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.     

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.     

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.     

On the Solution of Some Elliptic Difference Equations

Iterative methods for the solution of some nonlinear ellipticdifference systems, approximating the first boundary value problemare considered. If h > 0 is the network step in the spaceof variables x = (x₁, x₂,..., x_p) and 2m is the order of theoriginal boundary value problem, then the iterative methodsproposed give solution of accuracy with the expenditure ofO(|In | h^–(p+m–)) and O(|In | |In h| h^–p)arithmetic operations in the case of a general region and arectangular parallelepiped respectively. In the case p = 2 theestimate O(|In | h^{–[2+ (m/2)]}) is obtained if the regionis made up of rectangles with sides parallel to the co-ordinateaxes.     

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 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.     

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.     

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 Behaviour of the Coefficients in Stokes's Series for Surface Gravity Waves

It is shown that the asymptotic behaviour of the coefficientsa_n at high order n and at large wave steepness ak is determinedmainly by the limiting form of the wave crest. In a lower rangeof n, a_n, decreases like n^–, corresponding to the Stokes120° corner flow. In an upper range, a_n, decreases exponentiallywith n. The transition occurs when n³ is O(1) where is relatedto the steepness ak of the waves by ² = 2.0[(ak)_max –ak].     

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)].     

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.     

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.     

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.     

Rate of convergence of numerical approximations to homoclinic bifurcation points

For x=f (x, ), x Rⁿ, R, having a hyperbolic or semihyperbolicequilibrium p(), we study the numerical approximation of parametervalues ^* at which there is an orbit homoclinic to p(). We approximate^* by solving a finite-interval boundary value problem on J=[T_–,T₊], T_–<0<T₊, with boundary conditions that sayx(T_–) and x(T₊) are in approximations to appropriate invariantmanifolds of p(). A phase condition is also necessary to makethe solution unique. Using a lemma of Xiao-Biao Lin, we improve,for certain phase conditions, existing estimates on the rateof convergence of the computed homoclinic bifurcation parametervalue , to the true value ^*. The estimates we obtain agree withthe rates of convergence observed in numerical experiments.Unfortunately, the phase condition most commonly used in numericalwork is not covered by our results.     

On Positive Multipeak Solutions of a Nonlinear Elliptic Problem

In this paper we continue our investigation in [5, 7, 8] onmultipeak solutions to the problem –²u+u=Q(x)|u|^q–2u, xR^N, uH¹(R^N) (1.1) where = ^N_i=1²/x²_i is the Laplace operator in R^N, 2 < q < for N = 1, 2, 2 < q < 2N/(N–2) for N3, and Q(x)is a bounded positive continuous function on R^N satisfying thefollowing conditions. (Q₁) Q has a strict local minimum at some point x₀R^N, that is,for some > 0 Q(x)>Q(x₀) for all 0 < |x–x₀| < . (Q₂) There are constants C, > 0 such that |Q(x)–Q(y)|C|x–y| for all |x–x₀| , |y–y₀| . Our aim here is to show that corresponding to each strict localminimum point x₀ of Q(x) in R^N, and for each positive integerk, (1.1) has a positive solution with k-peaks concentratingnear x₀, provided is sufficiently small, that is, a solutionwith k-maximum points converging to x₀, while vanishing as 0 everywhere else in R^N.     

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 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.     

The Set of {omega}-Limit Points of A Map of the Circle is Closed

Let f be a continuous self-map of the unit circle, S¹. The -limitpoints (x) of a point x are the set of all limit points of thesequence of iterates of f acting on x. We shall show that theset of all -limit points _xS¹(x) a closed set in S¹.     

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.