The problem of transient two-dimensional transport by diffusion and advection of a decaying contaminant in two adjacent porous media is solved using a boundary-integral method. The method requires the construction of appropriate Green's functions. Application of Green's theorem in the plane then yields representations for the contaminant concentration in both regions in terms of an integral of the initial concentration over the region's interior and integrals along the boundaries of known quantities and the unknown interfacial flux between the two adjacent media. This flux is given by a first-kind integral equation, which can be solved numerically by a discretisation technique. Examples of contaminant transport in fractured porous media systems are presented.

Competition between a finite number of searching insect parasites is modelled by differential equations and birth-death processes. In the one species case of intraspecific competition, the deterministic equilibrium is globally stable and, for large populations, approximates the mean of the stationary distribution of the process. For two species, both inter- and intraspecific competition occurs and the deterministic equilibrium is globally stable. When the birth-death process is reversible, it is shown that the mean of the stationary distribution is approximated by the equilibrium. Confluent hypergeometric functions of two variables are important to the theory.

The constancy in time of the ratio of unidirectional tracer fluxes, passing in opposite directions through a membrane that has transport properties varying arbitrarily with the distance from a boundary face, has been established recently for successively more sophisticated mathematical models of tracer transport within the membrane. Such results are important in that, when constancy is not observed experimentally, inferences can be drawn about the dimensionality of distributions of transport properties of the membrane. The known theoretical results are shown here to follow from much more general theorems, valid for a wide class of models based on linear-operator equations, including elliptic and hyperbolic partial differential equations as well as the essentially parabolic equations of interest in membrane transport problems. These theorems have the general character of “reciprocity theorems” known for a long time in other areas, such as mechanics, acoustics and elasticity. The general results obtained here clarify the conditions on membrane properties under which constancy of a flux ratio can be expected. In addition, flux ratio theorems of a new type are proved to hold under suitable conditions, for the normal components of flux vectors at points on either side of a membrane, as distinct from previously established theorems for total fluxes through membrane faces. Possible new experiments are suggested by the analysis.

The problem of determining a square integrable function from its modulus and that of its Fourier transform has been considered in an article by Corbett and Hurst ([1]). In this work we point out an error in one of the main results of the cited article concerning the Pauli uniqueness of real states, provide a proof of uniqueness for non-negative states, and present various related examples and discussion of the problem.

A point on a tree network space is said to be a distant point if it maximises its minimum weighted distance from any of its vertices. The median minimises the sum of its weighted distances from the vertices. In this paper two constrained problems are discussed. The first problem is to maximise the minimum of the weighted distances from the vertices subject to an upper bound value of the sum of the weighted distances from the vertices, while the second problem is to minimise the sum of the weighted distances subject to a lower bound value of the minimum weighted distance to any of its vertices. It is shown that these two constrained problems are duals of each other in a well defined sense.

The problem of obtaining explicit and exact solutions of soliton equations of the AKNS class is considered. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system; that is, a linear eigenvalue problem in the form of a system of first order partial differential equations. The method of characteristics is used and Bäcklund transformations are employed to generate new solutions from the old. Thus, families of new solutions for the KdV equation, the mKdV equation, the sine-Gordon equation and the nonlinear Schrôdinger equation are obtained, avoiding the solution of some Riccati equations. Our results in the KdV case include those obtained recently by other investigators.

We introduce assumptions in input optimisation that simplify the necessary conditions for an optimal input. These assumptions, in the context of nonlinear programming, give rise to conceptually new kinds of constraint qualifications.

A simple rigorous approach is given to finding boundary conditions for the adjoint differential equation in an optimal control problem. The boundary conditions for a time-optimal problem are calculated from the simpler conditions for a fixed-time problem.

In this paper, we consider a class of combined optimal parameter selection and optimal control problems with general constraints. The first aim is to provide a unified approach to the numerical solution of this general class of optimisation problems by using the control parametrisation technique. This approach is supported by some convergence results. The second aim is to show that several different classes of optimal control problems can all be transformed into special cases of the problem considered in this paper. For illustration, four numerical examples are presented.