Some multipoint iterative methods without memory, for approximating simple zeros of functions of one variable, are described. For m > 0, n ≧ 0, and k satisfying m + 1 ≧ k > 0, there exist methods which, for each iteration, use one evaluation of f, f′, … f(m) followed by n evaluations of f(k), and have order of convergence m + 2n + 1. In particular, there are methods of order 2(n + 1) which use one function evaluation and n + 1 derivative evaluations per iteration. These methods naturally generalize the known cases n = 0 (Newton's method) and n = 1 (Jarratt's fourth-order method), and are useful if derivative evaluations are less expensive than function evaluations. To establish the order of convergence of the methods we prove some results, which may be of independent interest, on orthogonal and “almost orthogonal” polynomials. Explicit, nonlinear, Runge-Kutta methods for the solution of a special class of ordinary differential equations may be derived from the methods for finding zeros of functions. The theoretical results are illustrated by several numerical examples.

The problem of planning the annual intakes to a university course, in which there are capacity constraints on the total enrolment, so as to produce a steady transition into an eventual no-growth situation is formulated as a linear program. The special structure of the problem is exploited to find a particular, optimal solution and to show that the addition of integrality constraints on the intakes poses no additional difficulty. The usefulness of the proposed methods is illustrated with an example from the University of Adelaide.

A plane strain or plane stress configuration of an inextensible transversely isotropic linear elastic solid with the axis of symmetry in the plane, leads to a harmonic lateral displacement field in stretched coordinates. Various displacement and mixed displacement-traction boundary conditions yield standard boundary-value problems of potential theory for which uniqueness and existence of solutions are well established. However, the important case of prescribed tractions at each boundary point gives a non-standard potential problem involving linking of boundary values at opposite ends of chords parallel to the axis of material symmetry. Uniqueness and existence of solutions, within arbitrary rigid motions, are now established for the traction problem for general domains.

In this paper, the procedure of the clinical measurement of blood pressure is modelled by the application of a uniform pressure band to a long, homogeneous, isotropic cylinder. The deformations are assumed to be infinitesimal, and transform methods are used to analyse the resulting equations. The inversion of the resulting transforms is carried out numerically. It is shown that, in spite of the fairly crude assumptions of the model, the actual load on the artery may be markedly different from that applied to the surface, leading to inaccuracies in the measured blood pressure. The parameter of importance is shown to be the ratio of pressure band width to arm diameter.

A correspondence is established between flows of air above stationary water, and flows of water below air at atmospheric pressure. Flows in the latter category are well studied, and all such hydrodynamic flows can be “turned upside-down” to generate flows of air in which the free surface deforms under gravity, due to a balance between aerodynamic and hydrostatic pressures. Examples are given of some exact inverse solutions, and a general semi-inverse approach is outlined for numerical solutions via an integral formulation.

The linear long-wave equations with (and without) small ground motion are considered. The governing equations are represented in a matrix from and transformations are sought which reduce the system to (for example) a form associated with the conventional wave equation. Integration of the system is then immediate. It is shown that such a reduction may be acheived provided the variation in water depth is specified by certain multi-parameter forms.

The hydrodynamic pressure forces acting upon a slender fish are derived for the case of a fish swimming in a non-uniform velocity field. Possible applications are the effects on fish propulsion of swimming in waves, in turbulent eddies, and in the presence of other fish or a moving ship. The fish is assumed to be a slender body, with no vorticity shed into the fluid except at a single abrupt trailing edge located at the posterior end of the fish, and to be performing small lateral swimming undulations of its body. The non-uniform field through which the fish swims is assumed to be irrotational, and this field as well as the body undulations must be slowly-varying on the length-scale of the lateral fish dimensions. Expressions are derived for the local force and the time-averaged total thrust force. These are applied to the study of steady-state bow-riding and wave-riding of porpoises.

In 1962 Lakshmikantham ([1], [2]) extended the concept of extreme stability (e.g. [4]) of a system described by an ordinary differential equation, not necessarily with uniqueness, to relative stability of two such systems. Here we show the restrictiveness of his definition of relative stability in that it implies not only are the solutions of two systems unique for each initial condition, they are in fact identical. We then introduce and give an example of a weaker version of relative stability which is of some interest for control systems. For greater simplicity and generality we use Roxin's attainability set defined General Control Systems [3] to describe the dynamics of our systems, as they subsume both ordinary differential equations without uniqueness and ordinary differential control equations.

It is shown that for the three dimensional Ising model with dipole-dipole interactions, the thermodynamic limit of the free energy with simple boundary conditions is not the same as the thermodynamic limit of the free energy with periodic boundary conditions. A variational principle is developed to connect the two free energies.