On Closure Conditions for the Moment Equations of a Bipolar Kinetic Model

We consider a bipolar kinetic model for charged media. In certain scalings the Debye length or the relaxation time are small. In addition different time scales are considered. These can be used in order to close the corresponding moment equations and leads to a (closed) set of macroscopic equations. We show three different scalings and obtain three completely different sets of macroscopic equations.     

Pattern formation in a 2D simple chemical system with general orders of autocatalysis and decay

In this paper, we investigate pattern formation in a coupledsystem of reaction–diffusion equations in two spatialdimensions. These equations arise as a model of isothermal chemicalautocatalysis with termination in which the orders of autocatalysisand termination, m and n, respectively, are such that 1 <n < m. We build on the preliminary work by Leach & Wei(2003, Physica D, 180, 185–209) for this coupled systemin one spatial dimension, by presenting rigorous stability analysisand detailed numerical simulations for the coupled system intwo spatial dimensions. We demonstrate that spotty patternsare observed over a wide parameter range.     

Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium

Several choices of scaling are investigated for a coupled system of parabolic partial differential equations in a two‐phase medium at the microscopic scale. This system may be regarded as modelling a reaction–diffusion problem, the Stokes problem of single‐phase flow of a slightly compressible fluid or as a heat conduction problem (with or without interfacial resistance), for example. It is shown that, starting with the same problem on the microscopic scale, different choices of scaling of the diffusion coefficients (resp. permeability or conductivity) and the interfacial‐exchange coefficient lead to different types of macroscopic systems of equations. The characterization of the limit problems in terms of the scaling parameters constitutes a modelling tool because it allows to determine the right type of limit problem. New macroscopic models, not previously dealt with, arise and, for some scalings, classical macroscopic models are recovered. Using the method of two‐scale convergence, a unified approach yielding rigorous proofs is given covering a very broad class of different scalings. Copyright © 2008 John Wiley & Sons, Ltd.     

Reaction-diffusion models from the Fokker-Planck formulation ofchemical processes

A Fokker-Planck-type model is proposed to describe the kineticsof certain chemical reactions. In particular, the competitionbetween transport and reaction processes is analysed. The studyis carried out considering various scalings of interaction,measured by the exponent of a small parameter related to themean free path. In the most significant case of competitionbetween both effects, the lowest-order density in the asymptoticexpansion obeys a reaction-diffusion equation. Such an equationwas earlier considered as the starting point in the study ofthese processes by other authors (e.g. by Schlgl). For otherinteraction scalings, the prevalence of chemical processes impliesthat the lowest-order density is determined by the (algebraic)equations of chemical equilibrium. In contrast, when transportprevails, the reaction terms affect only higher-orderdensities.     

A Two‐Parameter Stabilized Finite Element Method for Incompressible Flows

Based on two‐grid discretizations, a two‐parameter stabilized finite element method for the steady incompressible Navier–Stokes equations at high Reynolds numbers is presented and studied. In this method, a stabilized Navier–Stokes problem is first solved on a coarse grid, and then a correction is calculated on a fine grid by solving a stabilized linear problem. The stabilization term for the nonlinear Navier–Stokes equations on the coarse grid is based on an elliptic projection, which projects higher‐order finite element interpolants of the velocity into a lower‐order finite element interpolation space. For the linear problem on the fine grid, either the same stabilization approach (with a different stabilization parameter) as that for the coarse grid problem or a completely different stabilization approach could be employed. Error bounds for the discrete solutions are estimated. Algorithmic parameter scalings of the method are also derived. The theoretical results show that, with suitable scalings of the algorithmic parameters, this method can yield an optimal convergence rate. Numerical results are provided to verify the theoretical predictions and demonstrate the effectiveness of the proposed method. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 425–444, 2017     

A Fast Direct Method for the Least Squares Solution of Slightly Overdetermined Sets of Linear Equations

Noble (1969) has described a method for the solution of N+Mlinear equations in N unknowns, which is based on an initialpartitioning of the matrix A, and which requires only the solutionof square sets of equations. He assumed rank (A) = N. We describehere an efficient implementation of Noble's method, and showthat it generalizes in a simple way to cover also rank deficientproblems. In the common case that the equation is only slightlyoverdetermined (M << N) the resulting algorithm is muchfaster than the standard methods based on M.G.S. or Householderreduction of A, or on the normal equations, and has a very similaroperation count to the algorithm of Cline (1973). Slightly overdetermined systems arise from Galerkin methodsfor non-Hermitian partial differential equations. In these systems,rank (A) = N and advantage can be taken of the structure ofthe matrix A to yield a least squares solution in (N²) operations.     

A Fast Galerkin Scheme for Linear Integro-differential Equations

We describe an expansion method for the solution of first orderand second order ordinary integro-differential equations, whichis a generalization of the Fast Galerkin scheme for second kindintegral equations (Delves, 1977a; Delves, Abd-Elal & Hendry,1979). The method retains the O(N² In N) operation count ofthat scheme, and pays particular attention to the way in whichthe boundary conditions are incorporated, with the aim of retainingalso the stable structure of the Fast Galerkin equations, andits very rapid convergence. An error analysis, and numericalexamples, indicate that these aims are met.     

Non-local boundary value problems of arbitrary order

We give a new unified method of establishing the existence ofmultiple positive solutions for a large number of non-lineardifferential equations of arbitrary order with any allowed numberof non-local boundary conditions (BCs). In particular, we areable to determine the Green's function for these problems withvery little explicit calculation, which shows that studyinga more general version of a problem with appropriate notationcan lead to a simplification in approach. We obtain existenceand non-existence results, some of which are sharp, and givenew results for both non-local and local BCs. We illustratethe theory with a detailed account of a fourth-order problemthat models an elastic beam and also determine optimal valuesof constants that appear in the theory.     

The Distance Perspective of Generalized Biadditive Models: Scalings and Transformations

Recently two articles studied scalings in biplot models, and concluded that these have little impact on the interpretation. In this article again scalings are studied for generalized biadditive models and correspondence analysis, that is, special cases of the general biplot family, but from a different perspective. The generalized biadditive models, but also correspondence analysis, are often used for Gaussian ordination. In Gaussian ordination one takes a distance perspective for the interpretation of the relationship between a row and a column category. It is shown that scalings—but also nonsingular transformations—have a major impact on this interpretation. So, depending on the perspective one takes, the inner product or distance perspective, scalings and transformations do have (distance) or do not have (inner-product) impact on the interpretation. If one is willing to go along with the assumption of the author that diagrams are in practice often interpreted by a distance rule, the findings in this article influence all biplot models.     

Some asymptotic limits of reaction–diffusion systems appearing in reversible chemistry

This paper concerns reaction–diffusion systems consisting of three or four equations, which come out of reversible chemistry. We introduce different scalings for those systems, which make sense in various situations (species with very different concentrations or very different diffusion rates, chemical reactions with very different rates, etc.). We show how recently introduced mathematical tools allow to prove that the formal asymptotics associated to those scalings indeed hold at the rigorous level.     

High-accuracy Tridiagonal Finite Difference Approximations for Non-linear Two-point Boundary Value Problems

We discuss the construction of finite difference approximationsfor the non-linear two-point boundary value problem: y" = f(x,y), y(a)=A, y(b)=B. In the case of linear differential equations,the resulting finite difference schemes lead to tridiagonallinear systems. Approximations of orders higher than four involvederivatives of f. While several approximations of a particularorder are possible, we obtain the "simplest" of these approximationsleading to two high-accuracy methods of orders six and eight.These two methods are described and their convergence is established;numerical results are given to illustrate the order of accuracyachieved.     

Exact controllability of damped coupled Euler-Bernoulli and Timoshenko beam model

^** Email: marianna.shubov{at}euclid.unh.edu The zero controllability problem for the system of two coupledhyperbolic equations which governs the vibrations of the coupledEuler–Bernoulli and Timoshenko beam model is studied inthe paper. The system is considered on a finite interval witha two-parameter family of physically meaningful boundary conditionscontaining damping terms. The controls are introduced as separableforcing terms g_i(x)f_i(t), i = 1, 2, on the right-hand sidesof both equations. The force profile functions g_i(x), i = 1,2, are assumed to be given. To construct the controls f_i(t),i = 1, 2, which bring a given initial state of the system tozero on the specific time interval [0, T], the spectral decompositionmethod has been applied. The approach, used in the present paper,is based on the results obtained in the recent works by theauthor and the collaborators. In these works, the detailed asymptoticand spectral analyses of the non-self-adjoint operators generatingthe dynamics of the coupled beam have been carried out. It hasbeen shown that for each set of the boundary parameters, theaforementioned operator is Riesz spectral, i.e. its generalizedeigenvectors form a Riesz basis in the energy space. Explicitasymptotic formulas for the two-branch spectrum have also beenderived. Based on these spectral results, the control problemhas been reduced to the corresponding moment problem. To solvethis moment problem, the asymptotical representation of thespectrum and the Riesz basis property of the generalized eigenvectorshave been used. The necessary and/or sufficient conditions forthe exact controllability are proven in the paper and the explicitformulas for the control laws are given. The case of the approximatecontrollability is discussed in the paper as well.     

Equilibrium attractivity of Runge-Kutta methods

For dissipative differential equations y' = f (y) it is knownthat contractivity of the exact solution is reproduced by algebraicallystable Runge–Kutta methods. In this paper we investigatewhether a different property of the exact solution also holdsfor Runge–Kutta solutions. This property, called equilibriumattractivity, means that the norm of the righthand side f neverincreases. It is a property dual to algebraic stability sinceneither is sufficient for the other, in general. We derive sufficientalgebraic conditions for Runge–Kutta methods and proveequilibrium attractivity of the high-order algebraically stableRadau-IIA and Lobatto-IIIC methods and the Lobatto-IIIA collocationmethods (which are not algebraically stable). No smoothnessassumptions on f and no stepsize restrictions are required but,except for some simple cases, f has to satisfy certain additionalproperties which are generalizations of the simple one-sidedLipschitz condition using more than two argument points. Thesemultipoint conditions are discussed in detail.     

A Lower Bound for the Cost Functional for Control Problems in Hilbert Space

Linear quadratic problems for evolution equations in Hilbertspace are considered. Given any control u₀, a lower bound forthe ratio of the cost of performance of any other control tothe cost of u₀ is obtained. The significance of this lower boundis discussed and the results applied to some simple examplesof both ordinary and partial differential equations.     

Noise-induced on–off intermittency in mutually coupled semiconductor lasers

Intermittent switches between low-frequency fluctuations and steady-state emission are experimentally observed in two bidirectionally coupled semiconductor lasers subject to common Gaussian noise applied to the laser pump currents. The time series analysis reveals power-law scalings typical for on–off intermittency near its onset, with critical exponents of −1 for the mean turbulent length versus noise intensity and −3/2 for probability distribution of laminar phases versus the laminar length. The same −1 power-law scaling is found by the power spectrum analysis for the signal-to-noise ratio versus the noise intensity.     

A Class of Pencils of Matrices

A linear machine is one in which the time dependent input yis related to the output z by P(D). z = S(D). y where P andS are polynomials in D = d/dt with constant coefficients. Fornumerical computation it is necessary to replace this relationby a set of simultaneous first order differential equationsand this paper shows how to construct such equations by methodswhich extend the results of Gilder (1961). Attention is restrictedto those sets of equations that are of a special form (see (1))which is characterized by the matrix operating on the dependentvariables. This matrix forms a pencil, being linear in D, andthree theorems are given to show how such matrix pencils maybe constructed from the polynomials. The theorems also statethat any matrix pencil with the required properties can be transformedinto the canonical forms given in the theorems by pre- and post-multiplicationby suitable constant non-singular matrices. Thus the variablesof any set of equations having the required properties are linearcombinations of the variables of the equations given by thetheorems. In the paper it is assumed that the degree of P(D)is greater than that of S(D), as otherwise z would be replacedby z₁+Q(D) . y, where Q is the quotient of S(D)/P(D). Also,as the algebriac manipulations are independent of the natureof the polynomials, D is replaced by an indeterminate x andthe coefficients considered to be from an arbitrary field. Fortechnical reasons we rename y and z, y_o and y_n–_m respectively.     

Amplitude synchronization in a system of two coupled semiconductor lasers

We consider a system of ordinary differential equations used to describe the dynamics of two coupled single-mode semiconductor lasers. In particular, we study solutions corresponding to the amplitude synchronization. It is shown that the set of these solutions forms a three-dimensional invariant manifold in the phase space. We study the stability of trajectories on this manifold both in the tangential direction and in the transverse direction. We establish conditions for the existence of globally asymptotically stable solutions of equations on the manifold synchronized with respect to the amplitude. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 426–435, March, 2008.     

The Stability Properties of Singly-implicit General Linear Methods

In this paper we examine the linear stability properties ofsingly-implicit general linear methods. We show numericallythat there exist A-stable methods of up to order 14. Based onvarious stability and implementation considerations we proposea family of methods of orders two to ten to be incorporatedinto a variable order, variable stepsize package suitable forsolving stiff ordinary differential equations.     

Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of orderr is applied to a smooth system of ordinary differential equations.Given any integer m 1, the method may be shown to be of orderr + m as an approximation to a certain modified equation. Ifthe method and the system have a particular qualitative propertythen it is important to determine whether the modified equationsinherit this property. In this article, a technique is introducedfor proving that the modified equations inherit qualitativeproperties from the method and the underlying system. The techniqueuses a straightforward contradiction argument applicable toarbitrary one-step methods and does not rely on the detailedstructure of associated power series expansions. Hence the conclusionsapply, but are not restricted, to the case of Runge-Kutte methods.The new approach unifies and extends results of this type thathave been derived by other means: results are presented forintegral preservation, reversibility, inheritance of fixed points.Hamiltonian problems and volume preservation. The techniquealso applies when the system has an integral that the methodpreserves not exactly, but to order greater than r. Finally,a negative result is obtained by considering a gradient systemand gradient numerical method possessing a global property thatis not shared by the associated modified equations.