We prove the existence of a weak solution for a two-phase continuous casting Stefan problem with a general monotone nonlinear cooling condition. We establish a sufficient condition for stability, which yields uniqueness and comparison results for the evolutionary and the steady- state solutions. We also discuss the asymptotic behaviour as t←∞ of the corresponding temperatures and enthalpies.

Integral characteristics, such as elastic polarization matrices of elastic inclusions and cavities, are described. The matrix of elastic polarization of a finite cavity is constructed in the case of the two-dimensional Lamé operator under the assumption that the geometry of the domain occupied by the cavity is defined by a conformal mapping from the unit disk. Examples and applications of these integral characteristics in the theory of cracks are considered.

An experimental study using an analogue electronic model of the equation x‴ + x′ = є sin x, modified by the addition of small terms ax″ and βx with 0 < β < α ≫ є shows that these dissipative terms have a profound effect on the solutions for large time. If ∈ is not too large, experimental solutions tend to a simple periodic form, unlike the case α=β = 0. The existence of this limiting periodic form suggests the possibility of a simple analytic treatment using the method of harmonic balance, and this treatment leads to excellent agreement with the experimental results for a wide range of initial conditions and values of the parameters. The approach towards attracting limiting periodic solutions is analysed by using the method of multiple scales.

We are concerned with the problem of determining the diffusivity D of a diffusion process governed by the equation ut = (Dux)x', under the assumption that D depends on u. The main point consists in the observation that there exist solutions of travelling-wave type and that the dependence D = D(u) can be explicitly found in terms of the profile of such solutions. The property of finite propagation speed is required for this method to work. We propose two concrete implementations of the inverse problem, and give a rigorous mathematical proof of our statements. We also describe the application of the travelling-wave method to another interesting class of nonlinear parabolic equations.

We consider the equation of turbulent burst propagation

with m > 1 and k ≥ 0, or, in an equivalent formulation

We consider initial data u(x, 0) which are radially symmetric and have compact support; x = l(t) = r(T) describes the interface of the solution.

A practical method is given that can determine whether or not it is possible for all or part of a problem in crystal elasticity to be expressed in explicit rational or radical terms when the characteristic sextic polynomial is known to be unsolvable. It involves the determination of the group to which an elastic quantity (displacement, stress, strain, energy density) belongs under permutations of the roots of the sextic polynomial. If this is not the symmetric group then a rational or radical expression is impossible. It is applied to the three-dimensional problem of a point force (the Green's function) in a cubic crystal. It is proved that it is impossible that the Green's function could be given by any radical expression that is valid for all elastic constants and directions in the crystal, as the existence of such an expression would lead to the solution of a proven unsolvable polynomial. It follows that the rational expression given by Dikici (1986) for the Green's function is not just incorrect, but that it is impossible that there could be any such rational expression. The same impossibility is found to apply to the displacements, stresses and strains of a straight dislocation in a cubic crystal. The application to other elasticity problems is discussed.

The system of conservation laws governing heat and mass transfer processes in a continuous medium is obtained in a symmetric form on the basis of the successive application of fundamental thermodynamic principles. This approach involves reformulating the problem in intensive thermodynamic variables such as the temperature and chemical potential. The equations of capillary fluid mechanics and phase transitions with moving free boundaries are analysed in detail. The unsteady motion of a drop driven by buoyancy forces in an unbounded ambient fluid with dilute surfactants is investigated where the LeChatelier principle is established for an arbitrary surfactant. The general procedure for construction of self-similar solutions for the thermodiffusive Stefan problem with piecewise constant matrices of coefficients is described

A simple computational procedure is described for the evaluation of the modified Bennett functions, which arise in the multiple Fourier analysis for bang-bang controls with memory.

In this paper we introduce a class of nonlinear complex ordinary differential equations that arises in the removal of channel distortion in digital telecommunication signals. Techniques from dynamical systems theory show that Hopf bifurcations are possible in the simplest of these systems. Numerical integrations, however, show that such bifurcations are degenerate. When attempts are made to follow the periodic orbits, both the period of oscillation and the principal bifurcation parameter remain fixed at their values at bifurcation. The periodic orbits exist as a family at discrete parameter values, i.e. we have a nonlinear centre. A stability régime diagram is presented

We apply the method of matched asymptotic expansions to link the outgoing wave solution at infinity of the differential equation y″(x)+(lgr;+єxn)y(x) = 0, x∈(0, ∞),λ∈Cє>0, n∈N across the turning point x = (—λ/є)1/n nearest to the positive real axis to a linear homogeneous boundary condition at the origin. The equation with n = 1 models the leakage of energy from the core of a bent fibre optic waveguide, the rate of leakage corresponding to Im λ, which was shown to be exponentially small like O(exp[— 1/є]) by Paris & Wood (1989). The extension n = 2 by Brazel et al. (1990) obtained Im λ = O(exp[— 1/є1/2]). Both these papers involve delicate analysis of the asymptotics of special functions near to Stokes' lines. When n > 2 no special functions are available, and completely different methods must be employed to obtain the result Im λ = O(exp[— 1/є1/n]).

It is shown that for a class of pairs of energy wells the only Young measures having these wells as support must reduce to spatially constant Dirac masses. This implies the prediction that fine structures will be absent in certain crystal phase transitions.