We prove the existence of solutions of Maxwell's equations for a conducting medium whose constitutive parameters are piecewise constant on R3, and then examine the convergence of these solutions in the quasi-static limit in which displacement currents are neglected. Secondly, we examine the regularity of the limiting solution and the sense in which the classical boundary conditions hold, namely, continuity of the tangential electric field and the normal current density.

Microwave heating of porous solid materials has received considerable attention in recent years because of its widespread use in industry. In this study, the microwave power absorption term is modelled as the product of an exponential temperature function with function that decays exponentially with distance. The latter describes the penetration of material by the microwaves.

We investigate the phenomena of multiplicity in class A geometries, paying particular attention to how the penetration function affects the behaviour of the system. We explain why the phase-plane techniques which have been used in the case when the penetration term is constant do not extend to non-constant penetration.

Experimental evidence shows that plane Couette flow becomes unstable when the Reynolds number R reaches certain critical values. Linear stability theory does not predict these observations and has been unable to locate these instabilities. A Chebyshev/QR numerical technique is used to investigate much higher values of R than those previously tested. In particular, values of R up to 108 are confidently tested, whereas previously values of R up to only 2 × 104 have been considered.

Planar hinged segmented bodies have been used to represent models of biomechanical systems. One characteristic of a segmented body moving under gravitational acceleration and torques between segments is the possibility that the body's segments spin through more than a revolution or past a natural limit, and a computational mechanism to stop such behaviour should be provided. This could be done by introducing angle constraints between segments, and computational models utilising optimal control are studied here. Three models to maintain angle constraints between segments are proposed and compared. These models are: all-time angle constraints, a restoring torque in the state equations and an exponential penalty model. The models are applied to a 2-D three-segment body to test the behaviour of each model when optimising torques to minimise an objective. The optimisation is run to find torques so that the end effector of the body follows the trajectory of a half-circle. The result shows the behaviour of each model in maintaining the angle constraints. The all-time constraints case exhibits a behaviour of not allowing torques (at a solution) which makes segments move past the constraints, while the other two show a flexibility in handling the angle constraints which is more similar to what occurs in a real biomechanical system.

We study the sequence of monic polynomials orthogonal with respect to inner product

$$\begin{eqnarray*}\langle p, q\rangle = \int \nolimits \nolimits_{0}^{\infty } p(x)q(x){e}^{- x} {x}^{\alpha } \hspace{0.167em} dx+ Mp(\zeta )q(\zeta )+ N{p}^{\prime } (\zeta ){q}^{\prime } (\zeta ),\end{eqnarray*}$$

$\alpha \gt - 1$

$M\geq 0$

$N\geq 0$

$\zeta \lt 0$

$p$

$q$

This paper deals with the flux identification problem for scalar conservation laws. The problem is formulated as an optimization problem, where the objective function compares the solution of the direct problem with observed profiles at a fixed time. A finite volume scheme solves the direct problem and a continuous genetic algorithm solves the inverse problem. The numerical method is tested with synthetic experimental data. Simulation parameters are recovered approximately. The tested heuristic optimization technique turns out to be more robust than classical optimization techniques.

We study a nonpreemptive scheduling on two parallel identical machines with a dedicated loading server and a dedicated unloading server. Each job has to be loaded by the loading server before being processed on one of the machines and unloaded immediately by the unloading server after its processing. The loading and unloading times are both equal to one unit of time. The goal is to minimize the makespan. Since the problem is NP-hard, we apply the classical list scheduling and largest processing time heuristics, and show that they have worst-case ratios, $8/5$ and $6/5$, respectively.

Inexact Uzawa algorithms for solving nonlinear saddle-point problems are proposed. A simple sufficient condition for the convergence of the inexact Uzawa algorithms is obtained. Numerical experiments show that the inexact Uzawa algorithms are convergent.

A simple model for the propagation of a combustion wave is proposed and the speed of propagation is predicted. It is assumed that the reactant ignites at a specified temperature and then burns until depleted with reaction rate dependent on temperature and reactant concentration. The exact solution and linear stability are determined in the case of constant heat generation and a numerical scheme is developed to generate traveling wave solutions in the more general case. This numerical method is applied to the case where the temperature dependence of the reaction rate is modeled by the Arrhenius function.

When studying deep convection in a compressible medium the effects of viscous dissipation can become important and must be taken into account in any realistic model. But even in shallow convection, for which the Boussinesq approximation is valid, the viscous dissipation effects will become important at high Rayleigh numbers. These effects are estimated with the help of asymptotic methods and the results are compared with those obtained by numerical integration.

Sufficient conditions for the controllability of nonlinear neutral Volterra integrodifferential systems with implicit derivative are established. The results are a generalisation of the previous results, through the notions of condensing map and measure of noncompactness of a set.

We analyse the asymptotic behaviour of a biological system described by a stochastic competition model with $n$ species and $k$ resources (chemostat model), in which the species mortality rates are influenced by the fractional Brownian motion of the extrinsic noise environment. By constructing a Lyapunov functional, the persistence and extinction criteria are derived in the mean square sense. Some examples are given to illustrate the effectiveness of the theoretical result.

Financial contracts with options that allow the holder to extend the contract maturity by paying an additional fixed amount have found many applications in finance. Closed-form solutions for the price of these options have appeared in the literature for the case when the contract for the underlying asset follows a geometric Brownian motion with constant interest rate, volatility and nonnegative dividend yield. In this paper, option price is derived for the case of the underlying asset that follows a geometric Brownian motion with time-dependent drift and volatility, which is more important for real life applications. The option price formulae are derived for the case of a drift that includes nonnegative or negative dividend. The latter yields a solution type that is new to the literature. A negative dividend corresponds to a negative foreign interest rate for foreign exchange options, or storage costs for commodity options. It may also appear in pricing options with transaction costs or real options, where the drift is larger than the interest rate.

The nonlinear and weakly dispersive Serre equations contain higher-order dispersive terms. These include mixed spatial and temporal derivative flux terms which are difficult to handle numerically. These terms can be replaced by an alternative combination of equivalent temporal and spatial terms, so that the Serre equations can be written in conservation law form. The water depth and new conserved quantities are evolved using a second-order finite-volume scheme. The remaining primitive variable, the depth-averaged horizontal velocity, is obtained by solving a second-order elliptic equation using simple finite differences. Using an analytical solution and simulating the dam-break problem, the proposed scheme is shown to be accurate, simple to implement and stable for a range of problems, including flows with steep gradients. It is only slightly more computationally expensive than solving the shallow water wave equations.

In this study, Liénard equations in their general form are treated using the Adomian decomposition method. The special structure of the Liénard equation is exploited to obtain a numerically efficient algorithm suitable for solution by a computer program.

A uniform approximation to the description of a linear oscillator's slow resonant transition is calculated. If the time scale of the transition is ɛ−1, the approximation contains explicitly the 0(1) and 0(ɛ½) terms, and fixes a uniform 0(ɛ) error bound.

Mathematical modelling of Wairakei geothermal field is reviewed, both lumped-parameter and distributed-parameter models. In both cases it is found that reliable predictions require five to ten years of history for calibration. With such calibration distributed-parameter models are now used for field management. A prudent model of Wairakei, constructed without such historical data, would underestimate field capacity and provide only general projections of the type of changes in surface activity and subsidence.

The controllable set of a controlled ordinary differential dynamic system to a given set is defined. Under certain reasonable conditions, the controllable set is characterised by a level set of the unique viscosity solution to some Hamilton-Jacobi-Bellman equation. The result is used to determine the asymptotic stable set of nonlinear autonomous differential equations.

Recently we have developed a completely symmetric duality theory for mathematical programming problems involving convex functionals. Here we set our theory within the framework of a Lagrangian formalism which is significantly different to the conventional Lagrangian. This allows various new characterizations of optimality.

Necessary and sufficient conditions for optimality in the control of linear differential systems ẋ = Ax + Bu with Stieltjes boundary conditions , where ν is an r × n matrix valued measure of bounded variation, are obtained, Feedback-like control is given in the case of quadratic performance.