Long-time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems

The long-time behaviour of continuous time Galerkin (CTG) approximationsof some well-known one-dimensional non-linear evolution problemswhich model phase transitions are analyzed. These numericalschemes are fully discrete and of arbitrary order. Partially supported by the Army Research Office through grant28535-MA. Partially supported by the ONR Contract No N00014-90-J-1238.     

A numerical method using the Prfer transformation for the calculation of eigenpairs of two-parameter Sturm-Liouville problems

A shooting method for coupled Prfer equations is discussedfor numerical solution of two-parameter Sturm-Liouville problems. Research supported in part by grants from the NSERC of Canada.     

Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. I: reaction-diffusion-type problems

We consider singularly perturbed high-order elliptic two-pointboundary value problems of reaction-diffusion type. It is shownthat, on an equidistant mesh, polynomial schemes cannot achievea high order of convergence that is uniform in the perturbationparameter. Piecewise polynomial Galerkin finite-element methodsare then constructed on a Shishkin mesh. Almost optimal convergenceresults, which are uniform in the perturbation parameter, areobtained in various norms. Numerical results are presented fora fourth-order problem. e-mail address: stynes{at}bureau.ucc.ie.     

Pointwise Stabilization for a Chain of Coupled Vibrating Strings

For the system representing a chain of coupled vibrating strings,we show that the associated semigroup satisfies the assumptionof spectrum-determined growth, and hence obtain conditions forenergy to decay strongly or exponentially. We examine in detailthe three-string case, and our results include those obtainedby others for the two-string case. Permanent address: Beijing Institute of Information and Control,Beijing, China.     

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     

An Example of Only Linear Convergence of Trust Region Algorithms for Non-smooth Optimization

Most superlinear convergence results about trust region algorithmsfor non-smooth optimization are dependent on the inactivityof trust region restrictions. An example is constructed to showthat it is possible that at every iteration the trust regionbound is active and the rate of convergence is only linear,though strict complementarity and second order sufficiency conditionsare satisfied. Presented at the 1983 Dundee Conference on Numerical Analysis     

Estimates of the region of attraction of continuous-time cascade systems

Xiao-Song Yang Department of Mathematics, Huazhong University of Science and Technology Wuhan, Hubei 430074, People's Republic of China Huimin Li Departments of Mathematics and Control and Engineering, Huazhong University of Science and Technology Wuhan, Hubei 430074, People's Republic of China Corresponding author. Email: yangxs{at}cqupt.edu.cn Received on May 1, 2006; Accepted on October 6, 2006 In this paper, we first present some new sufficient conditionsfor global asymptotic stability of continuous-time cascade systems.Then, we give some detailed results on the region of attractionof continuous-time cascade systems. Some examples are presentedto illustrate these results.     

Remarks on Perturbations of the Equation

Perturbations of the equation are discussed. The structure of near-orthogonal matrices is shown, and implicationsfor certain control problems are pointed out. Examples are givenand the circumstances are noted in which the real part of aneigenvalue can be expected to change sign. This paper was written while the author was on a Leave of Absenceat the Department of Mathematics, University of Salford.     

Finite-difference schemes for scalar conservation laws with source terms

Explicit and semi-implicit finite-difference schemes approximatingnon-homogeneous scalar conservation laws are analyzed. Optimalerror bounds independent of the stiffness of the underlyingequation are presented. This author has been supported by The Norwegian Research Council(NFR), program No 100284/431. e-mail: schroll{at}igpm.rwth-aachen.de This author has been supported by The Norwegian Research Council(NFR), program Nos 100284/431 and STP.29643. e-mail: ragnar{at}ifi.uio.no     

Simpler Polynomial Solutions in Control and Filtering

Kuera's completely algebraic method is used to derive a polynomialsolution with only one Diophantine equation for least-squaresdeterministic control in discrete-time. The same method is used,again involving only one Diophantine equation, to derive themultivariable discrete-time Wiener filter. This paper was presented at the IMA Symposium on Control Theory,Somerville College,Oxford, September 1986.     

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).     

Convergence of Methods for the Numerical Solution of the Korteweg--de Vries Equation

We prove that a family of methods for the numerical solutionof the Korteweg—de Vries equation is convergent. Thisfamily includes as particular cases some known finite differenceand finite element schemes. It is also found that the stabilityproperties of the methods vary significantly with the treatmentof the non-linear term. On leave from Shanghai University of Science and Technology,Shanghai, China.     

An Ergodic Control Problem for Reflected Diffusion with Jump

The control of the drift of a stochastic differential equationwith jump term is considered for the long-run average cost.The convergence of the discounted problem is studied, as wellas the corresponding dynamic programming condition. This research was supported in part by U.S. Army Research Officeunder contract DAAG29-83-K-0014 and completed during a visitto INRIA.     

On approximate approximations using Gaussian kernels

This paper discusses quasi-interpolation and interpolation withGaussians. Estimates are obtained showing a high-order approximationup to some saturation error negligible in numerical applications.The construction of local high-order quasi-interpolation formulasis given. Supported in part by the International Centre for MathematicalSciences, Edinburgh.     

Numerical simulations of measurements of capillary contact angles

In a recent paper, Fischer and Finn have proposed a procedureto improve the accuracy in the measurement of capillary contactangles, based on the use of vessels with canonical cross-sections.We simulate numerically the behaviour of such shapes for a numberof cross-sections and fluid contact angles. Our approximationconsists of the minimization of a suitable convex functionaldiscretized by finite elements. e-mail: bellettini{at}sns.it e-mail: paolini{at}isa.mat.unimi.it     

A Chebyshev collocation method for solving Symm's integral equation for conformal mapping: a partial error analysis

A boundary integral representation for the conformal mappingfrom a simply connected domain onto a given canonical regionis presented. The conformal map can be constructed via the solutionof a first-kind integral equation with logarithmic kernel: Symm'sintegral equation. A novel method for solving this equationis presented and a partial error analysis, including convergencerates, is given. The analysis is also relevant to the solutionof Symm's equation on a smooth, open contour. A superconvergenceresult for the conformal mapping is proved. Now in the Department of Mathematics and Computer Science, Universtyof Leicester, Leicester LE1 7RH, UK.     

Structure of solution manifolds in a strongly coupled elliptic system

In this paper we study the steady-state solutions of a reaction-diffusionmodel, the Selkov scheme for glycolysis, under homogeneous Dirichletboundary conditions. Near to thermodynamic equilibrium, thestructure and stability of solutions are fully described. Abifurcation analysis is carried out, using the size of the regionin which the reaction takes place and one diffusion coefficientas main bifurcation parameters. The analysis helps us to understandthe nature of the bifurcation points, and determines the shapesand stability of the bifurcating manifolds in the neighbourhoodof the constant state. Local convergence of spectral methodsis shown, and some global pictures are calculated using path-followingtechniques. The framework we use can be applied to a wide varietyof reaction-diffusion systems. ^*Permanent address: Dpto. de Matemtica Aplicada, Fac. de CenciasQumicas, Universidad Complutense, 28040-Madrid, Spain. Current address: Department of Mathematics, Paisley Collegeof Technology, High Street, Paisley, Renfrewshire PA1 2BE, UK. Permanent address: Dpto. de Matemtica, Fac. de Matemtca, Astronomiay Fisica, Universidad National de Crdoba, 5000-Crdoba, R.Argentina.     

Minimal equivalence of differential IO systems

Differential algebraic controllability, observability, and minimalrealization are considered further, upon which a new equivalenceproblem, i.e. the minimal equivalence of two given differentialinput-output (IO) systems through state-space realization isconsidered. Some criteria for equivalence and minimal equivalenceof two IO systems are given. The algorithm for checking minimalequivalence of two IO systems with single input is implementedwith MACSYMA. Email: xyl{at}sun.engg.le.ac.uk David.Bell{at}umist.ac.uk     

An Algorithm with a Point-Linearized Input-Output Mapping for Integrated System Optimization and Parameter Estimation

An improved algorithm for centralized integrated system optimizationand parameter estimation based on a point-linearized I-O mappingand Newton step in updating is described. Two convergence theoremsunder different assumptions are investigated. Optimality ofthe method and the relationship between the algorithmic fixed-pointset and the optimum solution set of the system are presented.Simulation reveals that, because of the adoption of point-linearizedI-O mapping and a Newton step in updating, the proposed techniqueenjoys two benefits of easier computation and a higher convergencerate. On study leave from the Anhui Institute of Machinery & Electronics,Wuhu, Anhui Province, China On study leave from the Institute of Systems Engineering, Xi’anJiaotong University, Xi’an, China     

On the Stability Analysis of Nonlinear Systems

The stability of nonautonomous delay differential equationsis studied via a combination of the global linerization andnonlinear variation of parameter techniques. Permanent address: Department of Mathematics, University ofElectronics Science & Technology of China, Chengdu, China.