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.     

Asymptotic Convergences in a One-Phase Continuous-Casting Stefan Problem with High Extraction Velocity

Asymptotic convergence for variational solutions and for thefree boundaries in a one-phase continuous-casting problem ofthe Stefan type are derived when the Péclet number ß+. This corresponds, in particular, to a model with high extractionvelocity, where an ultraparabolic problem is obtained from aparabolic one and, in the stabilized case (i.e. when t +), aparabolic problem results from an elliptic one. The asymptoticstabilization in time as t + is also discussed in the limitß = +.     

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.     

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)].     

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.     

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.     

Numerical Evaluation of Fourier Integrals

A method is developed for evaluating Fourier integrals of theform A() = ¹_–1f(x) e^fax dx, 0. The method consists of expanding the function f in a seriesof Chebyshev polynomials and expressing the integral A() asa series of the Bessel functionsJ_r+(), r= 0, 1, 2,.... A partialsum A_N() of the series provides an approximant to A(). The principalfeature of the method is that one set of N+1 evaluations off(x) suffices for the calculation of A_N() for all , and alsothe truncation error A()–A_N() is essentially independentof . Numerical tests show that the method is accurate, economicaland reliable. An application to the inversion of Fourier andLaplace transforms is briefly described.     

Total Flux Estimates for a Finite-Element Approximation of Parabolic Equations

An initial-boundary-value problem for a parabolic equation ina domain x (0, T) with prescribed Dirichlet data on is approximatedusing a continuous-time Galerkin finite-element scheme. It isshown that the total flux across ₁= can be approximated withan error of O(h^k) when is a curved domain in Rⁿ (n = 2 or 3)and isoparametric elements having approximation power h^k inthe L² norm are used.     

An analogue of Berry's conjecture for the phase in fractal obstacle scattering

In this paper, the authors consider the high-frequency asymptoticsof the phase s() of acoustic waves scattered by an obstacleRⁿ with fractal boundary. Under certain conditions, it is provedthat if is –Minkowski measurable with –Minkowskimeasure µ then there exists a positive constant C_n, dependingonlyon n and such that where     

Injection of Ideal Fluid from a Slot into a Stream: Two Free Boundaries

A fluid is injected from a slot into a stream of another fluid.In a simple model this leads to a two-phase two-free-boundaryproblem with the jump relation |u^–|² – |u⁺|² = on the free boundary {u=0}, and |u| = ¹ on the free boundary{u > – Q}, where u is the stream function and Q isthe flux of the injected fluid. Using the variational theoryof Alt, Caffarelli & Friedman, we prove existence of (,₁, u) such that there is a smooth fit for both free boundaries.     

Asymptotic Solution of Hyperbolic Systems with Constant Coefficients

An integral representation of the exact solution of the initialvalue problem for the hyperbolic equation of the form is derived. Here A^o, A^v, B, and Care constant m x m matrices, u(t, X; ) is an m-component columnvector, and is a positive parameter. Various conditions areimposed on the coefficient matrices that permit the applicationof the method of stationary phase in several variables to theintegral representation of the exact solution. The leading termof the asymptotic expansion as of the exact solution is obtainedfor several types of initial data and source functions whichdepend on the parameter .     

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

OUR attention has been drawn to the fact that the criticalitycondition ^* = 1 of Adler & Herbert (1985), for well-stirredreactive systems, has been derived previously (Gray, 1975).It arises from an examination of trajectories in the temperaturereactant phase plane when a tube stability argument is employed.Using the criterion ^* = 1, values of the critical Semenov numberhave also been obtained numerically (Gray & Jones, 1981). Our work on criticality for systems with reactant consumptioncame about by trying to reconcile the inflection criterion ofBoddington et al. (1983), for finite B, with the correspondingmaximum criterion in the limit B . Our contribution was to showthat the Semenov number versus maximum temperature ^* curvehas a bifurcation at ^* = 1 for all B. Both Gray's work and ourown are attempting to resolve the same problem; the approachesare, however, quite distinct and complement each other     

Turning Point Problems and Resonance

Consider the boundary value problem: ²yⁿ + (xp(x) + ²f(x, ))y'+ g(x, )y = 0, y(a) = A, y(b) = B, where a < 0 < b, p(x)< p(x) < 0, and p, f, and g are analytic. We investigatethe solution of this problem for small positive values of theparameter . If-g(0, 0)/p(0) c where c N = {0, 1, 2, 3,...},then so-called resonance does not occur, and y = o(ⁿ) on closedsubintervals of (a, b), for any n N, with expected boundarylayer behaviour at the end-points. If -g(0, 0)/p(0) = c, c N, then further transformations of dependent and independentvariables may still expose resonance or non-resonance. The setof necessary conditions that is developed is compared to otherauthors' criteria, most notably, Olver's sufficiency condition,and the necessary conditions of Cook & Eckhaus, Lakin, andMatkowsky. Finally, it is proved that these conditions are necessaryfor resonance.     

On the behaviour of blow-up interfaces for an inhomogeneous filtration equation

We study the asymptotic behaviour of blow-up interfaces of thesolutions to the one-dimensional nonlinear filtration equationin inhomogeneous media where m>1 isa constant and (x) = |x|^– (for |x| 1, with > 2) isa bounded, positive, smooth, and symmetric function. The initialdata are assumed to be smooth, bounded, compactly supported,symmetric, and monotone. It is known that due to the fast decayof the density (x) as |x| the support of the solution increasesunboundedly in a finite time T. We prove that as tT^– theinterface behaves like O((T – t)^–b), where the exponentb > 0 (which depends on m and only) is given by a uniqueself-similar solution of the second kind satisfying the equation|x|^– u_t = (u^m)_xx. The corresponding rescaled profilesalso converge. We establish the stability of the self-similarsolution of the second kind for the exponential density (x)=e^–|x|for |x| 1. We give a formal asymptotic analysis of the blow-upbehaviour for the non-self-similar density (x) = e^–|x|2.Several exact self-similar solutions and their correspondingasymptotics are constructed.     

A Comparison Between Chebyshev and Ultraspherical Expansions

This paper examines the norms of the projections from C[–1,1] to P_M[–1, 1] which are obtained by truncating Chebyshevand ultraspherical expansions after n terms. The norms are evaluatednumerically for n = 1, 2, 3 , ... , 10 and –0.1 5 wherethe ultraspherical weight function is (1 – x²)^x–.In particular the data show that the norm of the Chebyshev projectionis not the smallest in this range for 1 n 10.     

The Structure of the Solution Set for Periodic Oscillations in a Suspension Bridge Model

Numerical results are reported for the computation of periodicsolution paths for a suspension bridge model represented bythe equation. u_n + EIu_xxxx + u_t, + Ku⁺ = W(^x) + sin x sin µt. with hinged-end boundary conditions, as the forcing amplitude and frequency µ are varied. The term Ku⁺ models the factthat there is restoring force due to the cables only when theyare being stretched. It is found that an S-shaped curve is obtainedwhen the displacement amplitudes are plotted against the forcingamplitudes for some frequency regimes. As a two-parameter problem,it appears that the solution set resembles a cusp-like surfacewith the singular point near linear resonance. While the effectof strengthening the cable (i.e. increasing K) will enhancethe occurrence of the multiple solutions, the effect of increasingthe damping coefficient gives the opposite effect.     

Supercooling and Superheating Effects in Phase Transitions

In the one-dimensional Stefan problem, the standard equilibriumcondition ; = 0 at the free boundary x = s(t) is here replacedby the kinetic law s'(t) = ß((s(t), t)), where ß:R R is continuous and increasing and ß(0) = 0. Thisrepresents supercooling and superheating effects. The standardStefan problem is then obtained in the limit as ß'(0) + A similar condition is considered for a radially symmetric system,taking also account of the surface tension effect. A kineticcondition is introduced also for phase transitions in binaryalloys, represented by means of the system of the Fourier'sand Fick's laws. In the case of several space dimensions, denoting by [0, 1]the concentration of the more energetic phase, the followinglaw is considered this is also extendedto binary systems. For all of the previous models of phase transitions, existenceresults are proved for the variational problems obtained bycoupling the free boundary condition with the energy conservationequation (and with the mass diffusion equation, for alloys).For heterogeneous systems, also a different model based on "non-equilibriumthermodynamics" is considered. This paper reviews the results of Visintin [IMA J. appl. Math.(1985) 34, 225–245] and announces those of Visintin (1985,to appear in Q appl. Math, and in Ann. Mat. pura appl.).     

Sharp Upper Bound to the First Nonzero Neumann Eigenvalue for Bounded Domains in Spaces of Constant Curvature

The main result of this paper is that for a domain containedin a hemisphere of the n-dimensional sphere Sⁿ the first nonzeroNeumann eigenvalue µ₁() is less than or equal to the firstnonzero Neumann eigenvalue µ₁(D) where D is a geodesicball in Sⁿ of the same measure as . Equality occurs if and onlyif is isometric to D. This result generalizes old results ofSzegö and Weinberger which gave the corresponding upperbound for µ₁() in the Euclidean case, and a result ofChavel for domains in Sⁿ which restricted to lie in a geodesicball of radius when n = 2and to even smaller geodesic balls for larger n. The techniquesused are analogous to those for our recent proof of the Payne-Pólya-Weinbergerconjecture: rearrangement inequalities and properties of specialfunctions are the key elements. The general approach is a directextension of Weinberger's for domains in Rⁿ.     

Superharmonic Extension from the Boundary of a Domain

A sharp geometric condition is given for a bounded domain tohave the property that for each superharmonic function u ona neighbourhood of , there is a superharmonic function on Rⁿsuch that = u on .     

Experimental Observation of the Large-Amplitude Solutions of Duffing's and Related Equations

Experiments with a nonlinear electronic model show that certainsimple features of the solutions of where f(u) is an odd monotonic function of u for example u³,repeat in a regular pattern as either is decreased or U isincreased. For fixed U, the position of these features is periodicin 1/ and, when f(u) has the form u|u|^k–1 a quantitativerelation between the period in 1/ and U can be found. The occurrenceof large-amplitude chaotic solutions is found to depend notonly on the nonlinearity of f(u) for large U but also on itsbehaviour near u = 0. For the Duffing equation, which can bereduced to the range of parameters accessible to experiment is 0<1 and0<F5000.