This paper deals with a minimax control problem for semilinear elliptic variational inequalities associated with bilateral constraints. The control domain is not necessarily convex. The cost functional, which is to be minimised, is the sup norm of some function of the state and the control. The major novelty of such a problem lies in the simultaneous presence of the nonsmooth state equation (variational inequality) and the nonsmooth cost functional (the sup norm). In this paper, the existence conditions and the Pontryagin-type necessary conditions for optimal controls are established.

This paper is concerned with a reinvestigation of the problem of water wave scattering by a wall with multiple gaps by using the solution of a singular integral equation with a combination of logarithmic and power (Cauchy-type) kernels in disjoint multiple intervals. Use of Havelock's expansion of water wave potential reduces the problem to such an integral equation in the horizontal velocity across the gaps. The solution of the integral equation is obtained by utilizing the solutions of Cauchy-type integral equations in (0,∞) and also in multiple disjoint intervals. An explicit expression for the reflection coefficient is obtained for a wall with n gaps and supplemented by numerical results for up to three gaps.

The leading-order interaction of short gravity waves with a dominant long-wave swell is calculated by means of Zakharov's [7] spectral formulation. Results are obtained both for a discrete train of short waves and for a localised wave packet comprising a spectrum of short waves.

The results for a discrete wavetrain agree with previous work of Longuet-Higgins & Stewart [5], and general agreement is found with parallel work of Grimshaw [4] which employed a very different wave-action approach.

Under the appropriate physical hypotheses, the problem of determining the pressure distribution in a gas-filled slider bearing becomes a singular perturbation problem as Λ, the bearing number, tends to infinity. This paper extends the results of an earlier one by the author to consider the case where the film profile has jump discontinuities in slope at points interior to the bearing. Application of the methods of general singular perturbation theory establishes the appropriate existence-uniqueness results for this problem, and a means is devised by which uniformly valid asymptotic approximations to the pressure distribution may be obtained for large values of Λ.

A new derivation of the averaged heat and mass transport equations for two-phase flows is presented. A volume averaging technique is used in which averaging is perform over both phases simultaneously in order to derive equations that describe transport the mixture, rather than transport in each phase. The derivation is particularly applicable to incompressible liquid/solid systems in which the two phases are tightly coupled. An example of the numerical solution of the equations is then presented in which a thermally convecting suspension is modelled. It is seen that large-scale instability can result from the interaction of thermal and compositional density gradients.

Continental shelf waves are examined for side band instability. It is shown that a modulated shelf wave is described by a nonlinear Schrödinger equation, from which the stability criterion is derived. Long shelf waves are stable to side band modulations, but as the wavenumber is increased there are regions of instability (in wavenumber space). A change of stability occurs at each long wave resonance, defined by the condition that the group velocity of the shelf wave equals a long wave speed. Equations describing the long wave resonance are derived.

In this paper various wave motions in water of infinite depth containing vertical porous boundaries are determined when the water is of infinite extent on one or both sides. Initially surface tension is ignored and simple solutions for incident waves are obtained before going on to harder wave source and wave-maker solutions. A reduction method is developed to obtain solutions for two-sided boundaries from those for one-sided, which are obtained by standard techniques. The effect of surface tension that precludes simple solutions is also considered, although a present lack of information on dynamical edge behaviour for porous boundaries means that the formal mathematical solutions must be left in terms of arbitrary edge constants. In conclusion, some solutions are noted for finite depth.

In this paper, a model for lateral dispersion in open-channel flow is studied involving a diffusion equation which has a nonlinear term describing the effect of buoyancy. The model is used to investigate the interaction of two buoyant pollutant plumes. An approximate analytic technique involving Hermite polynomials is applied to the resulting PDEs to reduce them to a system of ODEs for the centroids and widths of the two plumes. The ODEs are then solved numerically. A rich variety of behaviour occurs depending on the relative positions, widths and strengths of the initial discharges. It is found that for two plumes of equal strength and width discharged side-by-side, the plumes move apart and the rate of spreading is inhibited by their interaction, whereas when one plume is initially much wider than the other, both plumes tend to drift to the side of the narrower plume. Finally, the PDEs are solved numerically for two sets of initial conditions and a comparison is made with the ODE solutions. Agreement is found to be good.

By using fixed point index theory, we present the existence of positive solutions for a Sturm-Liouville singular boundary value problem with at least one positive solution. Our results significantly extend and improve many known results even for non-singular cases.

The authors have noticed an oversight in Section 3.1 of this paper. To correct this error it is necessary to assume a uniform differentiability condition on G(x, h). This is required, for example, to imply δ on line 4 of p. 178 can be chosen independent of h. For brevity we note that this oversight has been corrected in [1], which is available from the authors. Also, the results of subsequent sections are unaffected.

The paper analyes the dynamics of duopoly output game involving a warfare strategy proposed by Robert Bishop. Necessary and sufficient conditions are obtained for the stability of a duopoly warfare game.

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of O(N4) where N is the number of retained modes of polynomial approximations. This paper presents some efficient spectral algorithms, which have a condition number of O(N2), based on the ultraspherical-Galerkin methods for the integrated forms of second-order elliptic equations in one and two space variables. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of Nd+1 operations for a d-dimensional domain with (N – 1)d unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.

An efficient probabilistic algorithm is presented for the determination of the rate matrix of a block-GI/M/1 Markov chain. Recurrence of the chain is not assumed.

The Huygens' property is exploited to study propagation relations for solutions of certain types of linear higher order Cauchy problems. Motivated by the solution properties of the abstract wave problem, addition formulas are developed for the solution operators of these problems. The application of these alternative forms of the solution operators to data leads to connecting operator relations between distinct solutions of the problems at different times. We examine this solution behaviour for both analytic and abstract Cauchy problems. A basic algorithm for constructing addition formulas for solutions of ordinary differential equations is included.