The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Theory

The aim of this paper is to develop a straightforward analysisof the Galerkin method for two-dimensional boundary integralequations of the first kind with logarithmic kernels. A distinctivefeature of the analysis is that no appeal is made to ‘coercivity’,as a result of which some existence questions cannot be answereddirectly. In return, however, the analysis has no special difficultyin handling corners, cusps, or open arcs. Instead of coercivity,the central feature of the analysis is the positive-definiteproperty of the integral operator for small enough contours.Rates of convergence are predicted theoretically and, in particular,certain linear functionals are shown to exhibit ‘superconvergence’.Numerical results supporting the theory are given in the companionpaper Sloan & Spence (1987) for problems on both open andclosed polygonal arcs.     

On the Existence and Stability of Spatially Structured Solutions of the Reaction-Diffusion Equations

The reaction-diffusion equations for the well-known ‘Brusselator’chemical kinetic model are investigated when the model is madeconsistent with the principle of detailed balance. In contrastto the original model, the corrected one does not show solutionswith ‘spatial structure’ i.e. solutions with multipleinternal maxima and multiple internal global minima in bothdependent variables. Sufficient conditions for stability ofthe solutions are given in terms of a Rayleigh quotient forgeneral boundary conditions for nonlinear reaction-diffusionequations in general. For the particular case of the ‘Brusselator’it is shown that conditions for a change of stability are muchmore unlikely to be attained as a result of the restrictionsof detailed balancing. The detailed sufficiency condition forthe stability of any steady-state solution and for no branchingfrom the ‘equilibrium’ branch solution depends onwhether the solution has global extrema inside the region, onits boundary, or both     

A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domainswith piecewise smooth boundaries is presented. It is intendedfor problems describing the Stokes flow in the situations whereone has corners or cusps formed by parts of the domain boundaryand, due to the nature of the boundary conditions on these partsof the boundary, these regions have a global effect on the shapeof the whole domain and hence have to be resolved with sufficientaccuracy. The algorithm combines the boundary integral equationmethod for the main part of the flow domain and the finite-elementmethod which is used to resolve the corner/cusp regions. Twoparts of the solution are matched along a numerical ‘internalinterface’ or, as a variant, two interfaces, and theyare determined simultaneously by inverting a combined matrixin the course of iterations. The algorithm is illustrated byconsidering the flow configuration of ‘curtain coating’,a flow where a sheet of liquid impinges onto a moving solidsubstrate, which is particularly sensitive to what happens inthe corner region formed, physically, by the free surface andthe solid boundary. The ‘moving contact line problem’is addressed in the framework of an earlier developed interfaceformation model which treats the dynamic contact angle as partof the solution, as opposed to it being a prescribed functionof the contact line speed, as in the so-called ‘slip models’.     

A safeguarded dual weighted residual method

Andreas Veeser ^{The dual weighted residual (DWR) method yields reliable a posteriorierror bounds for linear output functionals provided that theerror incurred by the numerical approximation of the dual solutionis negligible. In that case, its performance is generally superiorthan that of global ‘energy norm’ error estimatorswhich are ‘unconditionally’ reliable. We presenta simple numerical example for which neglecting the approximationerror leads to severe underestimation of the functional error,thus showing that the DWR method may be unreliable. We proposea remedy that preserves the original performance, namely a DWRmethod safeguarded by additional asymptotically higher ordera posteriori terms. In particular, the enhanced estimator isunconditionally reliable and asymptotically coincides with theoriginal DWR method. These properties are illustrated via theaforementioned example.}     

Separating Sets as Envelopes

The set which forms the boundary of the regions of interferencewhen a family of curves is superposed on a parallel gratingis related to the envelopes of two ‘interference sets’in two fixed directions.     

Numerical analysis of boundary integral solution of the Helmholtz equation in domains with non-smooth boundaries

In the paper we carry out a complete analysis of several efficientnumerical methods for the solution of boundary integral equationsdefined on a non-smooth boundary. In particular the solutionof the Helmholtz equation in the exterior of a closed wedgeis studied. The analytical behaviour of the solution of theresulting boundary integral equation (with a non-compact operator)near the wedge is investigated. Numerical analysis of the collocationand iterated collocation method for the problem is presented.Graded meshes are used to reflect the ‘singular’behaviour of the analytical solution, as well as the degreeof the polynomial approximant, in order to yield results with‘optimal convergence rates’. Finally the convergenceanalysis of some modified two-grid iterative methods for thefast solution of the resulting linear systems is given and numericalresults are presented which agree with the theoretical predictions.     

Quadratisation de Certaines Structures de Poisson

On sait associer à certaines structures de Poisson surRⁿ, de 1-jet nul en 0, des actions de R² sur Rⁿ, donnéespar le ‘rotationnel’ de leur partie quadratiqueet un autre champ de vecteurs. Lorsque ces actions sont ‘nonrésonantes’ et ‘hyperboliques’, onmontre que ces structures sont ‘quadratisables’,en ce sens qu'il existe des coordonnées dans lesquelles,elles sont quadratiques. Dans le cas de la dimension 3, nosrésultats mènent à la ‘non-dégénérescence’générique des structures de Poisson quadratiquesà rotationnels inversibles. We can associate with some Poisson structures defined on Rⁿwith a zero 1-jet at zero, actions from R² on Rⁿ, given by the‘curl’ of their quadratic part and another vectorfield. Assuming that those actions are ‘hyperbolics’and without ‘resonances’, we give a normal formfor those structures. On R³, we prove that every quadratic Poissonstructure with invertible curl, is generically ‘non degenerate’.     

Bounds and Self-Consistent Estimates for the Overall Properties of Nonlinear Composites

Variational ‘self-consistent’ estimates for nonlinearproblems are formulated, building on a variational formulationpreviously developed by the authors. The formulation employsa linear ‘comparison medium’ for whose propertiessome ‘self-consistent’ choice is made. In contrastto linear problems, three possible self-consistent choices presentthemselves. The results that they give are analysed for twoparticular systems (a nonlinear dielectric and a nonlinear lossycomposite) for which bounds are already available. Estimatesbased on self-consistent embedding of a single inclusion ina homogeneous matrix composed of ‘comparison material’are also developed.     

Timeout control scheme for overloaded M/E2/1 queue

A timeout scheme is considered for controlling an infinite ‘firstcome, first served’ overloaded single-server queue. Inthe overload situation, a customer-rejection mechanism is usedfor timing out ‘older’ customers in the queue, i.e.excluding those who have waited longer than a certain time.Applying ‘level-crossing analysis’ to an M/E₂/1queue, exact analytic expressions of performance such as thedensity and distribution functions of waiting time of the customerswho get served, the mean delay of customers, successful throughput,and ‘goodput’ are determined for this queue.     

Contrary Statements About Mathematics

The preceding paper ‘Strong statements of analysis’by A. R. D. Mathias defends a so-called full-blooded set theorywithout full detail [3]. He again objects to a weak set theorywhich he calls ‘Mac’, in which the usual Zermelo–Fraenkelseparation scheme is required only for formulas with suitably‘restricted’ quantifiers. I had proposed that suchseparation is adequate for all standard uses of set theory inmathematics. But Mathias has not produced any counter examplesof actual mathematics which requires the use of a stronger separation.     

Transformations Between Sturm-Liouville Problems with Eigenvalue Dependent and Independent Boundary Conditions

Explicit relationships are given connecting ‘almost’isospectral Sturm-Liouville problems with eigen-value dependent,and independent, boundary conditions, respectively. Applicationis made to various direct and inverse spectral questions. 2000Mathematics Subject Classification 34B24, 34L05, 47E05.     

On the Robin problem in fractal domains

We study the solution to the Robin boundary problem for theLaplacian in a Euclidean domain. We present some families offractal domains where the infimum of the solution to the mixedDirichlet–Robin boundary problem is greater than 0, andsome other families of domains where it is equal to 0. We alsogive a new result on ‘trap domains’ defined in Burdzy,Chen and Marshall (Math. Z.), that is, domains where reflectingBrownian motion takes a long time to reach the center of thedomain.     

Two-parameter asymptotic approximations in the analysis of a thin solid fixed on a small part of its boundary

Planar elasticity problems are considered for thin domains fixedalong a small part of the end region boundary. The analysisinvolves two small parameters: the normalized thickness of thebody and the normalized length of the fixed part of the boundary.The aim of the paper is to derive an asymptotic approximationof the solution to a boundary-value problem in such a domainand, in particular, analyze the ‘effective boundary conditions’,which occur for the leading-order terms of the asymptotics.We include applications for problems of both anti-plane shearand plane strain elasticity.     

Reduced-basis output bound methods for parabolic problems

^** Email: rovas{at}uiuc.edu^*** Email: luc_machiels{at}mckinsey.com^**** Corresponding author. Email: maday{at}ann.jussieu.fr In this paper, we extend reduced-basis output bound methodsdeveloped earlier for elliptic problems, to problems describedby ‘parameterized parabolic’ partial differentialequations. The essential new ingredient and the novelty of thispaper consist in the presence of time in the formulation andsolution of the problem. First, without assuming a time discretization,a reduced-basis procedure is presented to ‘efficiently’compute accurate approximations to the solution of the parabolicproblem and ‘relevant’ outputs of interest. In addition,we develop an error estimation procedure to ‘a posteriorivalidate’ the accuracy of our output predictions. Second,using the discontinuous Galerkin method for the temporal discretization,the reduced-basis method and the output bound procedure areanalysed for the semi-discrete case. In both cases the reduced-basisis constructed by taking ‘snapshots’ of the solutionboth in time and in the parameters: in that sense the methodis close to Proper Orthogonal Decomposition (POD).     

Steady-state motion of a mode-III crack on imperfect interfaces

The aim of the present paper is to analyse the behaviour ofthe stress and displacement fields in the vicinity of the tipof a crack moving along a bi-material interface. For simplicity,we consider a straight interface of infinite extent. We assumethat the two phases are separated by a thin layer which is either‘soft’ or ‘stiff’ compared to the othertwo phases. We derive the transmission conditions which takeinto account the material properties of the layer and modelthe way the load is transferred across the layer from one phaseto the other. We assume that the point of interchange in theboundary/transmission conditions coincides with the crack tipthat moves along the interface boundary with a constant speed.We develop an integral equation formulation and derive asymptoticformulae for the out-of-plane displacement and the Mode-IIIstress intensity factor associated with such a motion of thecrack inside the interphase layer. The theoretical results areillustrated by numerical examples.     

Monodromy and Connectedness

The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is ‘mobile’ (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from ‘relative’ or total-space irreducibilityto ‘absolute’ or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05.     

Some exact results for nonlinear diffusion with absorption

Simple exact solutions and first integrals are obtained fornonlinear diffusion incorporating absorption. These are obtainedby the standard techniques of separation of variables and theuse of invariant one-parameter group transforma-tions to reducethe governing partial differential equation to various ordinarydifferential equations. For two of the equations so obtained,first integrals are deduced which subsequently give rise toa number of explicit simple solutions. As with all special solutionsof nonlinear partial differential equations, the associatedinitial and boundary conditions are imposed by the particularfunctional form of the solution and irrespective of their physicalapplicability, simple exact solutions are always important,because one of the key features of nonlinearity is the rangeand variety of response which is often bizarre and unexpected,but which is frequently embodied in the simplest of exact solutions.Many of the solutions obtained here are illustrated graphicallywith particular reference to the phenomena of ‘extinction’and ‘blow-up’ and in general demonstrate a widevariety of differing physical response embodied in the disposableconstants, which is characteristic of nonlinear theories.     

Axisymmetric isospectral annular plates and membranes

A simpler ‘conditional’ form of the vibrating plateequation for which all the physical parameters may be spatiallyvarying is found when the relationship D(1 – v) = constantis satisfied between flexural rigidity and Poisson's ratio.A method of coordinate inversion is used to find a family ofinhomogeneous clamped annular plates (with axisymmetric spatiallyvarying density, thickness, Young's modulus, and Poisson's ratio)which have the same complete frequency spectrum as a standardhomogeneous clamped annular plate of the same inner and outerradii. In particular, these inhomogeneous plates have inversefourthpower radial dependence of density. General radial power-lawtransformations are also considered, leading to conclusionsfor the axisymmetric vibration spectra. Various results forcircular and annular membranes also emerge from the analysis.     

A queueing model of activities in an intensive care unit

^** Email: griffiths{at}cardiff.ac.uk Activities in an intensive care unit (ICU) at a major teachinghospital are modelled by means of a queue-theoretic approach.Using data supplied by the ICU relating to the admissions process,the bed availability and the length of stay of patients, itwas possible to fit theoretical distributions to the observed‘arrival’ and ‘service’ distributions.Queueing equations relevant to a multi-channel system havingrandom arrivals and hyper-exponential service times for eachchannel are set up, and solved iteratively. Results obtainedmatch well with observations, and the model is then utilisedto investigate several ‘what if? ’ scenarios. Referenceis made to a simulation model developed in conjunction withthe queueing model.     

Some new aspects of the transient analysis of discrete-parameter Markov models with an application to the evaluation of repair events in power transmission

In Markov reliability modelling, a partitioned state space isused to describe the behaviour of a system each state of whichis associated with the system either being functional or underrepair. Such a system alternates between working and repairperiods indefinitely. Recent research results on the distributionof the sequences of the lengths of working and repair periodsafford the reliability analyst a set of system characteristicswhich can be used in addition to the traditional ones (reliability,point availability, etc.) to describe the system‘s transientbehaviour. In this paper, we present a concise derivation ofclosed-form expressions for the probability mass function andthe factorial moments of the total cumulative ’time‘spent in a subset of the state space by an irreducible or absorbingdiscrete-parameter Markov chain during the first n time instances.This result is then applied to analyse the sequence of repairevents categorized as ’minor‘ and ’major‘of a Markov model of a power transmission system. The numericalimplementation using the Macintosh version of MatLab is alsodiscussed.