Exact solutions are developed for instantaneous point sources subject to nonlinear diffusion and loss or gain proportional to nth power of concentration, with n > 1. The solutions for the loss give, at large times, power-law decrease to zero of slug central concentration and logarithmic increase of slug semi-width. Those for gain give concentration decreasing initially, going through a minimum, and then increasing, with blow-up to infinite concentration in finite time. Slug semi-width increases with time to a finite maximum in finite time at a blow-up. Taken in conjunction with previous studies, these new results provide an overall schema for instantaneous nonlinear diffusion point sources with nonlinear loss or gain for the total range n ≥ 0. Six distinct regimes of behaviour of slug semi-width and concentration are identified, depending on the range of n, 0 ≤ n < 1, n = 1, or n > 1. Three of them are for loss, and three for gain. The classical Barenblatt-Pattle nonlinear instantaneous point-source solutions with material concentration occupy a central place in the total schema.

Using the fact that a differentiable quasi-convex function is also pseudo-convex at every point x of its domain where ∇f(x) ≠ 0 recent results relating different forms of convexity and invexity are strengthened.

In this article, we generalise Newton's diagram method for finding small solutions ξ(λ) of equations f (ξ,λ) = 0 (0,0) = 0 with f analytic (see [1, 2, 4, 6]) to the case of a multi-dimensional function f, unknown variable ζ and small parameter λ. This method was briefly described in [1]. The method has many different applications and allows one to solve some inflexible problems. In particular, the method can be used in very difficult bifurcation problems, for example, for systems with small imperfections.

The essential aspects of the Boundary Integral Equation Method for the numerical solution of elliptic type boundary value problems are presented. A numerical example for a stress concentration problem in classical elasticity in three dimensions is given along with several examples for a class of scalar problems in elastic torsion of non-cylindrical bars. Some discussion and criticism of the method itself and in comparison with widely used field methods is also presented.

It is observed (among other things) that a theorem on bilinear and bilateral generating functions, which was given recently in the predecessor of this Journal, does not hold true as stated and proved earlier. Several possible remedies and generalizations, which indeed are relevant to the present investigation of various other results on bilinear and bilateral generating functions, are also considered.

A k-out-of-N:G reparable system with an arbitrarily distributed repair time is studied in this paper. We translate the system into an Abstract Cauchy Problem (ACP). Analysing the spectrum of the system operator helps us to prove the well-posedness and the asymptotic stability of the system.

The finite element method can be used to provide network models of distribution problems. In the present work ‘flow ratio design’ is applied to such models to obtain approximate minima and maxima for both the primal and dual FEM models. The resulting primal MIN and dual MAX solutions are equal to or close to the exact solutions but, intriguingly, the primal MAX and dual MIN solutions are approximately equal to an intermediate saddle point solution.

In this paper we consider the flow of an incompressible, inviscid and homogeneous fluid over two obstacles in succession. The flow is assumed irrotational and solutions are sought in which a hydraulic fall occurs over the first obstacle with supercritical flow over the second. The method used to solve the problem is capable of calculating flows over topography of any shape.

In a recent paper the authors give upper and lower bounds for the motion of the moving boundary for the classical Stefan problem for plane, cylindrical and spherical geometries. On comparison with the exact Neumann solution for the plane geometry and no surface radiation the bounds obtained are seen to be quite adequate for practical purposes except for the lower bound at small Stefan numbers. Here improved lower bounds are obtained which in some measure remove this inadequacy. Time dependent surface conditions are also examined and the new lower bounds obtained for the classical problem are illustrated numerically.

We study the propagation of electromagnetic waves (EMWs) in both isotropic and anisotropic ferromagnetic material media. As the EMW propagates through linear charge-free isotropic and anisotropic ferromagnetic media, it is found that the magnetic field and the magnetic induction components of the EMW and the magnetization excitations of the medium are in the form of solitons. However, the electromagnetic soliton gets damped and decelerates in the case of a charged medium. In the case of a charge-free nonlinear ferromagnetic medium we obtain results similar to those for the linear case.

Modified versions of the Euler midpoint formula are given for functions whose derivatives are either functions of bounded variation, Lipschitzian functions or functions in Lp-spaces. The results are applied to quadrature formulae.

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.

Diffusion in the presence of high-diffusivity paths is an important issue of current technology. In metals, high-diffusivity paths are identified with dislocations, grain boundaries, free surfaces and internal microcracks. In pourous media such as rocks, fissures provide a system of high-flow paths. Recently, based on a continuum approach, these phenomena have been modelled, resulting in coupled systems of partial differential equations of parabolic type for the concentrations in bulk and in the high-diffusivity paths. This theory assumes that each point of the medium is simultaneously occupied by more than one diffusion or flow path. Here a simple discrete random walk model of diffusion in a medium with double diffusivity is given.

The free surface due to a submerged source in a fluid of finite depth at infinite Froude number is reconsidered. A conformal transformation technique is used to formulate this problem as an integral equation for the free-surface angle. An elementary solution is found for the equation, which results in a closed form expression for the free-surface elevation. Comparison is made with previous numerical solutions.

This paper demonstrates feasibility of aerodynamically-supported motion of a thin sheet near to a plane wall. Steady equilibrium is possible for uniform sheets only if they are deformable, and a set of possible equilibrium shapes is determined.