The conserved phase field system with a small parameter in the n-dimensional case (n[les ]3) is considered. An asymptotic solution, describing the free interface dynamics, is constructed and justified. As the small parameter tends to zero, the limiting solution satisfies the modified Stefan problem with corrected Gibbs–Thomson law.

We prove the existence of generalized weak solutions of a new model for the process of in situ vitrification. The model describes the steady process of vitrification of soil, i.e. the melting of soil by means of electrical current and subsequent solidification as a rock. This is the core of a new technology developed to treat buried low-level radioactive or toxic waste. It is a free boundary problem consisting of a degenerate elliptic equation for the electric potential coupled with a stationary Stefan problem for melting the soil, which results from the Joule heating. The degeneracy reflects the smallness of the ratio of the electrical conductivity of the soil to that of the molten rock. The existence result is proved by considering a sequence of approximate nondegenerate problems, obtaining the necessary a priori estimates and passing to the limit. We also establish a sufficient condition for the solution to exhibit a molten region.

An unforced oscillator which obeys the equation x¨+α (x2+x˙2−1)x˙+x=0 has a pure sine-wave limit cycle which is attained rapidly for a range of values of α. In this paper, free and forced oscillation of this equation are examined experimentally and compared with those for the Van der Pol oscillator. Asymptotic solutions for small α confirm the experimental results.

A model is presented for the diffusion-driven drying of a polymeric solution such as liquid paint. Included is a stress build-up and relaxation in the polymer network of the viscoelastic material, which influences the diffusion process. The behaviour of the (one-dimensional) model is analysed by means of the maximum principle and illustrated with numerical calculations.

In this paper domain decomposition methods for radiative transfer problems including conductive heat transfer are treated. The paper focuses on semi-transparent materials, like glass, and the associated conditions at the interface between the materials. Using asymptotic analysis we derive conditions for the coupling of the radiative transfer equations and a diffusion approximation. Several test casts are treated and a problem appearing in glass manufacturing processes is computed. The results clearly show the advantages of a domain decomposition approach. Accuracy equivalent to the solution of the global radiative transfer solution is achieved, whereas computation time is strongly reduced.

In a recent paper devoted to the study of the superheating field attached to a semi-infinite superconductor, Chapman [1] constructs a family of approximate solutions of the Ginzburg–Landau system. This construction, based on a matching procedure, implicitly uses the existence of a family of solutions depending on a parameter c∈IR of the Painlevé equation in a semi-infinite interval (0, +∞)

with a Neumann condition at 0

and having a prescribed behaviour at +∞

In this paper we prove the existence of such a family of solutions and investigate its properties. Moreover, we prove that the second coefficient in Chapman's expansion of the superheating field is finite.

An algorithm, based on a discrete nonlinear model, is presented for evaluation of the critical shear stress required to move a dislocation through a lattice. The stability of solutions of the corresponding evolution problem is analysed. Numerical results provide upper and lower bounds for the critical shear stress.

A semi-infinite body, modelling the leading edge of a cutting tool or submerged hydrofoil, lies beneath a free surface in a uniform stream of infinitely-deep inviscid incompressible fluid flowing steadily under gravity. The body has horizontal upper and lower surfaces. The oncoming flow is partly diverted over, and partly under the body. The flow in that portion that travels over the body can be supercritical or subcritical. When it is supercritical it approaches a stream of some (to be determined) speed in a channel of some (to be determined) depth. When it is subcritical, there is also a train of waves on that stream, whose amplitude is also to be determined. Semi-analytic high-speed and low-speed solutions are obtained, and a numerical solution for finite speeds. There is a ‘forbidden’ intermediate speed range, within which steady flow may not be possible.

The direct Lyapunov method is used to investigate the stability of general equilibria of a nematic liquid crystal. First, we prove the converse Lagrange theorem stating that an equilibrium is unstable to small perturbations if the distortion energy has no minimum at this equilibrium (i.e. if the second variation of the distortion energy evaluated at the equilibrium is not positive definite). The proof is constructive rather than abstract: we explicitly construct a functional that grows exponentially with time by virtue of linearized equations of motion provided the condition of the theorem is satisfied. We obtain an explicit formula that gives the dependence of the perturbation growth rate upon the equilibrium considered and the initial data for the perturbation. Secondly, we obtain the upper and lower bounds for growing solutions of the linearized problem, and we identify the initial data corresponding to the most unstable mode (i.e. to the perturbation with maximal growth rate). All results are obtained in quite a general formulation: a nematic is inside a three-dimensional domain of an arbitrary shape and strong anchoring on the boundary is supposed; the standard equations of nematodynamics are employed as the governing equations.

We show how to find all the integrating factors and corresponding first integrals for any system of Ordinary Differential Equations (ODEs). Integrating factors are shown to be all solutions of both the adjoint system of the linearised system of ordinary differential equations and a system that represents an extra adjoint-invariance condition. We present an explicit construction formula to find the resulting first integrals in terms of integrating factors, and discuss techniques for finding integrating factors. In particular, we show how to utilize known first integrals and symmetries to find new integrating factors. Illustrative examples are given.

Analysing the motion of a driven, damped pendulum as a function of the amplitude of the driving force, we show, first, that for moderate values and larger of the amplitude, deviations from a simple motion with the period of the driving force are bounded by a constant times the inverse square root of the amplitude, for late times. For amplitudes above a larger threshold we are able to show that, for late times, the motion becomes a periodic motion with the period of the driving force. The manner in which this periodic motion is achieved with the passage of time is analysed.

For an external problem in IRd (d=2, 3) such that the unknown function satisfies the wave equation outside a finite domain, we generate artificial boundary conditions transparent to outgoing waves. These conditions permit an equivalent replacement of the original external problem by the problem inside the artificial boundary which is a circle (d=2) or a sphere (d=3): The questions of numerical implementation of the artificial conditions (that are non-local in both space and time) are considered. Special attention is paid to the reduction of necessary computational resources; in particular, a way of incorporating these conditions into numerical methods which makes the computational formulae local in time is suggested. The aspects of treating artificial boundaries of a non-spherical shape are discussed. Numerical examples of two- and three-dimensional scattering problems demonstrate the accuracy of proposed artificial boundary conditions.

The scattering of high-frequency sound waves by two-dimensional curved boundaries has received much attention over the past few decades, with particular interest in the effects of tangential ray incidence. In the event that the radius of curvature is not small, an analysis near the point of tangency gives rise to the Fock–Leontovič equation for the local field amplitude which, in turn, matches the creeping field of Keller's geometrical theory of diffraction. If the radius of curvature is sufficiently small, however, then this analysis is not valid and it is necessary to solve the full Helmholtz equation in the presence of a parabolic boundary. Under these conditions, which are canonical for diffraction by a sufficiently slender body, results are presented for the case of a plane wave impinging upon an acoustically hard parabolic cylinder. This diffraction process engenders a creeping field at one tip of the slender body, which then propagates around the body to the other tip. Here its energy is partially reflected, partially transmitted and partially radiated out in a detached field. A full description of this is given, along with a discussion of the ‘blunt’ limit in which we show that not only do we get the traditional creeping field of Keller's geometrical theory of diffraction, but also an exponentially small backward-propagating creeping field not predicted by traditional ray methods.

We revisit the theory of filtration (slow fluid motion) through a horizontal porous stratum under the usual conditions of gently sloping fluid height profile. We start by considering the model for flooding followed by natural outflow through the endwall of the stratum, which has an explicit dipole solution as generic intermediate asymptotics. We then propose a model for forced drainage which leads to a new kind of free boundary problem for the Boussinesq equation, where the flux is prescribed as well as the height h=0 on the new free boundary. Its qualitative behaviour is described in terms of its self-similar solutions. We point out that such a class of self-similar solutions corresponds to a continuous spectrum, to be compared with the discrete spectrum of the standard Cauchy problem for the porous medium equation. This difference is due to the freedom in the choice of the flux condition allowed in our problem setting. We also consider the modifications introduced in the above models by the consideration of capillary retention of a part of the fluid. In all cases we restrict consideration to one-dimensional geometries for convenience and brevity. It is to be noted however that similar problems can be naturally posed in multi-dimensional geometries. Finally, we propose a number of related control questions, which are most relevant in the application and need a careful analysis.

We study a unilateral equilibrium problem for the energy functional of a lipid tubule subject to an external field. These tubules, which constitute many biological systems, may form assemblies when they are brought in contact, and so made to adhere to one another along at interstices. The contact energy is taken to be proportional to the area of contact through a constant, which is called the adhesion potential. This competes against the external field in determining the stability of patterns with flat interstices. Though the equilibrium problem is highly nonlinear, we determine explicitly the stability diagram for the adhesion between tubules. We conclude that the higher the field, the lower the adhesion potential needed to make at interstices energetically favourable, though its critical value depends also on the surface tension of the interface between the tubules and the isotropic fluid around them.

We consider linear and nonlinear inverse source problems of sound radiation in an unbounded domain which models an oceanic waveguide. The method for analysing their solvability is based on analytical properties of generalized acoustic potentials and the theory of extremal problems.

This paper determines the asymptotic solution of certain initial-boundary value problems for singularly-perturbed reaction-diffusion equations, including the Allen–Cahn and Cahn–Hilliard equations, on bounded one-dimensional spatial domains for r[ges ]0. Attention is focused on the metastable evolution of a transition layer over an asymptotically exponentially-long time interval.

In this paper we consider a non-local anisotropic model for phase separation in two-phase fluids at equilibrium, and show that when the thickness of the interface tends to zero in a suitable way, the classical surface tension model is recovered. Relevant examples are given by continuum limits of ferromagnetic Ising systems in equilibrium statistical mechanics.

The paper is devoted to convective instability of reaction fronts. New approaches are developed to study some eigenvalue problems arising in chemical hydrodynamics. For gaseous combustion in the case of equality of transport coefficients, a linear stability analysis of an upward propagating front is carried out. A minimax representation of the stability boundary is obtained.

Steady incompressible inviscid flow past a three-dimensional multiconnected (toroidal) aerofoil with a sharp trailing edge TE is considered, adopting for simplicity a linearized analysis of the vortex sheets that collect the released vorticity and form the trailing wake. The main purpose of the paper is to discuss the uniqueness of the bounded flow solution and the role of the eigenfunction. A generic admissible flow velocity u has an unbounded singularity at TE; and the physical flow solution requires the removal of the divergent part of u (the Kutta condition). This process yields a linear functional equation along the trailing edge involving both the normal vorticity ω released into the wake, and the multiplicative factor of the eigenfunction, a1. Uniqueness is then shown to depend upon the topology of the trailing edge. If δTE=[empty ], as, for example, in an annular-aerofoil configuration, both ω and a1 are uniquely determined by the Kutta condition, and the bounded flow u is unique. If δTE≠[empty ], as, for example, in a connected-wing configuration, there is an infinity of bounded flows, parametrized by a1. Numerical results of relevance for these typical configurations are presented to show the different role of the eigenfunction in the two cases.