Spontaneous potential well-logging is an important technique in petroleum exploitation. The spontaneous potential satisfies an elliptic boundary value problem with jump conditions on interfaces. At the joint points of the interfaces, the jumps of the spontaneous potential do not, in general, satisfy the compatibility condition. It turns out that it is impossible to find a piecewise H1 solution to the problem, and the standard finite element method cannot be applied to get an approximate solution. In this paper, by means of a new method, we prove that the problem exists a unique piecewise solution for some p < 2. We give an estimate for the solution as well. This allows us to prove the convergence of a numerical scheme proposed in [3].

The bifurcation from a normally conducting to a superconducting state as an external magnetic field is lowered is examined using the Ginzburg-Landau theory. The results for three specific examples are reviewed, extended and unified in the framework of a systematic perturbation theory introduced in [1].

We study a system of two second-order differential equations with cubic nonlinearities which model a film of superconductor material subjected to a tangential magnetic field. We verify some recent conjectures of one of the authors about multiplicity of solutions. We show that for an appropriate range of parameter values the relevant boundary value problem has at least two symmetric solutions. It is also proved that a second range of parameters exists for which there are three symmetric solutions.

Nonlinear evolution equations including hysteresis functionals are studied. It is the purpose of this paper to investigate the asymptotic stability of the solution as time t → + ∞. In the case when the forcing term of the equation tends to a time-independent function as t → + ∞, we shall show that the solution stabilizes at + ∞ and the system is asymptotically stable (equilibrium stability). In the case when the forcing term is periodic in time, we shall show that there is at least one periodic solution and that in some restricted cases the periodic solution is unique.

In this paper we consider a superconductor free boundary problem. Under isothermal conditions, a superconductor material (of ‘type I’) will develop two phases separated by a sharp interface Γ(t). In the ‘normal’ conducting phase the magnetic field is divergence free and satisfies the heat equation, whereas on the interface Γ(t), curl , where n is the normal and Vn is the velocity of Γ(t) in the direction of n; further, (constant) on Γ(t). Existence and uniqueness of a classical solution locally in time are established by Newton's iteration method under assumptions which enable us to reduce the 3-dimensional problem to a problem depending on essentially two space variables.

This work treats the injection of certain thermoplastics into a planar mould cavity. The problem is to determine the filling pattern. It is assumed that the thermoplastic can be modelled as a non-Newtonian fluid of power-law type whose power-law exponent is relatively small (the pseudo-plastic case). The dependence of the viscosity on thermal variations is neglected. The mathematical description leads to a moving boundary problem, for which an asymptotic solution is found. According to this solution, the expansion of the polymer melt follows the level sets of an interior distance function, which is determined by the geometry of the mould, and the position of the injection point. The solution is easily computed and results of numerical experiments are given.

An analysis is given of brine transport through a porous medium, which incorporates the effect of volume changes due to variations in the salt concentration. Two specific situations are investigated which lead to self-similarity. We develop the existence and uniqueness theory for the corresponding ordinary differential equations, and give a number of qualitative properties of the solutions. In particular, we present an asymptotic expression for the solution in terms of the relative density difference (ρs−ρf)/ρf. Finally, we show some numerical results. It is found that the volume changes have a noticeable effect on the mass transport only when salt concentrations are large.

Ovsiannikov's method of partially invariant solutions of differential equations can be considered to be a special case of the method of differential constraints introduced by Yanenko and by Olver and Rosenau. Differential constraints are used to construct non-reducible partially invariant solutions of the boundary layer or Prandtl equations.

A stationary phase method is developed for the asymptotic evaluation, as R → ∞, of oscillatory sums of the form

It is extended to multidimensional sums. Numerical comparisons demonstrate the accuracy of the asymptotic approximations. The results are applied to the practical estimation of the number of lattice points in large domains in ℝ2.

A mean-field model for the motion of rectilinear vortices in the mixed state of a type-II superconductor is formulated. Steady-state solutions for some simple geometries are examined, and a local existence result is proved for an arbitrary smooth geometry. Finally, a variational formulation of the steady-state problem is given which shows the solution to be unique.

This paper deals with the spherically symmetric Stefan problem in three space dimensions. The melting temperature satisfies the Gibbs–Thomson law. The solution is obtained as a limit of solutions of similar problems containing a small additional kinetic term in the melting temperature. Under some structural assumptions we show that the phase-change boundary has at most one discontinuity point t = T0 (see the corresponding result for the planar Stefan problem in Götz & Zaltzman (1995)). In the one-phase problem the discontinuity point always exists. At the time T0 the whole solid phase melts instantaneously. We study also the asymptotical stability (t → ∞) of stationary solutions satisfying boundary conditions of thermostat type.

The linearized shape stability of melting and solidifying fronts with surface tension is discussed in this paper by using asymptotic analysis. We show that the melting problem is always linearly stable regardless of the presence of surface tension, and that the solidification problem is linearly unstable without surface tension, but with surface tension it is linearly stable for those modes whose wave numbers lie outside a certain finite interval determined by the parameters of the problem. We also show that if the perturbed initial data is zero in the vicinity of the front, but otherwise quite general, it does not affect the stability. The present results complement those in Chadam & Ortoleva [4] which are only valid asymptotically for large time or equivalently for slow-moving interfaces. The theoretical results are verified numerically.

This paper is concerned with some mathematical aspects of magnetic resonance imaging (MRI) of the beating human heart. In particular, we investigate the so-called retrospective gating technique which is a non-triggered technique for data acquisition and reconstruction of (approximately) periodically changing organs like the heart. We formulate the reconstruction problem as a moment problem in a Hilbert space and give the solution method. The stability of the solution is investigated and various error estimates are given. The reconstruction method consists of temporal interpolation followed by spatial Fourier inversion. Different choices for the Hilbert space ℋ of interpolating functions are possible. In particular, we study the case where ℋ is (i) the space of bandlimited functions, or (ii) the space of spline functions of odd degree. The theory is applied to reconstructions from synthetic data as well as real MRI data.

We study the development of concentration profiles in a semi-infinite slab of semi-conductor material, in which impurities have been implanted at a high concentration. When the implant is uniform throughout the slab, no impurities can pass through the face of the slab, and the vacancy concentration at the surface is kept at its equilibrium value, it is shown that the density profiles of impurities, vacancies and host atoms may have self-similar form. The analysis is constructive and yields qualitative properties of the profiles and the front.

This paper discusses the initial value problem

where A, Bi and Ci are d × d complex matrices, pi, qi ∈ (0, 1), i = 1, 2, …, and y0 is a column vector in ℂd. By using ideas from the theory of ordinary differential equations and the theory of functional equations, we give a comprehensive analysis of the asymptotic behaviour of analytic solutions of this initial value problem.

The elastic displacement of a thin cylinder subject to given forces is approximated by means of a function constructed from the solutions of certain one-dimensional problems. Estimates are given for the error in terms of a decreasing function of the radius of the cylinder.

We consider a family of problems involving two-dimensional Stokes flows with a time dependent free boundary for which exact analytic solutions can be found; the fluid initially occupies some bounded, simply-connected domain and is withdrawn from a fixed point within that domain. If we suppose there to be no surface tension acting, we find that cusps develop in the free surface before all the fluid has been extracted, and the mathematical solution ceases to be physically relevant after these have appeared. However, if we include a non-zero surface tension in the theory, no matter how small this may be, the cusp development is inhibited and the solution allows all the fluid to be removed.