A deferred correction procedure for the approximate solution of the second-kind equation is introduced, compared with an extrapolation procedure, and illustrated for integral and differential equations.

Two problems involving the “best” solution X for a matrix equation V×V* = M are discussed, together with methods for their solution, and a generalization of one of the methods beyond matrix equations.

There is presently considerable interest in the utilisation of microwave heating in areas such as cooking, sterilising, melting, smelting, sintering and drying. In general, such problems involve Maxwell's equations coupled with the heat equation, for which all thermal, electrical and magnetic properties of the material are nonlinear. The heat source arising from microwaves is proportional to the square of the modulus of the electric field intensity, and is known to increase with increasing temperature. In an attempt to find a simple model of microwave heating, we examine here simple transient temperature profiles corresponding to a heat source with spatial exponential decay but increasing with temperature, for which we assume either a power-law dependence or an exponential dependence. The spatial exponential decay is known to apply exactly when the electrical and magnetic properties of the material are assumed constant. A number of transient temperature profiles for this model are examined which arise from the invariance of the governing heat equation under simple one-parameter transformation groups. Some closed analytical expressions are obtained, but in general the resulting ordinary differential equations need to be solved numerically, and extensive numerical results are presented. For both models, these results indicate the appearance of moving fronts.

When multiple operators are connected to a single power source which is not large enough to simultaneously supply all users, interference can take place. This paper considers two models of a spot welding station which differ in the method of resolving interference between users. In the first model the system is allowed to become overloaded and the consequent deterioration in weld quality is accepted. In the second model any request which will overload the system is rejected. Expressions are found for (a) the proportion of poor quality welds in the first model and (b) the probability of operators being rejected in the second model. Numerical results are given which indicate how small the power supply can be made for a typical welding shop while keeping interference at a minimal level.

A finite element Galerkin method for a diffusion equation with constrained energy and nonlinear boundary condition is analysed and optimal error estimates in L2 and L∞-norms are derived. These results improve upon previously derived estimates by Cannon et al. [4].

Two eccentric rotating cylinders together with a permeable membrane surrounding the inner cylinder are used to model the flow around a modified viscometer. A perturbation method is used to solve for the flow between the membrane and the outer cylinder; the flow between the inner rotor and the membrane is assumed to be governed by Stoke's equation, and the two flow regimes are coupled by the through-flow across the membrane. For moderate values of Reynolds number and eccentricity, the permeability of the membrane plays a negligible role, and the flow through the membrane is found to be eccentricity dependent. High eccentricities result in the formation of eddies which, upon increasing the Reynolds number, move in a direction opposite to that of the rotation of the outer bowl.

In this paper a control problem with a cost functional depending on the number of switchings and on the speed of alterations of control is considered. Necessary conditions for the existence of an optimal solution are given.

This paper studies the impairing of flows in multi-index transportation problem with axial constraints. For any curtailed flow, the problem is shown to be equivalent to a standard axial sum problem, whose solution can be obtained by known methods. The equivalence is established only for specially defined solutions (referred to as M-feasible solutions) of the standard problem. It is also proved that an optimal solution of the impaired flow problem corresponds to such an M-feasible solution.

The quasi-linear infiltration problem of flow from a semi-infinite wetted region on a soil of finite depth above a horizontal water table is considered in the presence of linearised evaporative loss away from the region. The resulting equations are solved by the Wiener-Hopf technique in terms of certain infinite products. Expressions for the porosity and stream function are derived, and appropriately plotted throughout the layer.

This paper presents a two-step formulation for the dynamic analysis of generalised planar linkages. First, a rigid body is replaced by a dynamically equivalent constrained system of particles and Newton's second law is used to study the motion of the particles without introducing any rotational coordinates. The translational motion of the constrained particles represents the general motion of the rigid body both translationally and rotationally. The simplicity and the absence of any rotational coordinates from the final form of the equations of motion are considered the main advantages of this formulation. A velocity transformation is then used to transform the equations of motion to a reduced set in terms of selected relative joint variables. For an open-chain, this process automatically eliminates all of the non-working constraint forces and leads to efficient integration of the equations of motion. For a closed-chain, suitable joints should be cut and some cut-joint constraint equations should be included. An example of a closed-chain is used to demonstrate the generality and efficiency of the proposed method.

Simple proofs are given of improved results of Brown and Shepp which are useful in calculations with fractal sets. A new inequality for convex functions is also obtained.

We consider an optimal control problem with, possibly time-dependent, constraints on state and control variables, jointly. Using only elementary methods, we derive a sufficient condition for optimality. Although phrased in terms reminiscent of the necessary condition of Pontryagin, the sufficient condition is logically independent, as can be shown by a simple example.

An optimal control problem governed by a class of delay semilinear differential equations is studied. The existence of an optimal control is proven, and the maximum principle and approximating schemes are found. As applications, three examples are discussed.

In mathematical programming, an important tool is the use of active set strategies to update the current solution of a linear system after a rank one change in the constraint matrix. We show how to update the general solution of a linear system obtained by use of the scaled ABS method when the matrix coefficient is subjected to a rank one change.

Various initial and boundary value problems for a 2-dimensional reaction-diffusion equation are studied numerically by an explicit Finite Difference Method (FDM), a Galerkin and a Petrov-Galerkin Finite Element Method (FEM). The results not only show the transition processes from different local initial disturbances to quasitravelling waves, but also demonstrate the long term behaviour of the solutions, which is determined by the system itself and does not depend on the details of the initial disturbances.