The thermistor problem is a system of equations modelling the electric heating of a conducting material. We prove uniqueness of a solution to this system.

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.

We present several algorithms, executable in a finite number of steps, which have been implemented in the symbolic language maple. The standard form algorithm reduces a system of PDEs to a simplified standard form which has all of its integrability conditions satisfied (i.e. is involutive). The initial data algorithm uses a system's standard form to calculate a set of initial data that uniquely determines a local solution to the system without needing to solve the system. The number of arbitrary constants and arbitrary functions in the general solution to the system is directly calculable from this set. The taylor algorithm uses a system's standard form and initial data set to determine the Taylor series expansion of its solution about any point to any given finite degree. All systems of linear PDEs and many systems of nonlinear PDEs can be reduced to standard form in a finite number of steps. Our algorithms have simple geometric interpretations which are illustrated through the use of diagrams. The standard form algorithm is generally more efficient than the classical methods due to Janet and Cartan for reducing systems of PDEs to involutive form.

We investigate the stability of travelling wave solutions of the one-dimensional under-cooled Stefan problem. We find a necessary and sufficient condition on the initial datum under which the free boundary is asymptotic to a travelling wave front. The method applies also to other types of solutions.

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.

This work deals with the global instability mechanism of viscous fingering in a Hele–Shaw cell with the inclusion of surface tension. We investigate the interaction and propagation of travelling waves in the system, and obtain two discrete sets of global wave modes: symmetrical modes and anti-symmetrical modes. We call the instability mechanism determined by these global modes the global trapped wave (GTW) instability. A unique global, neutrally stable state of the system is found; it explains the formation of the narrow, oscillatory fingers discovered by Couder el al. (1986) and by Kopf-Sill & Homsy (1987).

We describe and analyze a model of filtration combustion, in which a gas is forced at high pressure into a porous solid matrix so that after ignition and under favourable conditions a combustion wave can propagate through the medium. We consider the case of counter flow, where the gas is forced into the reaction zone through the unreacted part of the porous solid. Relations are derived for the steady state propagation of a planar combustion wave or front in the limit of high-activation energy, from which the propagation speed, reaction temperature, and reacted mass fraction of the solid product can be found in terms of the mass flux of the injected gas, the gas pressure and mass flux on exit from the front, and other physico-chemical parameters describing the system. Two distinct modes of combustion are discussed, corresponding to the reaction being driven to completion by exhaustion of either the gaseous or the solid component, these being referred to as the gas-deficient and solid-deficient modes of burning respectively. For both homogeneous and heterogeneous forms of the reaction rate we find that there is a critical inlet mass flux for the incoming gas below which steady state solutions no longer exist and that there are parameter values for which multiple steady states occur.

There are symbolic programs based on heuristics that sometimes, but not always, explicitly integrate the determining equations for the infinitesimal Lie symmetries admitted by systems of differential equations. We present a heuristic-free algorithm ‘Structure constant’, which can always determine whether the Lie symmetry group of a given system of PDEs is finite- or infinite-dimensional. If the group is finite-dimensional then ‘Structure constant’ can determine the dimension and structure constants of its associated Lie algebra without the heuristics of integration involved in other methods. If the group is infinite-dimensional, then ‘Structure constant’ computes the number of arbitrary functions which determine the infinite-dimensional component of its Lie symmetry algebra and also calculates the dimension and structure of its associated finite-dimensional subalgebra. ‘Structure constant’ employs the algorithms ‘Standard form’ and ‘Taylor’, described elsewhere. ‘Standard form’ is a heuristic-free algorithm which brings any system of determining equations to a standard form by including all integrability conditions in the system. ‘Taylor’ uses the standard form of a system of differential equations to calculate its Taylor series solution. These algorithms have been implemented in the symbolic language MAPLE. ‘Structure constant’ can also automatically determine the dimension and structure constants of the Lie symmetry algebras of entire classes of differential equations dependent on variable coefficients. In particular, we obtain new group classification results for some physically interesting classes of nonlinear telegraph equations depending on two variable coefficients, one representing a nonlinear wave speed and the other representing a nonlinear dispersion.

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]).

This paper presents a detailed analysis of a model for a positive corona discharge, and discusses the relationship between the model and the space charge equations which predict the macroscopic motion of charged ions in a gas. The study is restricted to the asymptotic behaviour of the discharge for large time, and it does not consider its initial or burst phases in detail.

One prediction of this model is the existence of a steady state solution to the space charge equations that may lose stability to a travelling wave disturbance which then grows into a strongly pulsed oscillation.

We compare some numerical calculations with an asymptotic analysis of the discharge and find good agreement.

We investigate the Brusselator reaction–diffusion equations with periodic boundary conditions. We consider the range of values of the parameters used by Kuramoto in his study of chaotic concentration waves. We determine numerically the bifurcation diagram of the long-time travelling and standing wave solutions using a highly accurate Fourier pseudo-spectral method. For moderate values of the bifurcation parameter, we have found a sequence of instabilities leading either to periodic and quasiperiodic standing waves, or to chaotic regimes. However, for large values of the control parameter, we have found only uniform time-periodic solutions or time-periodic travelling wave solutions. Our numerical study motivates a new asymptotic analysis of the Brusselator equations for large values of the control parameter and small diffusion coefficients. This analysis explains the numerical predictions. The chaotic regime is limited to moderate values of the control parameter and periodic solutions are the only solutions for large values of the control parameter. We identify the stabilizing mechanism as the relaxation oscillations which appear when the control parameter is large. Our asymptotic result on the stability of periodic solutions is then generalized to a class of two-variable reaction-diffusion equations.

Low order differential equations typically have solutions which represent homoclinic or heteroclinic orbits between singular points in the phase plane. These orbits occur when the stable manifold of one singular point intersects or coincides with its unstable manifold, or the unstable manifold of another singular point. This paper investigates the persistence of these orbits when small dispersion is added to the system. In the perturbed system the stable manifold of a singular point passes through an exponentially small neighbourhood of a singular point and careful analysis is required to determine whether a homoclinic or heteroclinic connection is achieved.

Global smooth solutions are shown to exist for the system governing magneto-thermoviscoelastic phenomena in an electrically and thermally conducting isotropic solid immersed in an electromagnetic field. It is assumed that displacement currents are negligible, and that all field quantities depend on one space variable only; Joule heating is included.

The two-dimensional flow of a viscous incompressible fluid in a channel with accelerating walls is analysed by use of Hiemenz's similarity solution. Steady flows and their instabilities are calculated, and unsteady flows are computed by solving the initial-value problem for the governing partial-differential system. Thereby, these exact solutions of the Navier–Stokes equations are found to exhibit turning points, pitchfork bifurcations, Hopf bifurcations and Takens–Bogdanov bifurcations along the route to chaos. The substantial physical result is that the chaos previously found for flows with symmetrically accelerating walls is easily destroyed by a little asymmetry.

A mathematical analysis is carried out for the Cahn–Hilliard equation where the free energy takes the form of a double well potential function with infinite walls. Existence and uniqueness are proved for a weak formulation of the problem which possesses a Lyapunov functional. Regularity results are presented for the weak formulation, and consideration is given to the asymptotic behaviour as the time becomes infinite. An investigation of the associated stationary problem is undertaken proving the existence of a nontrivial stationary solution and further regularity results for any stationary solution. Stationary solutions are constructed in one and two dimensions; a formula for the number of stationary solutions in one dimension is derived. It is then natural to study the asymptotic behaviour as the phenomenological parameter λ→0, the main result being that the interface between the two phases has minimal area.

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.

The main objective of this work is to present, for practical use, some original results about several qualitative properties of the solutions of a large class of degenerate diffusion-convection equations arising from fluid mechanics. Current interest in models of the simultaneous motion of two immiscible incompressible liquids results from its significance for many applied fields such as, for instance, the theoretical modelling of oil reservoirs where the pores of a threedimensional porous medium contain some hydrocarbon component (oil). In secondary recovery, a second inexpensive fluid (water) is injected into the porous medium in order to push the oil towards the producing wells.

Following Krasnoselskii & Pokrovskii (1983) we express the constitutive law for the Prandtl–Reuss elastoplastic model in terms of a hysteresis operator, and introduce the vector Ishlinskii model. We investigate some properties (continuity, energy inequalities, dependence on spatial variables) of these operators.