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.     

A Mathematical Theory of Full-wave Rectification

The results of a systematic asymptotic analysis of two basicfull-wave rectifier circuits are presented in this paper. Thephysical circuits are modelled by two systems of nonlinear differentialequations which are both singularly perturbed in the limit RC.Here is the angular frequency of the source, R is the resistanceof the load, and C is the capacitance of the circuit. The methodof multiple scales is used to construct the asymptotic behaviourof the solutions as RC. The results contain transient, ripple,and steady-state information. The latter two components simplifyand yield the standard results when the diodes' forward impedanceis small. The analysis of more complicated full-wave rectifiersfollows from the procedure described.     

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)].     

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.     

Asymptotic Expansion of a Double Integral with a Curve of Stationary Points

An asymptotic expansion is constructed for the double integral as +, where f(x, y) hasa curve of stationary points in D. The first two coefficientsof the expansion are explicitly calculated.     

Contractions et Hyperdistributions a Spectre de Carleson

Let =(_n)_n1 be a log concave sequence such that lim inf_n+n/n^c>0for some c>0 and ((log _n)/n)_n1 is nonincreasing for some<1/2. We show that, if T is a contraction on the Hilbertspace with spectrum a Carleson set, and if ||T^–n||=O(_n)as n tends to + with _n11/(n log _n)=+, then T is unitary. Onthe other hand, if _n11/(n log _n)<+, then there exists a (non-unitary)contraction T on the Hilbert space such that the spectrum ofT is a Carleson set, ||T^–n||=O(_n) as n tends to +, andlim sup_n+||T^–n||=+.     

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 .     

Asymptotics of the m-Coefficient for Eigenvalue Problems with Eigenparameter in the Boundary Conditions

For Sturm-Liouville problems on [a, ) with a regular -dependentboundary condition at a, and the limit point case at , a techniqueof W. N. Everitt [1] is employed to obtain asymptotic formulaefor the associated m()-functions on rays and lines in the complex-plane. The method relies on asymptotic formulae for solutionsof the initial value problem for –u'+qu = u, as || ,which the author has given in [4]. For the case of the regularleft endpoint, the asymptotic formulae on vertical lines sufficeto provide a direct proof of the formula for the total variationof the associated spectral function, a question which the authorhad raised in [3; Remark 5.2].     

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.     

The large deformation of nonlinearly elastic rings in a two-dimensional compressible flow

The nonlinear nonlocal system of the equilibrium equations ofan elastic ring under the action of an external two-dimensionaluniformly subsonic potential barotropic steady-state gas flowis considered. The configurations of the elastic ring are identifiedby a pair of functions (, ). The simple curve represents theshape of the ring and the real-valued function identifies theorientation of the material sections of the ring. The pressurefield on the ring depends nonlocally on , and on two parametersU and P which represent the pressure and the velocity at infinity.The system is shown to be equivalent to a fixed-point problem,which is then treated with continuation methods. It is shownthat the solution branch ensuing from certain equilibrium states((₀, ₀), 0, P₀) in the solution-parameter space of ((₀, ₀),0, P₀) either approaches the boundary of the admissible ((,), U,p)'s in a well-defined sense, or is unbounded, or is homotopicallynontrivial in the sense that there exists a continuous map from the branch to a two-dimensional sphere which is not homotopicin the sphere to a constant, while restricted to the branchminus ((₀, ₀), 0, P₀) is homotopic to a constant in the sphere.Furthermore, by fixing the pressure parameter at P₀ and by consideringthe one-parameter problem in ((, ), U), the following holds.Every hyperplane in the solution-parameter space of the ((,), U)'s which contains the equilibrium state ((₀, ₀), 0) anddoes not include a welldetermined one-dimensional subspace intersectsthe solution branch above at a point different from ((₀, ₀),0).     

Numerical studies of the fourth Painlev equation

In this paper the authors investigate numerically solutionsof a special case of the fourth Painlev equation given by with a parameter, satisfying theboundary condition Equation (1a) arises asa symmetry reduction of the derivative nonlinear Schrdinger(DNLS) equation, which is a completely integrable soliton equationsolvable by inverse scattering techniques. Previous resultsconcerned with solutions of (1) are largely restricted to thecase when is an integer and very little has been proved when is a noninteger. Here a numerical approach to describing solutionsof (1) for noninteger is adopted, and information is obtainedcharacterizing connection formulae which describe how the asymptoticbehaviours of solutions as + relate to those as –.     

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     

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.     

Great time delay in a system with material cycling and delayed biomass growth

In this paper, we consider the effects of time delay on thelinear stability of the positive equilibrium of a reaction–diffusionmodel with time delayed nutrient recycling. We take the twodelay kernels as f(s) = (s), g(s) = p(0) + (1 – p)(r),s (0, +), r [0, +), 0 < p < 1, where is the Dirac deltafunction. When both N₁ and N₂ are space-independent, we canobserve that the increasing time delay destabilizes the positiveequilibrium, and stability-switches from stability to instabilityto stability occur. That is, for some special time delay kernels,the equilibrium is stable for small time delay, bifurcates towardsstable oscillations when the time delay is increased, then regainsstability for larger values. When both N₁ and N₂ are space-dependent,we are concerned with the effects of the delay kernels as wellas diffusion on the dynamics of the system.     

The Mesa Problem: Diffusion Patterns for ut = {nabla} {middle dot} (um{nabla}u) as m -> +{infty}

In this paper we consider the limit m+ of solutions of the porous-mediumequation u_t = · (u^mu) (xR^N), with N > 1. We conjecturethat, for initial data with a unique maximum, the evolutionis characterized by the onset of a ‘mesa’ region,in which the solution is nearly spatially independent, surroundedby a region in which u is nearly equal to its initial value.The transition between these regions occurs near a surface whichis identified with the free boundary in a certain Stefan problemwhich can be studied using variational inequalities. Moreover,singular-perturbation theory can be used to describe the structureof the transition region.     

Multivariate R-O Varying Measures Part I: Uniform Bounds

A finite Borel measure µ on R^d is called R-O varying withindex F if there exist a GL(R^d)-valued function f varying regularlywith index (–F), an increasing function k: (0, ) (0,) with k(t) and k(t + 1)/k(t) c 1 as t , and a -finitemeasure on R^d\0 such that R-O varying measures generalize regularly varying measures introducedby Meerschaert (see M. M. Meerschaert, ‘Regular variationin R^k’, Proc. Amer. Math. Soc. 102 (1988) 341–348)and have numerous applications in limit theorems for probabilitymeasures. For an R-O varying measure µ and – < let denote the tail- andtruncated moment functions of µ in the direction || =1. The purpose of this paper is to show that R-O variation ofa measure implies sharp bounds on the growth rate of the tail-and truncated moment functions depending on the real parts ofthe eigenvalues of the index F along a compact set of directions.Furthermore, bounds on the ratio of these functions for certainvalues of a and b are obtained. 1991 Mathematics Subject Classification:60B10, 28C15.     

Stability and Asymptotic Stability of Functional-Differential Equations

We investigate asymptotic behaviour of solutions of the functional-differentialequation where f and g arelocally Lipschitz functions, C is a continuous matrix and thesmooth lag function obeys 0 (t) t for t 0. We transformthe equation into a delay equation with an infinity of delaysand use a theorem of Söderlind to derive sufficient conditionsfor stability and for asymptotic stability in the case lim_t (t) = . The situation is qualitatively different when lim_t (t) = * < and we outline stability conditions for thatcase by employing direct techniques.     

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     

Blow up of solutions of a generalized Boussinesq equation

Consider the Cauchy problem u_tt = (f(u))_xx + u_xxtt x R, t 0, u(x,0) = u₀(x), u_t(x,0) = u₁(x),7rcub; where f : R R C, f(0) = ). After treatment of the local existenceproblem, we show the blow up of the solution of the equation(1) under the folowing assumptions. Let > 0 be real, such that 2(l + 2)F(u) uf(u), (v₀, Pv₀)_l² + _- F(u₀)dx < 0 where P = 1 - d²/dx², F'(s), and v₀ is given by u₁(x,0) = (v₀(x))_x. Then we focus on various perturbations of the question. We alsostudy the vectorial case in the same way, and finally we giveexamples.