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.     

Pseudospectral collocation methods for fourth-order differential equations

A multi-domain pseudospectral collocation scheme for the approximationof linear fourth-order differential equations in one and twodimensions is presented. A complete analysis of the scheme isprovided and error estimates are proved for the one-dimensionalproblem. An efficient preconditioner based on a low-order finite-differenceapproximation to the same differential operator is proposed.The extension of the method to the biharmonic equation in twodimensions is discussed and results are presented for a problemdefined in a non-rectangular domain. Present address: Department of Mathematics, Tarbiat ModarresUniversity, Tehran, iran     

The numerical solution of a non-linear boundary integral equation on smooth surfaces

We study a boundary integral equation method for solving Laplace'sequation u=0 with non-linear boundary conditions. This non-linearboundary value problem is reformulated as a non-linear boundaryintegral equation, with u on the boundary as the solution beingsought. The integral equation is solved numerically by usingthe collocation method, with piecewise quadratic functions usedas approximations to u. Convergence results are given for thecases where (1) the original surface is used, and (2) the surfaceis approximated by piecewise quadratic interpolation. In addition,we define and analyze a two-grid iteration method for solvingthe non-linear system that arises from the discretization ofthe boundary integral equation. Numerical examples are given;and the paper concludes with a short discussion of the relativecost of different parts of the method. This work was supported in part by NSF grant DMS-9003287.     

A new algorithm for finding numerical solutions of optimal feedback control

Bing Sun Department of Mathematics, Beijing Institute of Technology, Beijing 100081, People's Republic of China and School of Computational and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa Email: bzguo{at}iss.ac.cn Received on March 15, 2007; Revision received October 17, 2007. A new algorithm for finding numerical solutions of optimal feedbackcontrol based on dynamic programming is developed. The algorithmis based on two observations: (1) the value function of theoptimal control problem considered is the viscosity solutionof the associated Hamilton–Jacobi–Bellman (HJB)equation and (2) the appearance of the gradient of the valuefunction in the HJB equation is in the form of directional derivative.The algorithm proposes a discretization method for seeking optimalcontrol–trajectory pairs based on a finite-differencescheme in time through solving the HJB equation and state equation.We apply the algorithm to a simple optimal control problem,which can be solved analytically. The consistence of the numericalsolution obtained to its analytical counterpart indicates theeffectiveness of the algorithm.     

The Method of Fundamental Solutions for the Solution of Nonlinear Plane Potential Problems

The method of fundamental solutions is described for the solutionof elliptic boundary value problems governed by Laplace's equationin the plane subject to nonlinear radiation-type boundary conditions.The effectiveness of the method is demonstrated by examiningits performance on two problems from the literature, and comparisonsare made with published results obtained using boundary elementmethods. Portions of this work were conducted while this author wasa visiting assistant professor at the University of Kentucky. Partially supported by the National Science Foundation undergrant MCS-8303287 and grant RII-8610671, and by the Commonwealthof Kentucky through the Kentucky EPSCoR Program.     

On the Convergence of Some Approximate Methods of Conformal Mapping

A method was given in Ellacott (1978) for determining approximatelythe conformal mapping of a Jordan region on to a disc. Someresults on the convergence of this method are given, which canbe used to prove the result (conjectured in Ellacott, 1978)that if the boundary curve is analytic, then convergence isuniform. The corresponding result is also proved for the Bergmanand Szeg Kernel methods with polynomial basis functions. (Theresult is already known for the Szeg? Kernel method, but a differentproof is given.) Also discussed is the use of rational basisfunctions for the Bergman Kernel method. Currently visiting Forschungsinstitut fr Mathematik, ETH-Zentrum,CH-8092, Zurich.     

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.     

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     

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).     

Algebraic Manipulation of Polynomial Matrices: A Survey

A concise review is given of some recent methods for performingalgebraic operations on polynominal matrices. The problems addressedare division, inversion, and extraction of greatest common divisorof a pair of matrices. Based on a presentation at the International Symposium on theMathematical theory of Networks and Systems, Amsterdam, June1989.     

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.     

On Implicit Ordinary Differential Equations

In a recent paper Fox & Mayers discuss the numerical solutionof implicit ordinary differential equations of the form f(x,y(x), y'(x)) = 0. They find that numerical methods can be veryunreliable near the point where f_y' = 0. In this paper we givea theoretical analysis of the problem which enables us to explainwhen to expect numerical difficulties. We suggest a possibleline of approach for the solution of such problems, and discusssome numerical examples. Research supported by the National Science Foundation, the Officeof Naval Research, the Army Research, and the Air Force Officeof Scientific Research. Travel funding provided by the Universityof Toronto and the British Council.     

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     

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.     

On the stability of continuous quadrature rules for differential equations with several constant delays

The aim of the present paper is to study the stability propertiesof the numerical methods for pure delay differential equations.The methods we consider are based on a quadrature rule and aninterpolant (NCE) to get an approximation of the retarded part(continuous quadrature rule). As a test equation we consider and we give sufficient conditions for the boundedness of thesolutions. The same behaviour is preserved by the continuousquadrature rule under some restriction on the parameters. Asa conclusion we give some examples. This work was supported by CNR and M.U.R.S.T.     

On Symmetric Minimum Norm Updates

The theory of the 1-inverse of a matrix is used to derive allsymmetric updates which satisfy a matrix equation. This generalsolution is then used to obtain the solution which is minimumin a symmetrically weighted Frobenius norm. The minimum normsolution can be used to derive the minimum norm updates usedin quasi-Newton methods for unconstrained optimization. This research was supported in part by NIH Grant No. AM17593.     

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     

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-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.     

Differential algebraic control theory

Recent developments in differential algebraic control theoryare presented. The paper concentrates on the differential algebraiccontrollability, observability, elimination, realization, invertibility,and equivalence problem. Some other aspects can be found inthe references therein. The mathematical tool is differentialalgebra. This paper is an extended version of one presented at the IMAConference on Control: Modelling, Computation, and Information,UMIST, Manchester, September, 1992.