In industrial applications of microwave heating, it has been observed that rather than the heating taking place uniformly, regions of high temperature, called hot-spots, tend to form. Depending on the industrial application, these can be either desirable or undesirable, and hence a theoretical understanding of the properties of the material that lead to hotspot formation is necessary. It has been shown in previous studies that hotspot formation is a product of the nonlinear dependence of microwave energy absorption by the material on temperature. It is shown in the present work that the conductivity of the material can have a significant effect on hotspot formation and can, if large enough, stop a hotspot from forming.

In this paper we consider the asymptotics of extinction for the nonlinear diffusion reaction equation

with non-negative initial data possessing finite support. For t > 0, both solution and support vanish as t → T and x → x0. With T as the extinction time we construct the asymptotic solution as τ = T – t → 0 near the extinction point x0 using matched expansions. Taking x0= 0, we first form an outer expansion valid when η =xt–(m–p)/2 (1–p) = 0(1). This is nonuniformly valid for large |η| and has to be replaced by an intermediate expansion valid for |x| = O(τ−1/l0) where l0 is an even integer greater than unity. If p + m ≥ 2 this expansion is uniformly valid while for p + m < 2, there are regions near the edge of the support where diffusion becomes important. The zero order solution in these inner regions is discussed numerically.

We present new exact solutions for the flow of liquid during constant-rate expression from a finite thickness of liquid-saturated porous material with nonlinear properties. By varying a single nonlinearity parameter and a dimensionless expression rate, we systematically investigate the effect of nonlinearity and of an impermeable barrier (e.g. a piston). We illustrate the water profile shape and the water ratio deficit at the expression surface (e.g. a filter membrane) as a function of time.

In this paper, a model for lateral dispersion in open-channel flow is studied involving a diffusion equation which has a nonlinear term describing the effect of buoyancy. The model is used to investigate the interaction of two buoyant pollutant plumes. An approximate analytic technique involving Hermite polynomials is applied to the resulting PDEs to reduce them to a system of ODEs for the centroids and widths of the two plumes. The ODEs are then solved numerically. A rich variety of behaviour occurs depending on the relative positions, widths and strengths of the initial discharges. It is found that for two plumes of equal strength and width discharged side-by-side, the plumes move apart and the rate of spreading is inhibited by their interaction, whereas when one plume is initially much wider than the other, both plumes tend to drift to the side of the narrower plume. Finally, the PDEs are solved numerically for two sets of initial conditions and a comparison is made with the ODE solutions. Agreement is found to be good.

A mixed boundary-valued problem associated with the diffusion equation, that involves the physical problem of cooling of an infinite slab in a two-fluid medium, is solved completely by using the Wiener-Hopf technique. An analytical solution is derived for the temperature distribution at the quench fronts being created by two different layers of cold fluids having different cooling abilities moving on the upper surface of the slab at constant speed. Simple expressions are derived for the values of the sputtering temperatures of the slab at the points of contact with the respective layers, assuming one layer of the fluid to be of finite extent and the other of infinite extent. The main problem is solved through a three-part Wiener-Hopf problem of a special type, and the numerical results under certain special circumstances are obtained and presented in the form of a table.

We solve the problem of finding a simultaneous matrix normal form for an element of the Lie algebra o(p, q) and the underlying indefinite inner product. The results are used to determine several classes of classical Hamiltonian dynamical systems which possess a first integral linear in the momentum variables.

In this paper, we study the existence of periodic solutions of the NDDE (neutral differential difference equation):

where τ > 0 and c is a real number. We obtain a sufficient condition under which (*) has at least k nonconstant oscillatory periodic solutions.

In a paper by Teo and Jennings, a constraint transcription is used together with the concept of control parametrisation to devise a computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints of inequality type. The aim of this paper is to extend the results to a more general class of constrained optimal control problems, where the problem is also subject to terminal equality state constraints. For illustration, a numerical example is included.

Extending earlier duality results for multiobjective programs, this paper defines dual problems for convex and generalised convex multiobjective programs without requiring a constraint qualification. The duals provide multiobjective extensions of the classical duals of Wolfe and Schechter and some of the more recent duals of Mond and Weir.