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.     

Computer extended series solution for unsteady flow in a contracting or expanding pipe

Unsteady flow in a semi-infinite contracting or expanding pipeis reinvestigated using long series analysis. The proposed seriesmethod is useful in analysing the problem for a moderately largeconstant ( = aa/, where a = a(t), the radius of the pipe isa function of time, a(t) is the velocity of the wall, and iskinematic viscosity). For positive values of (expansion ofthe pipe) accuracy of the series representing shear stress andpressure gradient is increased from = 2.89 to = 6.0 by extractingthe singularity followed by completion of the series. For negativevalues of (contraction of the pipe), we revert the series whichresults into the increase of the region of validity of the transposedseries from = -25.0 to = -2.89. Later we use Padé approximantsfor summing them. Also, the asymptotic solution for large valuesof is obtained and it agrees closely with pure numerical valuesof shear stress at the wall and pressure gradient.     

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 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.     

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.     

Short-time Asymptotic Solutions of the Heat Conduction Equation with Spatially Varying Coefficients

An asymptotic solution of the heat conduction equation withspatially varying coefficients is developed for small times.The method followed consists of an application of the Laplacetransformation and use of the Liouville-Green approximationin the subdominant solution of the resulting second-order differentialequation. The approximate solution is inverted by contour integration.The resulting asymptotic expression has a time-dependence identicalto that applying to the case of constant properties providedthat an appropriately averaged value of the thermal diffusivity is used, namely, The error term is of the order of t where is a measure of spatialvariability of the coefficients in the heat equation.     

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 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.     

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.     

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     

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)].     

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.     

Exact Remainders for Asymptotic Expansions of Fractional Integrals

This is a continuation of work begun in an earlier paper inwhich we used the theory of distributions to derive explicitexpressions for the remainder terms associated with the asymptoticexpansions of the Stieltjes transform. In this paper similarresults are obtained for the fractional integral of order definedby 1f(x)=1/f()^x_o(x-)^x-1 f(t)dt, >. Heref(t) is a locally integrable function on [0, ) and satisfies f(t) a_st-5–0(ó >0), s=0 as      

Analysis of Certain Slowly Converging Series

Slowly converging series frequently arise in the solution ofdifferential equations. The slow convergence is commonly dueto the presence of some kind of singularity. This paper demonstratesthat, if sufficient information is available as to the natureof the singularity, it is possible to accelerate massively theconvergence by extracting most singular terms. The prime illustration is a separating laminar boundary layer,where there is a weak singularity at separation if the pressureis independently prescribed. This singularity was extensivelystudied by Goldstein (1948), Stewartson (1958) and Terrill (1960),who showed that the leading terms in an expansion about thesingularity depend upon a single parameter ₁, with later termsdepending upon parameters ₅, ₉, etc. In a particular case, first studied by Howarth (1938), a computersolution by Leigh (1955) gave ₁ = 0.492, and more recently Davies& Walker (1977) gave the (presumably) more accurate ₁ =0.502. The present work uses knowledge of the singularity toaccelerate the convergence of a series expansion (in powersof distance from the leading edge) and yields ₁ = 0.50287;though the last figure may not be reliable. Further, thoughthe computer solutions are not sufficiently accurate to estimate₅, the present method yields ₅ =– 2.0(6). Other detailsof the flow are predicted, and compared favourably with thecomputer solutions. The above major claims assume that the position of the singularityis known precisely. Additional calculations, where the positionof the singularity is first to be estimated, still yield excellentresults, except very close to the singularity. Even here, theyare remarkably accurate.     

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.     

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.     

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.