It is well known that transcritical flow past an obstacle may generate undular bores propagating away from the obstacle. This flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical simulations and asymptotic analyses have shown that the unsteady undular bores are connected by a locally steady solution over the obstacle. In this paper we present an overview of the underlying theory, together with some recent work on the case where the obstacle has a large width.

High tuberculosis (TB) prevalence in Papua New Guinea (PNG) is a serious public health concern. The epidemic in this region is exacerbated by the presence of drug-resistant TB strains as well as HIV infection. This presents a public health threat not only locally but also to Australia due to the high potential for cross-border transmission between PNG’s Western Province and the Australian Torres Strait Islands. We present two mathematical models of TB in the Western Province: a simple model of the underlying TB dynamics, and a detailed model which accounts for the additional effects of HIV and drug resistance. The detailed model is used to make quantitative predictions about the impact of expanding the TB case detection rate under the Directly Observed Treatment, Short-course treatment regimen. This paper provides a framework for future investigation into the economic costs and public health benefits of potential TB interventions in this region, with the eventual aim of providing recommendations to guide policy makers in both PNG and Australia.

A boundary-value problem for cell growth leads to an eigenvalue problem. In this paper some properties of the eigenfunctions are studied. The first eigenfunction is a probability density function and is of importance in the cell growth model. We sharpen an earlier uniqueness result and show that the distribution is unimodal. We then show that the higher eigenfunctions have nested zeros. We show that the eigenfunctions are not mutually orthogonal, but that there are certain orthogonality relations that effectively partition the set of eigenfunctions into two sets.

We investigate natural convection cooling of the fluid in a drink can placed in a refrigerator by simulating the full combined boundary layer system on the can wall. The cylindrical can is filled with water at initial nondimensional temperature 0, and located within a larger cylindrical container filled with air at initial temperature −1. The outer container walls are maintained at constant temperature −1. Initially both fluids are at rest. Two configurations are examined: the first has the inner can placed vertically in the middle of the outer container with no contact with the outer container walls, and the second has the inner can placed vertically at the bottom of the outer container. The results are compared to those obtained by assuming that the inner can walls are maintained at a constant temperature, showing similar basic flow features and scaling relations, but with very different proportionality constants.

This paper examines an antiplane crack problem for a functionally graded anisotropic elastic material in which the elastic moduli vary quadratically with the spatial coordinates. A solution to the crack problem is obtained in terms of a pair of integral equations. An iterative solution to the integral equations is used to examine the effect of the anisotropy and varying elastic moduli on the crack tip stress intensity factors and the crack displacement.

A preconditioned iterative method for the two-dimensional Helmholtz equation with Robbins boundary conditions is discussed. Using a finite-difference method to discretize the Helmholtz equation leads to a sparse system of equations which is too large to solve directly. The approach taken in this paper is to precondition this linear system with a sine transform based preconditioner and then solve it using the generalized minimum residual method (GMRES). An analytical formula for the eigenvalues of the preconditioned matrix is derived and it is shown that the eigenvalues are clustered around 1 except for some outliers. Numerical results are reported to demonstrate the effectiveness of the proposed method.

This paper concerns a nonlinear doubly degenerate reaction–diffusion equation which appears in a bacterial growth model and is also of considerable mathematical interest. A travelling wave analysis for the equation is carried out. In particular, the qualitative behaviour of both sharp and smooth travelling wave solutions is analysed. This travelling wave behaviour is also verified by some numerical computations for a special case.

The Exp-function method is applied to construct a new type of solution of the coupled (2+1)-dimensional nonlinear system of Schrödinger equations. It is shown that the method provides a powerful mathematical tool for solving nonlinear evolution equations in mathematical physics.