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.     

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.     

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.     

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).     

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.     

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     

An Asymptotic Analysis of the Delayed Logistic Equation when the Delay is Large

We show how to construct an asymptotic solution to the delayedlogistic equation = y(1–y₁), corresponding to the asymptoticlimit . The results of the analysis are compared with a numericalcomputation, and found to be comparatively accurate for >2. Since the approach adopted is novel, we comment on some featureswhich may be relevant in other problems.     

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)].     

Boundary integral methods for singularly perturbed boundary value problems

In this paper we consider boundary integral methods appliedto boundary value problems for the positive definite Helmholtz-typeproblem –U + ²U = 0 in a bounded or unbounded domain,with the parameter real and possibly large. Applications arisein the implementation of space–time boundary integralmethods for the heat equation, where is proportional to 1/(t),and t is the time step. The corresponding layer potentials arisingfrom this problem depend nonlinearly on the parameter and havekernels which become highly peaked as , causing standard discretizationschemes to fail. We propose a new collocation method with arobust convergence rate as . Numerical experiments on a modelproblem verify the theoretical results.     

A rivulet of perfectly wetting fluid draining steadily down a slowly varying substrate

We use the lubrication approximation to investigate the steadylocally unidirectional gravity-driven draining of a thin rivuletof a perfectly wetting Newtonian fluid with prescribed volumeflux down both a locally planar and a locally non-planar slowlyvarying substrate inclined at an angle to the horizontal. Weinterpret our results as describing a slowly varying rivuletdraining in the azimuthal direction some or all of the way fromthe top ( = 0) to the bottom ( = ) of a large horizontal circularcylinder with a non-uniform transverse profile. In particular,we show that the behaviour of a rivulet of perfectly wettingfluid is qualitatively different from that of a rivulet of anon-perfectly wetting fluid. In the case of a locally planar substrate we find that thereare no rivulets possible in 0 /2 (i.e. there are no sessilerivulets or rivulets on a vertical substrate), but that thereare infinitely many pendent rivulets running continuously from = /2 (where they become infinitely wide and vanishingly thin)to = (where they become infinitely deep with finite semi-width). In the case of a locally non-planar substrate with a power-lawtransverse profile with exponent p > 0 we find, rather unexpectedly,that the behaviour of the possible rivulets is qualitativelydifferent in the cases p < 2, p = 2 and p > 2 as wellas in the cases of locally concave and locally convex substrates.In the case of a locally concave substrate there is always asolution near the top of the cylinder representing a rivuletthat becomes infinitely wide and deep, whereas in the case ofa locally convex substrate there is always a solution near thebottom of the cylinder representing a rivulet that becomes infinitelydeep with finite semi-width. In both cases the extent of therivulet around the cylinder and its qualitative behaviour dependon the value of p. In the special case p = 2 the solution representsa rivulet on a locally parabolic substrate that becomes infinitelywide and vanishingly thin in the limit /2. We also determinethe behaviour of the solutions in the physically important limitsof a weakly non-planar substrate, a strongly concave substrate,a strongly convex substrate, a small volume flux, and a largevolume flux.     

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 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 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.     

Thermodynamic Inequalities in Continuum Mechanics

The constitutive relations for the transport of heat, stress,electric charge, etc., in a continuum must be chosen so thatthe second law of thermodynamics is not violated; the constraintstake the form of inequalities, typically requiring the entropygenerated within a material element to be non-negative. Thepaper is concerned with this concept—its history, thephysical principles on which it depends, how to apply it whensecond-order or non-linear effects are important and how itis widely misused in modern continuum mechanics. The history is reduced to the contributions of five leadingthermodynamicists—Clausius, Maxwell, Gibbs, Boltzmannand Duhem. The object here was to try to discover which formof the inequality one should regard as being fundamental. Oneimportant conclusion is that entropy S must be defined simultaneouslywith the identification of the inequality, and that in generalthis cannot be done until the constitutive equations are known.The empirical element enters with the notion of irreversibility,which is given a precise meaning with the aid of the motionreversed parity (x), a variable x having = +1 or = –1if, when time and motions are reversed, x x or x –x.The macroscopic parity of x, _*(x), is obtained by first replacingx by the constitutive equation for x. The entropy production rate has both irreversible (f) and reversible(r) parts. It is shown that the reciprocal relations followfrom the requirement that the macroscopic parity of (i) mustbe +1. Continuum thermodynamics is based on various principles extractedfrom theory developed for uniform systems, the example chosento illustrate the ideas being the simple monatomic gas. Second-orderconstitutive relations are introduced, and the expressions forentropy and its production rate per unit volume, , obtained.It is shown that the stability condition 0 cannot, in general,be satisfied merely by imposing constraints on the constitutiverelations. To second-order = ₁ + ₂, where ₁ is the usual bilinearform, and the terms in ₂ have an additional derivative. Thesecond-order term ₂ can have both signs, and is not dissipative.The relation between this fact and the frame-dependence of constitutiverelations is explained. The final section illustrates the errors frequently found inthe thermodynamic arguments appearing in books and papers onrational continuum mechanics. The principle of these is that 0 is interpreted as being a constraint on the constitutiverelations alone. Another is the idea that the balance equationscan be set aside as constraints by regarding them as mere definitionsof a heat source and a body force, an error based partly onthe misconception that constitutive relations should be frame-indifferent.Finally, an inequality due to Glansdorff & Prigogine isexamined and found to be in error.     

Application of JMN Approximants to Numerical Initial-value Problems in Linear Differential-algebraic Systems

For f: [0,) R, the J_MN approximant of f(t) is where M and N are non-negative integers and ß_k, _i,K_iare defined constants. Under appropriate conditions on f andprovided Re(_i) > 0 The approximants are the bases of recursions for numerical initial-valueproblems in linear differential-algebraic systems with constantcoefficients. The recursions are stable when N M N–2.Each step of a recursion involves mainly the solution of N/2uncoupled algebraic systems.     

Extremal Subgroups in Chevalley Groups

Throughout this paper G(k) denotes a Chevalley group of rankn defined over the field k, where n3. Let be the root systemassociated with G(k) and let ={₁, ₂, ..., _n} be a set of fundamentalroots of , with ⁺ being the set of positive roots of with respectto . For and ⁺, let n() be the coefficient of in the expressionof as a sum of fundamental roots; so =n(). Also we recall thatht(), the height of , is given by ht()=n(). The highest rootin ⁺ will be denoted by . We additionally assume that the Dynkindiagram of G(k) is connected.     

Asymptotic Error Formulae for Numerical Quadrature

A method, due to Fox, is used to derive asymptotic error formulaefor numerical procedures having the form (z+h, h).–(z,h)=(f,z,h).These procedures correspond to numerical quadrature for theintegrand (/h)(f,z,0)and compact expressions are given for determiningthe order of convergence as h 0, and the leading term in theerror. It is shown that a natural generalization of the Euler-Maclaurinexpansion is available. These results are applied to the particularcase where the Pt are polynomials in the differentiation operator.A related interpolation problem is also studied, and it is shownthat in certain cases higher order quadrature formulae are possiblewhen this interpolation problem is not poised.     

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.     

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 = +.     

Bivariational Methods for Hammerstein Integral Equations

Bivariational methods are presented for nonlinear integral equationsof Ham-merstein type = Kf(). With appropriate conditions onK and f, various upper and lower bounding functionals are derivedfor inner products and associated with the solution . Inthe latter case, suitable choices for g lead to point wise boundson both and its derivative. The methods are tested on a pendulumequation, and encouraging accuracy is obtained using simpletrial vectors.