An integral equation approach to electromagnetic diffraction by a cylinder

A two-dimensional perturbation theory of electromagnetic diffractionby a cylindrical conductor is developed, based upon Robin’sintegral equation of electrostatics. An innovative featureof the theory is of not requiring a boundary condition of zerotangential electric field at the conducting surface, but requiringexplicitly that the bulk conductivity be taken into account,and solutions of Maxwell’s equations are obtained for both the interior and exterior regions. The theory is a near-field, low-frequency one, being perhaps valid for the highest radio frequencies, but not for optical frequencies or for great distances from the cylinder. Received 3 June, 1999. ⁺ antoine@mip.ups-tlse.fr     

Hot spot formation in microwave heated ceramic fibres

An analysis of microwave heating of a thin ceramic cylinderin a single mode, highly resonant cavity is presented. Realisticassumptions regarding the effective electrical conductivity,thermal parameters, and physical dimensions are adhered to throughout.Consequently, the model developed herein incorporates most ofdetuning and a local electric feild perturbation on the heatingprocess. The model presented takes the form of a one-dimensionalreaction–diffusion equation which contains a functional.The developement of this equation is the product of a systematicmodelling process that involves S-matrix theory, a small-Biot-numberasymptotic analysis, and a matched asymptotic analysis of anon-standard electromagnetic scattering problem. This equationreveals both the mathematical structure and physical mechanismfor the formation of hot spots. An accurate numerical methodwhich approximates the solution of this equation is presented.The results agree qualitatively with experiments and predictobserved trends.     

Microwave heating of carbon-coated ceramic fibres: A mathematical model

The microwave heating of a thinly carbon-coated ceramic fibreis modelled and analysed in the small Biot number regime. Theelectric field is assumed known and uniform throughout the cross-section,and constant along the axis of the cylinder. The mathematicalmodel consists of a nonlinear heat equation with an idealizedsource term that models the thin carbon coating and simplifiedreaction kinetics. The analysis yields an asymptotic approximationof the heating process on two time scales. The first capturesthe initial heating of the carbon coating and the ceramic, andthe carbon reaction, while the second determines the long timebehaviour of the sample. The results show a qualitative relationshipbetween the coating thickness and the final temperature of thefibre. If the coating thickness is not uniform along the fibreaxis, then the model explains the mechanism for the formationand propagation of hot-spots.     

The general packed column: an analytical solution

The transient behaviour of a packed column is considered. The column, uniformly packed on a macroscopic scale, is multi-structured on the microscopic level: the solid phase consists of particles, which may differ in incidence, shape or size, and other relevant physical properties. Transport in the fluid phase happens by convective flow and longitudinal diffusion. The adsorptionrate in the column is determined by the combined effects ofmass transfer through a thin liquid film around the particles,diffusion into the particles and first-order loss in the particles.An analytical solution to the problem is given in the formof an infinite integral. The solution method is worked outin detail for three special cases, each with only one typeof particle in the solid phase, in the shape of a sphere, ora finite cylinder, or a rectangular parallelepiped. Received 7 September, 1998. Revised 4 March, 2000. ⁺ E-mail: jo.gielen@ztw.wk.wau.nl     

Riccati equation for infinite-dimensional discrete bilinear systems

The problem of quadratic optimal feedback control for infinite-dimensionaldiscrete bilinear systems is associated with an algebraic Riccati-likeoperator equation. The existence and uniqueness of a nonnegativesolution to such a Riccati-like operator equation is establishedas the strong limit of a sequence of solutions to Lyapunov-likeoperator equations. This research was supported in part by CNPq (Brazilian NationalResearch Council) and FAPESP (Research Council of the Stateof sào Paulo).     

Two-dimensional nonlinear advection-diffusion in a model of surfactant spreading on a thin liquid film

The spreading of a localized monolayer of dilute, insoluble surfactant, discharged from a point source that moves at constant speed over a thin liquid film coating a planar substrate, is described according to lubrication theory by a pair of coupled nonlinear evolution equations for the monolayer concentration and the film depth h. Numerical and asymptotic techniquesare here used to show that the extent and structure of sucha spreading asymmetric monolayer can be well approximated bya single nonlinear advection–diffusion equation involving alone. At large times the solution is composed of three, spatiallydistinct, asymptotic regions: (i) a quasi-steady ‘nose’region (containing the source), in which there is a dominantbalance between two-dimensional nonlinear diffusion and advection;(ii) an ‘advective’ region, in which longitudinaladvection balances transverse diffusion; and (iii) a ‘tail’region, in which unsteady diffusion is dominant. In each region,local similarity solutions are obtained either exactly (inthe advective region) or approximately (elsewhere) by rescalingnumerical solutions of the initial-value problem. If the sourceconcentration decreases with time, it is demonstrated that the monolayer’s width is greatest in the tail region, whereasfor a source of increasing concentration the monolayer is widestin the advective region. For the simpler one-dimensional problemof a monolayer spreading from a line source, the same balanceshold but with transverse diffusion eliminated; here self-similarsolutions are found in all three regions that agree closelywith numerical solutions of the initial-value problem. Received 7 October, 1998. Revised 11 April, 2000. ⁺ antoine@mip.ups-tlse.fr Present address: Division of Theoretical Mechanics, Schoolof Mathematical Sciences, University of Nottingham , UniversityPark, Nottingham NG7 2RD, UK. Oliver.Jensen@nottingham.ac.uk.     

Newton additive and multiplicative Schwarz iterative methods

Daniel B. Szyld^¶ Department of Mathematics, Temple University, Philadelphia, PA 19122, USA ^{Convergence properties are presented for Newton additive andmultiplicative Schwarz (AS and MS) iterative methods for thesolution of nonlinear systems in several variables. These methodsconsist of approximate solutions of the linear Newton step usingeither AS or MS iterations, where overlap between subdomainscan be used. Restricted versions of these methods are also considered.These Schwarz methods can also be used to precondition a Krylovsubspace method for the solution of the linear Newton steps.Numerical experiments on parallel computers are presented, indicatingthe effectiveness of these methods.}     

A posteriori estimates for approximations of time-dependent Stokes equations

Charalambos Makridakis ^{In this paper, we derive a posteriori error estimates for space-discreteapproximations of the time-dependent Stokes equations. By usingan appropriate Stokes reconstruction operator, we are able towrite an auxiliary error equation, in pointwise form, that satisfiesthe exact divergence-free condition. Thus, standard energy estimatesfrom partial differential equation theory can be applied directly,and yield a posteriori estimates that rely on available correspondingestimates for the stationary Stokes equation. Estimates of optimalorder in L(L2) and L(H1) for the velocity are derived for finite-elementand finite-volume approximations.}     

Optimal recovery using thin plate splines in finite volume methods for the numerical solution of hyperbolic conservation laws

The theory of optimal recovery is applied to finite volume methodsfor the numerical solution of conservation laws in multiplespace dimensions. Classical polynomial EN0 (essentially non-oscillatory)algorithms can be interpreted as trivial recovery operatorsfor the point functional in the light of this theory. Thin platesplines are identified as optimal recovery functions in Beppo-Levispaces and can be seen as multi-dimensional analogues of cubicsplines. Recovery algorithms of ENO-type based on thin platesplines are developed and applied to test problems includingthe Euler equations of compressible gas dynamics. E-mail: Thomas.Sonar{at}ts.go.dlr.de     

Thermal runaway and microwave heating in thin cylindrical domains

The behaviour of the solution to two nonlinear heating problemsin a thin cylinder of revolution of variable cross-sectionalarea is analysed using asymptotic and numerical methods. Thefirst problem is to calculate the fold point, correspondingto the onset of thermal runaway, for a steady-state nonlinearelliptic equation that arises in combustion theory. In the limitof thin cylindrical domains, it is shown that the onset of thermalrunaway can be delayed when a circular cylindrical domain isperturbed into a dumbell shape. Numerical values for the foldpoint for different domain shapes are obtained asymptoticallyand numerically. The second problem that is analysed is a nonlinearparabolic equation modelling the microwave heating of a ceramiccylinder by a known electric field. The basic model in a thincircular cylindrical domain was analysed in Booty & Kriegsmann(Meth. Appl. Anal. 4 (1994) p. 403). Their analysis is extendedto treat thin cylindrical domains of variable cross-section.It is shown that the steady-state and dynamic behaviours oflocalized regions of high temperature, called hot-spots, dependon a competition between the maxima of the electric field andthe maximum deformation of the circular cylinder. For a dumbell-shapedregion it is shown that two disconnected hot-spot regions canoccur. Depending on the parameters in the model, these regions,ultimately, either merge as time increases or else remain asdisconnected regions for all time.     

A modified Newton method for solving non-symmetric algebraic Riccati equations arising in transport theory

Liang Bao ^{The non-symmetric algebraic Riccati equation arising in transporttheory can be rewritten as a vector equation and the minimalpositive solution of the non-symmetric algebraic Riccati equationcan be obtained by solving the vector equation. In this paper,we apply the modified Newton method to solve the vector equation.Some convergence results are presented. Numerical tests showthat the modified Newton method is feasible and effective, andoutperforms the Newton method.}     

The Caratheodory--Fejer Extension of a Finite Geometric Series

It is shown that the Caratheodory—Fejer extension of afinite geometric series can be given explicitly up to a simplepolynomial equation in an auxiliary variable. This result allowsus to analyse the Caratheodory-Fejer approximation method inthe case where the quotients of successive Maclaurin coefficientsof the given function tend to a limit. ^*Research carried out while this author was at ETH Zurich partiallysupported by a Royal Society European Visiting Fellowship.     

On the Application of Extrapolation, Deferred Correction and Defect Correction to Discrete-time Galerkin Methods for Parabolic Problems

Conditions which ensure the applicability of extrapolation-in-timeof the backward-difference Galerkin and Crank—NicolsonGalerkin methods for linear parabolic initial-boundary problemsare discussed. Deferred correction and defect correction proceduresfor these discrete-time Galerkin methods are formulated andtheir theoretical properties summarized. These procedures yieldapproximations which are up to fourth-order correct in time,and whose computation requires less work per time step thanthat required by extrapolation. The results of numerical experiments,which demonstrate the expected rates of convergence with respectto the time discretization of the various techniques, are presented. ^*The work of this author was supported in part by the NationalScience Foundation under grant MCS-8102295     

Asymptotic expansion of highly conductive thin sheets

Sensitive measurement and control equipment are protected from disturbing electromagnetic fields by thin shielding sheets. Alternatively to discretisation of the sheets, the electromagnetic fields are modeled only in the surrounding of the layer taking them into account with the so called Generalised Impedance Boundary Conditions. We study the shielding effect by means of the model problem of a diffusion equation with additional dissipation in the curved thin sheet. We use the asymptotic expansion techniques to derive a limit problem, when the thickness of the sheet ε tends to zero, as well as the models for contribution to the solution of higher order in ε. These problems are posed in limit area of vanishing ε with condition for the jump of the solution and it's normal derivative, which avoid to mesh the computational domain, even locally, at the scale of ε. We derive the problems for arbitrary order and show their existence and uniqueness. Numerical experiments for the problems up to second order show the asymptotic convergence of the solution of right order in mean of the thickness parameter ε. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mixed block elimination for linear systems with wider borders

The paper is about the stable solution of possibly ill-conditionedbordered linear systems. Given stable solvers for matrix A andfor A^T, we prove that the Govaerts Mixed Block Elimination (BEM)method constitutes a stable solver for the matrix consistingof A or A^T with a border of width 1, and hence by recursionfor a border of any width. We express the algorithm in an efficient,iterative, form. We analyse its operation count, and verifythe theory by extensive numerical experiments. ^*Senior Research Associate of the Belgian National Fund of ScientificResearch NFWO.     

A calibration algorithm for simulation-based pricing models

Don L. McLeish Department of Statistics and Actuarial Science, University of Waterloo, Waterloo,Ontario N2L 3G1, Canada Corresponding author. Email: mreesor{at}uwo.ca Email: dlmcleis{at}math.uwaterloo.ca Received on 20 February 2007. Accepted on 15 March 2007. Derivative pricing models require calibration to market conditionsin order to determine quantities such as hedging positions andthe prices of other instruments. For stochastic models and/orcomplex derivatives whose prices are not of an analytic form,prices must be computed via simulation and the calibration ismore difficult. A method to facilitate the calibration of simulation-basedpricing models is proposed. The algorithm uses a statisticallydesigned experiment to select the points at which simulationsare performed. The method is quite general as it is independentof the stochastic model for the underlying and allows for differentobjective functions that can incorporate information such asopen interest and volume. Furthermore, market prices from European-and/or American-style derivatives covering a range of strikeprices and maturities can be handled by this technique. Examplesshow the procedure is successful at calibrating well-known assetpricing models to both simulated and market data.