We use the “Brownian Bridge” of Schrödinger to model a statistical search problem in which the initial and final distributions of a random motion are given. We raise the question of how to use this information to optimally reconstruct a likely past event.

The calculation of flows in pipe networks and in networks of mine shafts and the calculations of the currents in electrical circuits can be represented as variational problems. There are two approaches: the nodal method and the loop method. There is a variational representation for each of these. This paper describes the relationship between the two representations and in particular shows that the loop formulation is the Wolfe dual of the nodal formulation after the application of Legendre transformations to the variables and to the objective function.

We consider the epidemic model with subpopulations introduced in Hethcote [5]. It is shown that if the endemic equilibrium exists, then the system is uniformly persistent. Moreover, the endemic equilibrium is globally asymptotically stable under the assumption of small effective contact rates between different subpopulations.

Following work in an earlier paper, the theory of finite deformation of elastic membranes is applied to the problem of two initially-circular semi-infinite cylindrical membranes of the same radius but of different material, joined longitudinally at a cross-section. The body is inflated by constant interior pressure and is also extended longitudinally. The exact solution found for an arbitrary material is now specialised to the orthotropic case, and the results are interpreted for forms of the strain-energy function introduced by Vaishnav and by How and Clarke in connection with the study of arteries. Also considered in this context is the similar problem where two semi-infinite cylindrical membranes of the same material are separated by a cuff of different material. Numerical solutions are obtained for various pressures and longitudinal extensions. It is shown that discontinuities in the circumferential stress at the joint can be reduced by suitable choice of certain coefficients in the expression defining the strain-energy function. The results obtained here thus solve the problem of static internal pressure loading in extended dissimilar thin orthotropic tubes, and may also be useful in the preliminary study of surgical implants in arteries.

By a systematic search for Lie-Bäetcklund symmetries, a class of linearisable reaction-diffusion equations is obtained that has, as a canonical form, ut = u2uxx + 2u2. One such nonlinear equation is θt = ∂x[a(b - θ)-2 θx] - ma(b -θ)-2 θx - q exp(-mx). This represents an extension of Fokas-Yortsos-Rosen equation (q = 0) to incorporate a reaction term. It is relevant to the modelling of unsaturated flow in a soil with a volumetric extraction mechanism, such as a web of plant roots. Here, a reciprocal transformation is used to solve a nonlinear boundary-value problem for transient flow into a finite layer of a soil subject to a constant flux boundary condition to compensate for such water extraction.

The problem of radially directed fluid flow through a deformable porous shell is considered. General nonlinear diffusion equations are developed for spherical, cylindrical and planar geometries. Solutions for steady flow are found in terms of an exact integral and perturbation solutions are also developed. For unsteady flow, perturbation methods are used to find approximate small-time solutions and a solution valid for slow compression rates. These solutions are used to investigate the deformation of the porous material with comparisons made between the planar and the cylindrical geometries.

The Wiener-Hopf technique is applied to the quasi-linear infiltration problem of flow from a shallow half-plane pond. Fully-saturated conditions hold immediately under the pond, while on the surface away from the pond the linearised evaporative loss is assumed to be proportional to the local relative permeability.

Evaporation from the non-wetted region increases the water flow from the pond into the soil, thereby coupling to the effects from capillarity. Linearised evaporation introduces an additional length scale and additional logarithmic expressions to those derived previously. The total rate of volumetric flow into the soil from the pond per unit length of perimeter, in addition to the usual gravity flow, increases somewhat slowly as evaporation increases. The most extreme case considered in this paper yielded an additional flow rate 63% greater than that obtained in the absence of evaporation.

The interaction between evaporation and capillarity is enhanced in poorlydraining soils, where the reduced ability to transmit liquid water need not be compensated by a corresponding reduction in evaporative losses. However, in freelydraining soils the interaction between evaporation and capillarity is probably small.

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.

When a line sink is placed beneath the free surface of an otherwise quiescent fluid of infinite depth, two different flow types are now known to be possible. One type of flow involves the fluid being drawn down toward the sink, and in the other type, a stagnation point forms at the surface immediately above the position of the sink.

This paper investigates the second of these two flow types, which involves a free-surface stagnation point. The effects of surface tension are included, and even when small, these are shown to have a very significant effect on the overall solution behaviour. We demonstrate by direct numerical calculation that there are regions of genuine non-uniqueness in the nonlinear solution, when the surface-tension parameter does not vanish. In addition, an asymptotic solution valid for small Froude number is derived.