Simple and systematic algorithms for relating differential equations are given. They are based on comparing the local symmetries admitted by the equations. Comparisons of the infinitesimal generators and their Lie algebras of given and target equations lead to necessary conditions for the existence of mappings which relate them. Necessary and sufficient conditions are presented for the existence of invertible mappings from a given nonlinear system of partial differential equations to some linear system of equations with examples including the hodograph and Legendre transformations, and the linearizations of a nonlinear telegraph equation, a nonlinear diffusion equation, and nonlinear fluid flow equations. Necessary and sufficient conditions are also given for the existence of an invertible point transformation which maps a linear partial differential equation with variable coefficients to a linear equation with constant coefficients. Other types of mappings are also considered including the Miura transformation and the invertible mapping which relates the cylindrical KdV and the KdV equations.

This paper deals with the motion of a fluid in a closed loop under the effect of natural convection and a given external heat flux. More precisely, we show that the stationary solutions of a system describing the intermediate asymptotics of the previous problem are structurally linearly unstable.

We consider the two-phase one-dimensional Stefan problem in a finite interval, with initial and boundary conditions such that the solid phase vanishes at a finite time T and at a single point. We show that the temperature in the solid phase decreases to zero and is bounded by c exp (α/(t − T)) as extinction approaches (C, α > 0) and that phase boundaries at extinction have finite speeds.

The coupling of the Stefan equation for the heat flow with the Gibbs–Thomson law relating the melting temperature to the mean curvature of the phase interface is considered. Solutions, global in time, are constructed which satisfy the natural a priori estimates. Mathematically the main difficulty is to prove a certain regularity in time for the temperature and the indicator function of the phase separately. A capacity type estimate is used to give an L1 bound for fractional time derivatives.

We consider a polymer melt in a domain Ω whereby each polymer chain is attached at one endpoint to a fixed surface S contained in ∂Ω. Denote by G(x, t;y) the normalized number density of all subchains from x to y of length t. Then, according to the selfconsistent mean field theory, G satisfies, for each y: Gt - Δ2G + σϕG = 0, where σ is a real parameter, and ϕ is a functional of G(·, ·; ·) both non-local and nonlinear. We establish the existence of G and C∞ regularity of ϕ, as a function of x.

Low permeability formations are often hydrofractured to increase the production rate of oil and gas. This process creates a thin, but highly permeable, fracture which provides an easy path for oil and gas to flow through the reservoir to the borehole. Here we examine the payoff of hydrofracturing by determining the increased production rate of a hydrofractured well. We find explicit formulas for the steady production rate in the three regimes of small, intermediate, and large (dimensionless) fracture conductivities. Previously, only the formula for the large fracture conductivity case was known.

We assume that Darcy flow pertains throughout the reservoir. Then, the steady fluid flow through the reservoir is governed by Laplace's equation with a second-order boundary condition along the fracture. We first analyze this boundary value problem for the case of small fracture conductivities. An explicit formula for the production rate is obtained for this case, essentially by combining singular perturbation methods with spectral methods in a function space which places the second-order boundary condition on the same footing as Laplace's equation. Next, we re-cast Laplace's equation as a variational principle which has the second-order boundary condition as its natural boundary condition. This allows us to use simple trial functions to derive accurate estimates of the production rate in the intermediate conductivity case. Then, an asymptotic analysis is used to find the production rate for the large fracture conductivity case. Finally, the asymptotic and variationally-derived production rate formulasare compared to exact values of the production rate, which have been obtained numerically.

It may be feasible to create more than a single fracture about a borehole. So we also develop similar asymptotic and variational formulas for the production rate of a well with multiple fractures.

An algorithm is presented to linearize nonlinear partial differential equations by non-invertible mappings. The algorithm depends on finding nonlocal symmetries of the given equations which are realized as appropriate local symmetries of a related auxiliary system. Examples include the Hopf-Cole transformation and the linearizations of a nonlinear heat conduction equation, a nonlinear telegraph equation, and the Thomas equations.

The goal of this note is to explain the process of computing the lowest energy level achieved by an elastic crystal subject to homogeneous boundary deformation. This analysis follows mainly the lines of Chipot and Kinderlehrer or Fonesca but we expect that it will lead to some new results for some other energy functional having different invariance properties. The mathematical interest of the result also lies in the fact that the lowest energy level computed this way coincides with the relaxed energy functional (see Fonesca). This relaxed energy is determined by a known thermodynamic quantity, Ericksen' subenergy. Within this review we will also analyse some of the mathematical and physical aspects of these highly oscillatory problems.

This note deals with a new model of elastic–perfectly plastic materials in which the yield stress is regarded as a threshold above which plastic flow occurs rather than a constraint which cannot be violated. This modelling change allows us to treat a number of signalling and impact problems not solvable within the classic framework of elastic–perfectly plastic materials.

This note deals with a new model of elastic–perfectly plastic materials in which the yield stress is regarded as a threshold above which plastic flow occurs rather than a constraint which cannot be violated. This modelling change allows us to treat a number of signalling and impact problems not solvable within the classic framework of elastic–perfectly plastic materials.

The main result of this paper is a non-uniqueness theorem for the self-similar solutions of a model for phase transitions in binary alloys. The reason for this non-uniqueness is the discontinuity in the coefficients of heat conduction and diffusion at the inter-phase. Also the existence of a self-similar solution and the stability criterion are discussed.

Numerical methods for initial-value problems which develop singularities in finite time are analyzed. The objective is to determine simple strategies which produce the correct asymptotic behaviour and give an accurate approximation of the blow-up time. Fixed step methods for scalar ordinary differential equations are studied first and it is shown that there is a natural embedding of the discrete process in a continuous one. This shows clearly how and why the fixed-step strategy fails. A class of time-stepping strategies that correspond to a time- continuous re-scaling of the underlying differential equation is then proposed; this class is analyzed and criteria established to determine suitable choices for the re-scaling. Finally the ideas are applied to a partial differential equation arising from the study of a fluid with temperature-dependent viscosity. The numerical method involves re-formulating the equationas a moving boundary problem for the peak value and applying the ODE time-steppingstrategies based on this peak value.

The coupled nonlinear partial differential equations obeyed by the electrical potential and temperature distribution for a medium undergoing steady state electrical heating are applied to a one-dimensional rod having its surface temperature held constant as current is conducted along its length due to a potential difference maintained between its ends. Extension is given to the previously discussed class of solutions by the inclusion of a thermal conductivity which varies linearly with temperature. The resulting electrical current and resistance are found to be significantly influenced by the thermal conductivity of the medium. Molybdenum is identified as a material exemplifying such a thermal conductivity and the general effects are then numerically illustrated.

This article is concerned with the structure and stability properties of a combustion front that propagates in the axial direction along the surface of a cylindrical solid fuel element. The fuel consists of a mixture of two finely ground metallic powders, which combine upon ignition in a one-step chemical reaction. The reaction is accompanied by a melting process, which in turn enhances the reaction rate. The combustion products are in the solid state. The reaction zone, inside which the melting occurs, is modelled as a front that propagates along the surface of the cylinder. The different modes of propagation that have been observed experimentally (such as single- and multiheaded spin combustion and multiple-point combustion) are explained as the results of bifurcations from a uniformly propagating plane circular front. The stability properties of the various modes are discussed.

Phosphorus diffusion in silicon shows a number of anomalous effects, and we apply asymptotic methods to a model problem which includes most of these. Both constant surface concentration problems and the diffusion of implanted dopant are considered. An unusual feature of the model is the non-local dependence of the tail diffusivity on the peak concentration.

A concentrated capacity problem is posed for the heat equation in a multidimensional domain. In the concentrated capacity (i.e. in a portion of the boundary of the domain) a change of phase takes place, and a Stefan-like problem is posed. This scheme has been introduced in the literature as the formal limiting case of a certain class of diffusion problems.

Our main result is a theorem of continuous dependence of the solution on the data. It is also used to prove the existence of the solution (in a weak sense), assuming only integrability of the data. The solution is found as the limit of the solutions of the approximating problems.

An analysis is made of the diffraction of sound by a scratch (or groove) in the nominally plane surface of a fluid-loaded elastic solid. The solid is modelled as an isotropic half-space, and an exact representation of the diffracted sound is obtained as an integral over the surface of the scratch involving a Green's function and the acoustic field for the homogeneous solid. The restriction of the range of integration to the surface of the scratch permits the use of an approximate form of Green's function which is valid for source positions confined to a limited region of space. Details are worked out for the case in which the acoustic wavelength is much larger than the diameter of the scratch cross-section (Rayleigh approximation). The results are illustrated by application to sound waves in water inicident on a rectilinear scratch of circular cross-section in steel or rubber. In the former case an important rôle in diffraction is played by leaky Rayleigh waves on the fluid-solid interface.

This paper is concerned with the mathematical study of a nonlinear system modelling an irreversible phase change problem. Uniqueness of the solution is proved using the accretivity of the system in (L1)2. Expressing one of the two unknowns as an explicit functional of the other reduces the system to a single nonlinear evolution equation and ultimately leads to an existence theorem.

In this paper the existence and uniqueness of the solution of a nonlinear system modelling some irreversible phase changes is established.

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.

We prove that in the Stefan problem with planar, cylindrical or spherical symmetry, with vanishing heat capacity and constant boundary temperature, no mushy region can appear, even in the presence of constant volumetric heat sources, if the initial data are consistent with the presence of just two pure phases. If the boundary temperature is not constant, a mushy region may or may not appear; we find some general conditions ensuring one case or the other and we give a specific example illustrating the appearance of a mushy region.