This paper investigates the effect of heat transfer on the motion of a spherical bubble in the vicinity of a rigid boundary. The effects of heat transfer between the bubble and the surrounding fluid, and the resulting loss of energy from the bubble, can be incorporated into the simple spherical bubble model with the addition of a single extra ordinary differential equation. The numerical results show that for a bubble close to an infiniterigid boundary there are significant differences in both the radius and Kelvin impulse of the bubble when the heat transfer effects are included.

A discrete spatial model of a multi-species environment is formulated, and the behaviour of the system is studied. The model is used to explore stability and resilience of biological systems and discuss how they are dependent on spatial scale chosen.

A finite transverse shock wave propagates through an unbounded medium consisting of two joined incompressible elastic half-spaces of different material properties, in the direction normal to the plane interface. A semi-inverse method is used to examine the reflection-transmission of this wave at the interface. It is found that, depending on the material properties, the reflected wave is either a simple wave or a shock; the transmitted wave is always a shock.

A variation-of-constants formula is obtained for a linear abstract evolution equation in Hilbert space with unbounded perturbation and free term. As an application, the state of the abstract controlled system with unbounded mixed controls is explicitly given.

Many gravity driven flows can be modelled as homogeneous layers of inviscid fluid with a hydrostatic pressure distribution. There are examples throughout oceanography, meteorology, and many engineering applications, yet there are areas which require further investigation. Analytical and numerical results for two-layer shallow-water formulations of time dependent gravity currents travelling in one spatial dimension are presented. Model equations for three physical limits are developed from the hydraulic equations, and numerical solutions are produced using a relaxation scheme for conservation laws developed recently by S. Jin and X. Zin [6]. Hyperbolicity of the model equations is examined in conjunction with the stability Froude number, and shock formation at the interface of the two layers is investigated using the theory of weakly nonlinear hyperbolic waves.

The general solution of the rth inhomogeneous linear difference equation is given in the form

The coefficients , i = 2, …, r, and b(n−r)(n) can be evaluated from n values , k = 0, …, n − 1, which santisfy an rth order homogenous linear difference equation. In the rth order homogeneous case and if n ≥ 2r, the method requires the evaluation of r determinants of successive orders n − 2r + 1, n − 2r + 2, …, n − r. If r ≤ n ≤ 2r − 1, only n − r determinants are required, with orders varying from 1 to n − r. In the second order ihnomogenous case, can be evaluated from a continued fraction amd a simple product.

An integral equation for the normal velocity of the interface between two immiscible fluids flowing in a two-dimensional porous medium or Hele-Shaw cell (one fluid displaces the other) is derived in terms of the physical parameters (including interfacial tension), a Green's function and the given interface. When the displacement is unstable, ‘fingering’ of the interface occurs. The Saffman-Taylor interface solutions for the steady advance of a single parallel-sided finger in the absence of interfacial tension are seen to satisfy the integral equation, and the error incurred in that equation by the corresponding Pitts approximating profile, when interfacial tension is included, is shown. In addition, the numerical solution of the integral equation is illustrated for a sinusoidal and a semicircular interface and, in each case, the amplitude behaviour inferred from the velocity distribution is consistent with conclusions based on the stability of an initially flat interface.

The Wiener-Hopf technique is applied to the quasi-linear infiltration problem of flow from a shallow half-plane pond. Fully-saturated conditions hold immediately under the pond, while on the surface away from the pond the linearised evaporative loss is assumed to be proportional to the local relative permeability.

Evaporation from the non-wetted region increases the water flow from the pond into the soil, thereby coupling to the effects from capillarity. Linearised evaporation introduces an additional length scale and additional logarithmic expressions to those derived previously. The total rate of volumetric flow into the soil from the pond per unit length of perimeter, in addition to the usual gravity flow, increases somewhat slowly as evaporation increases. The most extreme case considered in this paper yielded an additional flow rate 63% greater than that obtained in the absence of evaporation.

The interaction between evaporation and capillarity is enhanced in poorlydraining soils, where the reduced ability to transmit liquid water need not be compensated by a corresponding reduction in evaporative losses. However, in freelydraining soils the interaction between evaporation and capillarity is probably small.

We analyse the various integrability criteria which have been proposed for discrete systems, focusing on the singularity confinement method. We present the exact procedure used for the derivation of discrete Painlevé equations based on the deautonomisation of integrable autonomous mappings. This procedure is then examined in the light of more recent criteria based on the notion of the complexity of the mapping. We show that the low-growth requirements lead, in the case of the discrete Painlevé equations, to exactly the same results as singularity confinement. The analysis of linearisable mappings shows that they have special growth properties which can be used in order to identify them. A working strategy for the study of discrete integrability based on singularity confinement and low-growth considerations is also proposed.

Standard results for boundary value problems involving second-order ordinary differential equations ensure that the existence of a well-ordered pair of lower and upper solutions together with a Nagumo condition imply existence of a solution. In this note we introduce some examples which show that existence is not guaranteed if no Nagumo condition is satisfied.

A reaction-diffusion equation with non-constant diffusivity,

ut = (D(x, t)ux)x + F(u),

is studied for D(x, t) a continuous function. The conditions under which the equation can be reduced to an equivalent constant diffusion equation are derived. Some exact forms for D(x, t) are given. For D(x, t) a stochastic function, an explicit finite difference method is used to numerically determine the effect of randomness in D(x, t) upon the speed of the reaction wave solution to Fisher's equation. The extension to two spatial dimensions is considered.

A minimum Hamiltonian completion of a graph G is a minimum-size set of edges that, when added to G, guarantee a Hamiltonian path. Finding a Hamiltonian completion has applications to frequency assignment as well as distributed computing. If the new edges are deleted from the Hamiltonian path, one is left with a minimum path cover, a minimum-size set of vertex-disjoint paths that cover the vertices of G. For arbitrary graphs, constructing a minimum Hamiltonian completion or path cover is clearly NP-hard, but there exists a linear-time algorithm for trees. In this paper we first give a description and proof of correctness for this linear-time algorithm that is simpler and more intuitive than those given previously. We show that the algorithm extends also to unicyclic graphs. We then give a new method for finding an optimal path cover or Hamiltonian completion for a tree that uses a reduction to a maximum flow problem. In addition, we show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles, that is, a 2-factor.

A continuous one-parameter set of binary operators on L2(R) called fractional convolution operators and which includes those of multiplication and convolution as particular cases is constructed by means of the Condon-Bargmann fractional Fourier transform. A fractional convolution theorem generalizes the standard Fourier convolution theorems and a fractional unit distribution generalizes the unit and delta distributions. Some explicit double-integral formulas for the fractional convolution between two functions are given and the induced operation between their corresponding Wigner distributions is found.

A spectral method is used to consider the porous medium combustion in an infinite slab. The infinite system of ordinary differential equations for the amplitude functions is truncated and comparisons are made for different numbers of modes included in the numerical computation. It is shown that the qualitative behaviour of the solution is captured by the first eigenmode. Dependence of the solution on initial data and a parameter is also considered.

Applying Ekeland's variational principle in this paper, we obtain a maximum principle for optimal control for a class of two-point boundary value controlled systems. The control domain need not be convex. For a special case, that is the so called LQ-type problem, we obtain the optimal control in the closed loop form and a corresponding Riccati type differential equation.

Knowledge about the foliage angle density g(α) of the leaves in the canopy of trees is crucial in foresty mangement, modelling canopy reflectance, and environmental monitoring. It is usually determined from observations of the contact frequency f(β) by solving a version of the first kind Fredholm integral equation derived by Reeve (Appendix in Warren Wilson [22]). However, for inference purposes, the practitioner uses functionals defined on g(α), such as the leaf area index F, rather than g(α) itself. Miller [12] has shown that F can be computed directly from f(β) without solving the integral equation. In this paper, we show that his result is a special case of a general transformation for linear functionals defined on g(α). The key is the existence of an alternative inversion formula for the integral equation to that derived by Miller [11].

In this paper the scattered progressive waves are determined due to progressive waves incident normally on certain types of partially immersed and completely submerged vertical porous barriers in water of infinite depth. The forms are approximate only, and are obtained using perturbation theory for nearly hard or soft barriers having high and low porosities respectively. The results for arbitrary porosity are difficult to obtain, in contrast to the well known hard limit of impermeable barriers.