Non-linear Theory of Wave-front Propagation

Present address: Division of Mathematical Sciences, Perdue University, Lafayette, Indiana, U.S.A. Equations are derived which govern the mode of propagation andchange in strengths of wave fronts whose behaviours are controlledby quite general systems of quasi-linear hyperbolic equations.The general theory is applied to discuss conditions at the headof a sound wave of arbitrary initial form and strength and alsoat the head of gravity waves on sloping beaches. Conditionsunder which shocks and bores occur are derived. At expansivewaves the non-linearity produces "forgetfulness" at "large"times.     

A Perturbation Solution for a Two-dimensional Annular Electrochemical Machining Problem

Present address: Deparment of Applied Mathematics, University of Western Ontario, London, Canada. A two-dimensional electrochemical machining problem which consistsof an almost circular anode placed inside a circular cathodeis solved using a perturbation procedure. The smoothing of roughmetal parts is studied by solving the resulting perturbationequations numerically.     

An analysis of the coupling of finite-element and Nystrm methods in acoustic scattering

In this paper we analyze the use of a combined Nystrm and finite-elementprocedure for approximating the scattered time-harmonic acousticfield produced when an incident wave interacts with a boundedinhomogeneity. The coupling technique uses a smooth artificialboundary and explicitly decouples the integral equation andfinite-element computations. We prove convergence of the method.One highlight of this analysis is that we prove Sobolev spaceerror estimates for the Nystrm scheme (which is usually analyzedin Hlder spaces). We also present some numerical results. E-mail: kirsch{at}am.uni-erlangen.de E-mail: monk{at}math.udel.edu     

Realistic a Priori and a Posteriori Error Bounds for Computed Eigenvalues

Permanent address: Department of Engineering Mathematics, Cairo University, Giza, Egypt. A priori and a posteriori error bounds are given for the computedeigenpair (, ) of the eigenvalue problem Ax = x, which are shownto be more realistic than some of the available ones. A simplemethod is also presented for computing the backward error. Finallya scaling procedure is explained for reducing the residual error.     

The Frictional Unloading Problem on a Linear Elastic Half-Space

Present Address: ICI Agricultural Divison, P.O. Box 1, Billingham, Cleveland, TS23 ILB. A variational formulation is derived for contact problems withfriction. The problem of contact between a rigid, circular,cylindrical indentor and an isotropic, homogeneous, linearlyelastic half-space is then considered. Using the above variationalformulation numerical solutions are obtained to the problemsof frictionless contact, adhesive contact, and frictional loading,for which there are known analytical solutions, as well as thefrictional unloading problem, for which there is no previousknown solution. Comparisons are made with experiment and predictionsfrom physical considerations.     

Direct approximation of observer error linearization for nonlinear forced systems

Email: klaus{at}roebenack.de In this paper, we discuss a generalization of the extended Luenbergerobserver. Our approach can be interpreted as an approximateerror linearization. The design procedure which is formulatedin terms of Lie derivatives and Lie brackets can easily be implementedusing computer algebra software.     

Pointwise error estimates for a streamline diffusion scheme on a Shishkin mesh for a convection-diffusion problem

We analyze a streamline diffusion scheme on a special piecewiseunform mesh for a model time-dependent convection–diffusionproblem. The method with piecewise linear elements is shownto be convergent, independently of the diffusion parameter,with a pointwise accuracy of almost order 5/4 outside the boundarylayer and almost order 3/4 inside the boundary layer. Numericalresults are also given. stynes{at}bureau.ucc.ie     

Optimal boundary feedback stabilization of a string with moving boundary

Email: gugat{at}am.uni-erlangen.de Received on April 30, 2006; We consider a finite string that is fixed at one end and subjectto a feedback control at the other end which is allowed to move.We show that the behaviour is similar to the situation whereboth ends are fixed: As long as the movement is not too fast,the energy decays exponentially and for a certain parameterin the feedback law it vanishes in finite time. We considermovements of the boundary that are continuously differentiablewith a derivative whose absolute value is smaller than the wavespeed. We solve a problem of worst-case optimal feedback control,where the parameter in the feedback law is chosen such thatthe worst-case L^p-norm of the space derivative at the fixedend of the string is minimized (p [1, )). We consider the worstcase both with respect to the initial conditions and with respectto the boundary movement. It turns out that the parameter forwhich the energy vanishes in finite time is optimal in thissense for all p.     

Conjugate-gradient method for computing the Moore-Penrose inverse and rank of a matrix

A conjugate-gradient method is developed for computing the Moore-Penrose generalized inverseA of a matrix and the associated projectors, by using the least-square characteristics of both the method and the inverseA . Two dual algorithms are introduced for computing the least-square and the minimum-norm generalized inverses, as well asA . It is shown that (i) these algorithms converge for any starting approximation; (ii) if they are started from the zero matrix, they converge toA ; and (iii) the trace of a sequence of approximations multiplied byA is a monotone increasing function converging to the rank ofA. A practical way of compensating the self-correcting feature in the computation ofA is devised by using the duality of the algorithms. Comparison with Ben-Israel's method is made through numerical examples. The conjugate-gradient method has an advantage over Ben-Israel's method.After having completed the present paper, the author received from Professor M. R. Hestenes his paper entitledPseudo Inverses and Conjugate Gradients. This paper treated the same subject and appeared in Communications of the ACM, Vol. 18, pp. 40–43, 1975.     

Optimal control for the thermistor problem with a current limiting device

Email: cimatti{at}dm.unipi.it Received on September 26, 2005; Accepted on August 2, 2006 The problem of finding in the thermistor problem the optimaldifference of applied potential in order to minimize a suitablecost functional involving the temperature is studied. We findthe optimal system in the elementary case of constant electricconductivity. In addition, a theorem of existence of the optimalsolution is given in the general case of conductivities dependingon the temperature.     

The Transport Equation of Radiative Transfer in a Three-dimensional Space with Anisotropic Scattering

Present address: Science Research Council, Atlas Computer Laboratory, Chilton, Didcot, Berkshire, England. The exact solution of a radiative transfer problem requiresthe reduction of the integro-differential transport equationto an integral equation. This paper considers the transportequation in a general three-dimensional space, when the scatteringfunction consists of the first three terms of a series of Legendrepolynomials. The resulting integral equations may then be usedfor the solution of Radiative Transfer problems in any space.where the radiation is restricted to normal incidence.     

Stability of the approximation of a regular solution branch

In this paper we extend the so-called inf-sup conditions tononlinear problems having regular solution branches. The inf-supconditions for nonlinear problems have been introduced, forinstance, by Caloz and Rappaz in 1994. Here we present a moregeneral approach to include turning points on a solution branch.Our abstract results are applied to model examples. E-mail: Gabriel.Caloz{at}univ-rennesl.fr     

Error estimates for the finite-element solution of an elliptic singularly perturbed problem

e-mail:ang{at}am.uni-erlangen.de The paper deals with an upwind discretization of finite-volume-typefor singularly perturbed elliptic boundary value problems. Stability,inverse monotonicity and convergence are considered. The mainobjective is to pursue the dependence of the error estimateon the perturbation parameter.     

A new non-conforming Petrov-Galerkin finite-element method with triangular elements for a singularly perturbed advection-diffusion problem

A new non-conforming exponentially fitted Petrov-Galerkin finite-elementmethod based on Delaunay triangulation and its Dirichlet tessellationis constructed for the numerical solution of singularly perturbedstationary advectiondiffusion problems with a singular perturbationparameter . The method is analyzed mathematically and its stabilityis shown to be independent of . The error estimate in an -independentdiscrete energy norm for the approximate solution is shown todepend on first-order seminorms of the flux and the zero-orderterm of the equation, the sup norm of the exact solution, thefirst-order seminorm of the coefficient of the advection term,and the approximation error of the inhomogeneous term. Sincethe first two seminorms are not bounded uniformly in , the -uniformconvergence of the method still remains an open question. Noassumption is required that the angles in the triangulationare all acute. Since the system matrix for this method is identicalto that for the exponentially fitted box method, the theoreticalresults also provide an analysis of that box method. The newmethod also contains the central-difference and upwind methodsas two limiting cases. It can be regarded as a weighted finite-differencemethod on a triangular mesh. Numerical results are presentedto show the superior performance of the method in comparisonwith the upwind and central-difference methods for a small increasein the computation cost. Present address: School of Mathematics, The University of NewSouth Wales, Kensington, NSW 2033, Australia.     

New conditions on absolute stability of uncertain Lur'e systems and the maximum admissible perturbed bound

Email: fhao{at}buaa.edu.cn Received on April 5, 2006; Accepted on September 10, 2006 The robust absolute stability problem for structured uncertainLur'e systems is considered in this paper by using Popov criterionand extended strictly positive real lemma. The conditions onrobust absolute stability for Lur'e systems with structureduncertainties are established in terms of multilinear matrixinequalities. An estimate of the maximum bound of all admissibleperturbations is given by a generalized eigenvalue problem.Finally, a numerical example is worked out to illustrate theefficiency of the main results.     

The Solution of a Statistical Decision Problem with Moving Boundaries by the Image Method

Temporary address: Department of Theoretical Physics, University of Manchester. A point executes Brownian motion in one dimension, startingfrom a known initial position. Its velocity at any time is asuperposition of a constant "drift" speed and a random, diffusivecomponent, the motion taking place between two absorbing boundarieswhich move towards one another with constant speeds. The probabilitydistribution for the particle position at any time is foundby the method of images. The problem occurred originally inthe detection of a small d.c. signal in a large amount of "white"noise, and this application is discussed.     

Estimation of nonlinear systems without a priori structural information and its application in Hammerstein model identification with dynamic nonlinearities

Email: er-wei-bai{at}uiowa.edu Received on July 30, 2005; Accepted on July 23, 2006 In the paper, we discuss identification of a nonlinear systemwithout structural information and propose two methods, thekernel method and the orthonormal basis method. The convergenceresults are established for both methods without a priori structuralinformation. We then apply the results to identification ofHammerstein models with an unknown dynamic nonlinearity. Itis also shown that identification of the linear part in Hammersteinmodels is possible with no knowledge of the dynamic nonlinearity.     

The dynamic control of risk in optimised portfolios

E-mail: c.j.adcock{at}sheffield.ac.uk or adcock{at}intonet.co.uk This paper presents a theoretical treatment of the statisticalproperties of optimal portfolios. The results demonstrate theway in which the statistical properties of forecast returnsaffect the performance of optimised portfolios. The paper thenindicates how these methods may be used in principle to controlthe performance of an optimised portfolio. This is exemplifiedby an optimal portfolio of foreign currencies.     

Optimal control problems of parabolic equations with an infinite number of variables and with equality constraints

Email: bahaa_gm{at}hotmail.com Received on December 6, 2005; Accepted on December 7, 2006 Optimal control problems of systems governed by parabolic equationswith an infinite number of variables and with additional equalityconstraints are considered. The extremum principle, as wellas sufficient condition of optimality, is formulated for theNeumann problem by using certain extensions of Dubovitskii–Milyutinmethod.