Optimal Parameter Choice in the {theta}-Method for Parabolic Equations

The general first-order method, known as the -method, is appliedto the semi-discrete form of a parabolic equation. It is shownthat to every required local accuracy there corresponds a valueof the parameter that is optimal in the sense of allowing thelargest step for which the error remains bounded below . Anasymptotic formula for in terms of is obtained, showing thatthe maximum step-size for the optimal -method is more than twiceas large as that for the Crank-Nicolson method. A numericalexample is given, showing good agreement between theory andpractice.     

The Critical Conditions in a Reactive Slab with Slowly Varying Surface Temperature

Liouville's non-linear partial differential equation is consideredfor an infinite rectangular strip domain with a slowly varyingboundary condition. The equation describes a layer of chemicallyreactive material under conditions where the resistance to surfaceheat transfer is negligible and the ambient temperature variesslowly along the surface. Symmetrical heating by a zero orderexothermic reaction is assumed. If is a small dimensionlesstemperature difference between regions where the surface temperatureis effectively constant, a perturbation series solution in may be determined provided the Frank-Kamenetskii parameter satisfies _c(). It is shown that a plausible value for thecritical parameter is _c() = _c(0) e^–e,where _c(0) = 0.878.The corresponding critical temperature distribution is shownto have a dependence on different from that for subcriticalcases.     

On a Penalty-Perturbation Theory for Plate Problems

A penalty-perturbation method previously proposed by Westbrook(J. Inst. Maths Applics (1974) 14, 79–82) for the solutionsof static bending problems for elastic plates is analysed here.The method replaces the single fourth-order biharmonic equationby a system of three second-order equations which is "singularly"perturbed with respect to a small penalty parameter . The existenceof solutions of the perturbed problem for each > 0 is establishedand the behaviour of these solutions as 0 0 is studied. Inparticular, the results show that while these solutions arecontinuous in at = 0, analyticity in at = 0 is lost exceptin special cases.     

Elliptic Problems Involving Phase Boundaries Satisfying a Curvature Condition

Two theorems related to equilibrium free-boundary problems arepresented. One arises as a time-independent solution to thephase-field equations. The other is the relevant time-independentproblem for the Stefan model, modified for the surface tensioneffect. It also serves as a preliminary result for the phase-fieldformulation. Under appropriate conditions, we prove that, givenan appropriate positive constant and a smooth function u: R;,where is an annular domain in R², there exists a curve suchthat u(x)=—K(x) for all x , where K is the curvature.Using this result, we prove the existence of solutions to O=²+ ?(—³) + 2u that have a transition layer behaviour (from=—1 to =+1) for small and make the transition on thecurve . This proves there exist solutions to the phase fieldmodel that satisfy a Gibbs-Thompson relation.     

The onset of thermal runaway in partially insulated or cooled reactors

The conditions for the onset of thermal runaway in partiallyinsulated or cooled reactors are investigated. The temperaturein the reactor is taken to satisfy a nonlinear elliptic equationand the reaction is modelled by an Arrhenius heat generationterm with finite activation energy. To determine the onset ofthermal runaway, the method of matched asymptotic expansionsis used to derive expressions for the critical Frank-Kamenetskiiparameter _c() for reactors containing either a small coolingrod or having a small cooling patch on their boundary. The theoryused to determine _c() is an extension of the results of Wardand Keller (1991). These previous results of Ward and Kellerare also extended to the case of finite activation energiesby using a numerical scheme to evaluate the coefficients inthe asymptotic results for _c(). In some special cases, the asymptoticexpansions for _c() are compared with numerical results for _c(),and clear agreement is found.     

The Magnetogasdynamic Boundary Layer on a Thermally Insulated Flat Plate

This paper examines the effect of compressibility on the flowin the boundary layer on a semi-infinite, thermally insulatedflat plate placed at zero incidence to a uniform stream of electricallyconducting gas, with an aligned magnetic field at large distancesfrom the plate. The present discussion is limited to small values of the conductivityparameter = 4µv, and the Prandtl number is taken to beunity. The latter assumption permits a simplification of theanalysis, and the former allows the dependence of the flow onthe parameters ß = µH²/4U² and M = U/cto beadequately illustrated without excessive computation. A seriessolution valid for small values of the conductivity parameterand for Mach numbers not too large is derived. Values of ß = 0.3 and 0.5, = 0.01 and 0.1 are consideredand for those values the skin friction decreases with increasingMach number, similar to the case when ß = 0. The analysissuggests that for larger values of ß the skin frictionmight even increase with the Mach number initially. This iscertainly the case with the tangential component of the magneticfield, which for ß = 0.5 exhibits a maximum at approximatelyunit Mach number. The reason for this behaviour lies in thefact that, in view of the temperature changes taking place inthe flow, the electrical conductivity and thereby the localvalue of can change by more than an order of magnitude. Thishas the effect of giving results which are akin to those forarbitrary large in incompressible flow even though the valueof based on the main stream gas properties remains low.     

BOUNDARY CONCENTRATION IN RADIAL SOLUTIONS TO A SYSTEM OF SEMILINEAR ELLIPTIC EQUATIONS

We study concentration phenomena for the system in the unit ball B₁ of ³ with Dirichlet boundaryconditions. Here , , > 0 and p > 1. We prove the existenceof positive radial solutions (, ) such that concentrates ata distance (/2)|log | away from the boundary B₁ as the parameter tends to 0. The approach is based on a combination of Lyapunov–Schmidtreduction procedure together with a variational method.     

The asymptotic solution of a family of boundary value problems involving exponentially small terms

In this paper, the authors consider the family of boundary valueproblems in the limit || 0. This problem has recently appeared as a modelfor magnetic field annihilation but the equation itself, withvariously different boundary conditions, has an extensive literature.Using a combination of asymptotic and numerical analyses, thepaper gives a comprehensive treatment of the small || problem,paying particular attention to the question of duality of solutions.For |0, this is intimately connected with the occurrence ofexponentially small terms in the asymptotic solution. When =0(1) these termsz are forced by the boundary layer at y = 1,and the techniques used to deal with this case are well knownfrom previous work on the equation. However, for small ||, acase which reveals the true nature of the duality propertiesof the asymptotic solution, these well-known methods are notapplicable, and a new approach via the initial value formulationof (*) is used. The approach is based on a scaling method whichenables the problem to be reduced to a one-parameter familyof problems of initial value type. This considerably simplifiesthe search for and construction of numerical solutions thatare used to support the asymptotic analysis. For 0, it is shownthat convergence to the =0 solution only takes place for a restrictedrange of values of a and that, for sufficiently small || thereis only one solution to the given boundary value problem.     

Initial-layer theory and model equations of Volterra type

^** Email: amb16{at}nyu.edu It is demonstrated here that there exist initial layers to singularlyperturbed Volterra equations whose thicknesses are not of theorder of magnitude of O(), 0. It is also shown that the initial-layertheory is extremely useful because it allows one to constructthe approximate solution to an equation, which is almost identicalto the exact solution.     

On the Slow Rotation of a Sphere about a Diameter Parallel to a Nearby Plane Wall

A method using a matched asymptotic expansions technique ispresented for obtaining the Stokes flow solution for a rigidspherical body of radius a rotating uniformly about a diameterparallel to a fixed plane wall when the minimum clearance ais very much smaller than a. An inner solution is constructedwhich is valid for the region in the neighbourhood of the nearestpoints of the sphere and the wall where the flow is stronglysheared with large velocity gradients and pressure; in thisregion the leading term of the asymptotic expansion of the solutionsatisfies the equations of lubrication theory. A matching outersolution is constructed which is valid in the remainder of thefluid where the flow is weakly sheared and it is possible toassume = 0. The forces and couples acting on the sphere andthe wall are shown to be of the form (₀+₁) log +ß₀+0(,where ₀, ₁ and ß₀ are constants which have been determinedexplicitly. By use of these results it is shown that the problemwhen the sphere rolls on the wall is not well posed.     

Approximation of Ground State Eigenvalues and Eigenfunctions of Dirichlet Laplacians

Let be a bounded connected open set in R^N, N 2, and let –0be the Dirichlet Laplacian defined in L²(). Let > 0 be thesmallest eigenvalue of –, and let > 0 be its correspondingeigenfunction, normalized by ||||₂ = 1. For sufficiently small>0 we let R() be a connected open subset of satisfying Let – 0 be the Dirichlet Laplacian on R(), and let >0and >0 be its ground state eigenvalue and ground state eigenfunction,respectively, normalized by ||||₂=1. For functions f definedon , we let Sf denote the restriction of f to R(). For functionsg defined on R(), we let Tg be the extension of g to satisfying 1991 Mathematics SubjectClassification 47F05.     

Two-timing and One-dimensional Random Wave Propagation

This paper discusses the propagation of waves in a one-dimensionalrandom medium. We solve two first order coupled stochastic non-lineardifferential equations of the Riccati type governing the boundaryvalue Green's function associated with the refracting mediumby a stochastic two-timing perturbation approach. Expressionsvalid for propagation path length O(^-2) are obtained for themean-power reflected and transmitted for the case of small refractiveindex variations. It is demonstrated that there exists a layerof thickness O(^-1), roughly the region for which the Bornapproximation is valid, which accounts for the major contributionto the energy scattered.     

A Criticality Criterion for Thermal Explosions in Well-stirred Systems with Reactant Consumption

The autonomous differential equations for the temperature andreactant consumption in a first-order well-stirred exothermicreaction are considered. An examination of the phase-plane solutionsallows the qualitative behaviour of the Semenov number as afunction of maximum temperature rise ^* to be established. Inthe limit of infinite adiabatic temperature rise (B) and zeroactivation energy parameter ( = 0), the relationship between and stationary temperature _s is known to be e_¹ = _s. Criticalityarises at the maximum of (_s) and leads to the critical Semenovvalues (_s)_cr = 1, _cr = e^–1. For sufficiently large B,it is shown that the (^*) curve has a bifurcation at ^* = 1, withthe upper branch monotonically increasing and the lower branchmonotonically decreasing for ^* > 1. In the limit B thesebecome respectively the straight line = e^–1, _s 1 andthe unstable branch of = _se^–1, _s 1 and the unstablebranch of = _s e^–. Criticality for finite B is definedas occurring at the bifurcation, namely ^*_cr = 1, with _cr(B)the value of at this point. Values of these Semenoy numbersare obtainable from the numerical calculations of Boddingtonet al. [Proc. R. Soc. Lond. (1983), 390, 13–30]. The newcriterion is applied to an approximate phase-plane solution.The corresponding critical parameter is found to be _cr = e^–1[1+B^–(2–e^–1)+O(B^–1)].     

Metric Entropy of Convex Hulls in Hilbert Spaces

We show in this note the following statement which is an improvementover a result of R. M. Dudley and which is also of independentinterest. Let X be a set of a Hilbert space with the propertythat there are constants , >0, and for each n N, the setX can be covered by at most n balls of radius n^–. Then,for each n N, the convex hull of X can be covered by 2ⁿ ballsof radius . The estimate is best possible for all n N, apart from the value c=c(, , X).In other words, let N(, X), >0, be the minimal number ofballs of radius covering the set X. Then the above result isequivalent to saying that if N(, X)=O(^–1/) as 0, thenfor the convex hull conv (X) of X, N(, conv (X)) =O(exp(^–2/(12))). Moreover, we give an interplay between several coveringparameters based on coverings by balls (entropy numbers) andcoverings by cylindrical sets (Kolmogorov numbers). 1991 MathematicsSubject Classification 41A46.     

On hearing the shape of a bounded domain with Robin boundary conditions

The asymptotic expansions of the trace of the heat kernel (t)= [sum ]_j=1 exp(-t_j) for small positive t, where {_j} _j=1 arethe eigenvalues of the negative Laplacian -_n = -[sum ]ⁿ_k=1 (/x^k)²in Rⁿ (n = 2 or 3), are studied for a general multiply connectedbounded domain which is surrounded by simply connected boundeddomains _i with smooth boundaries _i (i = 1,...,m), where smoothfunctions Y_i (i = 1,...,m) are assuming the Robin boundary conditions(n_i + Y_i) = 0 on _i. Here /n_i denote differentiations along theinward-pointing normals to _i (i = 1,...,m). Some applicationsof an ideal gas enclosed in the multiply connected bounded containerwith Neumann or Robin boundary conditions are given.     

Spectra of graph neighborhoods and scattering

Let (G)_>0 be a family of ‘-thin’ Riemannian manifoldsmodeled on a finite metric graph G, for example, the -neighborhoodof an embedding of G in some Euclidean space with straight edges.We study the asymptotic behavior of the spectrum of the Laplace–Beltramioperator on G, as 0, for various boundary conditions. We obtaincomplete asymptotic expansions for the kth eigenvalue and theeigenfunctions, uniformly for kC^–1, in terms of scatteringdata on a non-compact limit space. We then use this to determinethe quantum graph which is to be regarded as the limit object,in a spectral sense, of the family (G). Our method is a directconstruction of approximate eigenfunctions from the scatteringand graph data, and the use of a priori estimates to show thatall eigenfunctions are obtained in this way.     

The Model Plasma Problem: Existence of a Solution Branch

We are interested in the model plasma problem –u = u⁺in ,u = –d on , _au⁺ dx=j where is a bounded domain in with boundary ; here, j isa given positive number, the function u and the positive number are the unknowns of the problem, and d is a real parameter.Using a variant of the implicit function theorem, we can provethe existence of a global solution branch parametrized by d.The method has the advantage that it can be used for analysingthe approximation of the above problem by a finite-element method.     

Sparse Systems of Functions Closed on Large Sets in RN

Let E(Z) = {e^inx}_nZ denote the trigonometrical exponential system.It is well known that E(Z) forms an orthogonal basis in thespace L²(0, 2). In 1964, H. Landau discovered that the trigonometricalsystem has the following property: certain small perturbationsof E(Z) yield exponential systems which are complete in L² onany finite union of 2-periodic translations of any interval(, 2–), 0 < < .     

On the Non-linear Evolution of Gortler Vortices in Non-parallel Boundary Layers

The non-linear development of finite amplitude Görtlervortices in a non-parallel boundary layer on a curved wall isinvestigated using perturbation methods based on the smallnessof e, the non-dimensional wavelength of the vortices. The crucialstage in the growth or decay of the vortices takes place inan interior viscous layer of thickness O(²) and length O().In this region the downstream velocity component of the perturbationcontains a mean flow correction of the same order of magnitudeas the fundamental which is driving it. Moreover, these functionssatisfy a pair of coupled non-linear partial differential equationswhich must be solved subject to some initial conditions imposedat a given downstream location. It is found that, dependingon whether the boundary layer is more or less unstable downstreamof this location, the initial disturbance either grows intoa finite amplitude Görtler vortex or decays to zero. Forthe Blasius boundary layer on a concave wall it is found thatGörtler vortices can only develop if the rate of increaseof curvature of the wall is sufficiently large. In this casethe finite amplitude solution which develops initially in an-neighbourhood of the position where the disturbance is introducedchanges its structure further downstream. This structure isinvestigated at a distance O() (with 0< <1) downstreamof the above -neighbourhood. In this régime the downstreamfundamental velocity component has an elliptical profile overmost of the flow field. However, in two thin boundary layerslocated symmetrically either side of the centre of the viscouslayer the fundamental velocity component decays exponentiallyto zero. The locations of these layers are determined by aneigenvalue problem associated with the one-dimensional diffusionequation. The mean flow correction persists both sides of theboundary layer and ultimately decays exponentially to zero. This large amplitude motion is not sensitive to the imposedinitial conditions and appears to be the ultimate state of anyinitial disturbance. However, in the initial stages of the growthof the vortex, some surprising flows are possible. For example,it is possible to set up a vortex flow similar to that observedby Wortmann (1969) which consists of a sequence of cells inclinedat an angle to the vertical.