Equations are derived which describe the evolution of the mean flow generated by a progressing water wave packet. The effect of friction is included, and so the equations are subject to the boundary conditions first derived by Longuet-Higgins [10]. Solutions of the equations are obtained for a wave packet of finite length, and also for a uniform wave train. The latter solution is compared with experiments.

This paper examines the predictions of shallow water theory for steady and unsteady withdrawal flows through an extended sink from fluid of finite depth. Two-dimensional plane flows and three-dimensional axi-symmetric flow through a circular drain are examined. Shallow water theory indicates the presence of limiting configurations, where the surface of the fluid collapses directly into the sink. In addition, this theory suggests that some previously computed steady solutions may be unstable.

The flow between two eccentric rotating cylinders with a slotted sleeve placed around the inner cylinder is determined numerically using an exponentially fitted finite-volume method. The flow field is determined for various Reynolds numbers, eccentricities and rotational speeds for the cases when the cylinders rotate in the same sense and rotate in opposite senses. The flow field developed when both cylinders rotate in the same sense is characterised, for sufficiently large eccentricity and rotational rate, by two counter-rotating eddies. Only one eddy is observed when the cylinders rotate in opposite senses. The presence of these eddies restricts the flow through the slotted sleeve in the former case but encourages through flow in the latter. For both cases, the eccentricity affects the location of the eddies, while changing the relative rotational rate only affects the eddy location for the case when the cylinders rotate in opposite directions. The change in Reynolds number has little effect on the flow field for the problems considered here. The vorticity generated by the slotted sleeve is convected into the main body of the flow field. No inviscid core within the main body of the flow field is observed for the range of Reynolds number considered.

This work deals with low-frequency asymptotic solutions using the method of matched asymptotic expansions. It is based on two papers by Buchwald [3] and Buchwald and Tran Cong [4] who studied the diffraction of elastic waves by a small circular cavity and a small elliptic cavity, respectively, in an otherwise unbounded domain. Here we clarify and systematize some aspects of their work and extend it to the diffraction of elastic waves by a small cylindrical cavity with a hypotrochoidal boundary. Results for the case of an incident P-wave are compared, in the special case of an elliptic boundary, with the results from the numerical solution of the boundary integral equation method.

Finite amplitude oscillatory convection rolls in the form of travelling waves are studied for a horizontal layer of a low Prandtl number fluid heated from below and rotating rapidly about a vertical axis. The results of the stability and nonlinear analyses indicate that there is no subcritical instability and that the oscillatory rolls are unstable for the ranges of the Prandtl number and the rotation rate considered in this paper.

In this paper, we consider the existence of a family of periodic solutions of large amplitude when a pair of eigenvalues of the linear part of a first-order system of ordinary differential equations crosses the imaginary axis. We refer to this problem as a Hopf bifurcation problem at infinity. In our work, the nonlinearities may be discontinuous at the origin, and the proof of existence of periodic solutions is arrived at through the corresponding system of integral equations. The applicability of the result is demonstrated by the study of the dynamics of a train truck wheelset system.

We prove a continuous version of a relaxation theorem for the nonconvex Darboux problem xlt ε F(t, τ, x, xt, xτ). This result allows us to use Warga's open mapping theorem for deriving necessary conditions in the form of a maximum principle for optimization problems with endpoint constraints. Neither constraint qualification nor regularity assumption is supposed.

Two sums given by J. B. Miller are evaluated in terms of classical hypergeometric results.

A method for solving quasilinear parabolic equations of the types

that differs radically from previously known methods is proposed. For each initial-boundary-value problem of one of these types that has boundary conditions of the first kind (second kind), a conjugate initial-boundary-value problem of the other type that has boundary conditions of the second kind (first kind) is defined. Based on the relations connecting the solutions of a pair of conjugate problems, a series of parabolic equations with constant coefficients that do not change step to step is constructed. The method proposed consists in calculating the solutions of the equations of this series. It is shown to have linear convergence. Results of a series of numerical experiments in a finite-difference setting show that one particular implementation of the proposed method has a smaller domain of convergence than Newton's method but that it sometimes converges faster within that domain.

The existence of stable periodic oscillatory solutions in a two species competition model with time delays is established using a combination of Hopf-bifurcation theory and the asymptotic method of Krylov, Bogoliuboff and Mitropoisky.

In this paper we consider a projective connection as defined by the nth-order Adler-Gelfand-Dikii (AGD) operator on the circle. It is well-known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar nth-order differential (AGD) operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for n ≤ 4) under the action of Vect(S1). The solutions of the AGD operator define an immersion R → RPn−1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ(n), for n ≤ 4.

A system of new integral equations is presented. They are derived from Maxwell's equations and describe radio-frequency (RF) current densities on a two-dimensional flat plate. The equations are generalisations of Pocklington's integral equation showing phase-retardation in two dimensions. These singular equations are solved, numerically, for the case of one-dimensional geometry. The solutions are shown to display effects which correspond to damped resonance when the wavelength of the current matches aspects of the geometry of the conductor.

A class of linear systems subject to sudden jumps in parameter values is considered. To solve this class of stochastic control problem, we try to seek the best feedback control law depending only on the measurable output. Based on this idea, we convert the original problem into an approximate constrained deterministic optimization problem, which can be easily solved by any existing nonlinear programming technique. An example is solved to illustrate the efficiency of the method.

Two-dimensional free-surface flows produced by a submerged source in a fluid of infinite depth are considered. It is assumed that the point on the free surface just above the source is a stagnation point and that the fluid outside two shear layers is at rest. The free-surface profile and the shape of the shear layers are determined numerically by using a series-truncation method. It is shown that there is a solution for each value of the Froude number F > 0. When F tends to infinity, the flow also describes a thin jet impinging in a fluid at rest.