject to the boundary conditions y(a) = A, y(b) = B. In thecase of linear differential equations, our finite differencescheme leads to tridiagonal linear systems. We establish, underappropriate conditions, O(h⁶)-convergence of the finite differencescheme. Numerical examples are given to illustrate the methodand its sixth-order convergence.     

Formulas for Liapunov Functions for Systems of Linear Difference Equations

Explicit quadratic Liapunov functions that provide necessaryand sufficient conditions for the asymptotic stability of thesystem of linear difference equations x (t + 1) = Ax(t) areconstructed by transforming the original systems to y (t + 1)= Gy(t), where G is a companion matrix associated with the characteristicpolynomial of A. A necessary and sufficient condition for allroots of the characteristic polynomial to lie in the unit circle|z| < 1 on the complex plane is also derived. 2000 MathematicalsSubject Classification 39A11, 93D05.     

Formulas of Bendixson and Alekseev for Difference Equations

A well-known formula of Bendixson states that solutions of first-orderdifferential equations, as functions of their initial conditions,satisfy a certain partial differential equation. A consequenceis Alekseev's nonlinear variation of parameters formula. Inthis paper, corresponding results are proved for differenceequations. To achieve this, use is made of the recently introducedconcept of alpha derivatives, rather than of differences orof the usual derivatives. This technique allows the resultsto be generalized to alpha dynamic equations, which includeamong others ordinary differential and difference equations.2000 Mathematics Subject Classification 39A12, 39A13.     

Preconditioning by inverting the Laplacian: an analysis of the eigenvalues

Wolfgang Hackbusch ^{We study the eigenvalues of the operator generated by usingthe inverse of the Laplacian as a preconditioner for self-adjointsecond-order elliptic partial differential equations with smoothcoefficients. It is well-known that the spectral condition numberof the preconditioned operator can be bounded by , where k is the uniformly positive coefficientof the second-order elliptic equation. The purpose of this paperis to study the spectrum of the preconditioned operator. Wewill show that there is a strong relation between the spectrumof this operator and the range of the coefficient function.In the continuous case, we prove, both for mappings definedon Sobolev spaces and in terms of generalized functions, thatthe spectrum of the preconditioned operator contains the rangeof the coefficient function k. In the discrete case, we indicateby numerical examples that the entire discrete spectrum is approximatelygiven by values of k.}     

Viscoelastic boundary stabilization for a transmission problem in elasticity

We consider an anisotropic body which is constituted of twodifferent types of materials supporting a memory boundary conditionand we show that its energy decays uniformly as time goes toinfinity with the same rate as the relaxation function g, thatis, the energy decays exponentially when g decays exponentially,and polynomially when g decays polynomially.     

Classification of linear time-varying difference equations under kinematic similarity

This paper concerns the problem to classify linear time-varying finite dimensional systems of difference equations under kinematic similarity, i.e., under a uniformly bounded time-varying change of variables of which the inverse is also uniformly bounded. Also the problem of reducing difference equations by using such similarity transformations is studied. Both problems are solved for a number of subclasses, including equations with scalar coefficients, time-invariant equations, finitely supported equations, and equations with one jump. For the general case an open problem is formulated.The research has been carried out during the third author's Ph.-D. period at the Faculteit der Wiskunde en Informatica, Vrije Universiteit, Amsterdam, The Netherlands     

COARSE AND UNIFORM EMBEDDINGS INTO REFLEXIVE SPACES

Answering an old problem in nonlinear theory, we show that c₀cannot be coarsely or uniformly embedded into a reflexive Banachspace, but that any stable metric space can be coarsely anduniformly embedded into a reflexive space. We also show thatcertain quasi-reflexive spaces (such as the James space) alsocannot be coarsely embedded into a reflexive space and thatthe unit ball of these spaces cannot be uniformly embedded intoa reflexive space. We give a necessary condition for a metricspace to be coarsely or uniformly embeddable in a uniformlyconvex space.     

Weak Laws of Large Numbers in Some Non-Commutative Spaces

We prove weak laws of large numbers for freely independent,uniformly bounded and non-identically distributed random variablesbelonging to non-commutative Ciach and, in particular, LorentzL^p-spaces. 2000 Mathematics Subject Classification 60B11, 60F25,46L54, 47N30.     

Multistep Discretization of Linear Hyperbolic Equations

Order and stability of multistep finite-difference discretizationsof the first-order linear hyperbolic equation u₁ = a(x)u_x areconsidered. We prove that if a stable method uses s upwind andrdownwind points and the coefficients depend only on the Courantnumber and on a(x) and its derivatives at the underlying gridpoint, then the order may not exceed r + s. This bound on orderis exactly half the bound of Strang & Iserles (1983) forconstant a. Furthermore, we prove that if r = s and a(x) isboth uniformly bounded and uniformly positive for x R thenthe new order barrier is attainable for every s 1.     

On the Least Q-order of Convergence of Variable Metric Algorithms

It is shown in this paper that the infimum of the Q-order ofthe convergence of variable metric algorithms is only 1, eventhough the objective function is twice continuously differentiableand uniformly convex. It is shown by example that the Q-ordercan be 1 + 1/N for any large N, though the R-order is (1+N)^1/2.