The orthogonal coordinate systems ξi(i = 1, 2, 3) are determined, in which the gneralised toroidal and poloidal fields, defined respectively by T{T} = ∇ × {T∇ξ1} and S{S} = ∇ × T{S}, have the following three properties:

GP1 Decoupling of the vector Helmholtz equation: There exist linear differential operators L1 and L2 such that Hu = 0, where H is the vector Helmholtz operator [see equation (1)] and u = T{T} + S {S}, if and only if L1T = 0 and L2S = 0.

GP2 Orthogonality

GP3 Closure: ∇ × S{S} is a T field.

Two choices of T and S fields are considered: type I fields with potentials T and S, which may depend on ξ1, ξ2 and ξ3, and type II fields with ξ1-independent potentials. It is shown that properties GP1–GP3 only hold for type I fields in spherical and cylindrical coordinate systems, and for type II fields in azimuthal and cylindrical coordinate systems with axisymmetric and two-dimensional potentials, respectively. Analogues of GP1 for the vector wave and diffusion equations, and the Navier equation of linear elasticity, are also only true in the same four cases. Generalisations of type I and II T and S fields to arbitrary coordinate systems are indicated.

The problem of obtaining explicit and exact solutions of soliton equations of the AKNS class is considered. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system; that is, a linear eigenvalue problem in the form of a system of first order partial differential equations. The method of characteristics is used and Bäcklund transformations are employed to generate new solutions from the old. Thus, families of new solutions for the KdV equation, the mKdV equation, the sine-Gordon equation and the nonlinear Schrôdinger equation are obtained, avoiding the solution of some Riccati equations. Our results in the KdV case include those obtained recently by other investigators.

The behaviour of duopolists is considered within a framework that allows for flexibility of the adopted strategy against the rival. In a difficult external climate, a firm may concentrate on its own profit, whereas in a more favourable external climate, it may adopt a more aggressive attitude towards the rival. The strategy considered in this paper permits this flexible approach. The market functions are kept general to allow the widest interpretation of the results.

The paper asks and answers the question “When does dominance of a particular strategy play a role in the search for evolutionary stable strategies?” The answer is much less obvious than would appear at first glance.

When there is strict dominance of a pure strategy, it is clear that the dominated strategy should never be employed in any conflict. However, when the dominance is not strict it is less obvious that the strategy should not be used. The research was originally intended to clear up this grey area in the theory of evolutionary stable strategies, but it has turned out to be of more than simply academic interest. The result can be used, with varying degrees of success, to simplify the search procedure for these evolutionary stable strategies when a reward matrix is given.

In this paper we consider an infinite horizon, continuous time model of economic growth. We prove two theorems; one on the existence of optimal paths of capital accumulation and the other on the dependence of the set of optimal paths on the initial capital stock (sensitivity analysis). In the existence result the underlying technology set is nonconvex and only its “investment’ slices are convex. The proof is direct, without any use of necessary conditions. In the sensitivity analysis, the technology set is convex and so we have that the value function is concave. Then having that, we show that the set of optimal paths is an upper semicaontinuous multifunction of the initial capital stock.

The modelling of the combustion of dust suspensions leads to a nonlinear eigenvalue problem for a system of ordinary differential equations defined over an infinite interval. The equations contain a number of parameters. In this study, the shooting method is used to prove the existence of a solution. Linearisation is then used to provide an approximate solution, from which an estimate of the eigenvalue and its dependence on the given parameters can be obtained.

In this paper we shall derive some asymptotic formulae for spectra of the third boundary value problem in Rn, n = 2 or 3, linked with variation of a positive function entering the boundary conditions. Further results may be obtained.

We introduce assumptions in input optimisation that simplify the necessary conditions for an optimal input. These assumptions, in the context of nonlinear programming, give rise to conceptually new kinds of constraint qualifications.

A method based on the method of images is described for the solution of the linear equation modelling diffusion and elimination of substrate in a fluid flowing through a chemical reactor of finite length, when the influx of substrate is prescribed at the point of entry and Danckwerts' zero-gradient condition is imposed at the point of exit. The problem is shown to be transformable to an equivalent problem in heat conduction. Associated with the images appearing in the method of solution is a sequence of functions which form a vector space carrying a representation of the Lie group SO(2, 1) generated by three third-order differential operators. The functions are eigenfunctions of one of these operators, with integer-spaced eigenvalues, and they satisfy a third-order recurrence relation which simplifies their successive determination, and hence the determination of the Green's function for the problem, to any desired degree of approximation. Consequently, the method has considerable computational advantages over commonly used methods based on the use of Laplace and related transforms. Associated with these functions is a sequence of polynomials satisfying the same third-order differential equation and recurrence relation. The polynomials are shown to bear a simple relationship to Laguerre polynomials and to satisfy the ordinary diffusion equation, for which SO(2, 1) is therefore revealed as an invariance group. These diffusion polynomials are distinct from the well-known heat polynomials, but a relationship between them is derived. A generalised set of diffusion polynomials, based on the associated Laguerre polynomials, is also described, having similar properties.

The homotopy method is used to find all eigenpairs of a generalised symmetric eigenvalue problem Ax = λBx with positive definite B. The determination of n eigenpairs (x, λ) is reduced to curve tracing of n disjoint smooth curves in Rn × R × [0, 1]. The method might be attractive if A and B are sparse symmetric. In this paper it is shown that the method will work for almost all symmetric tridiagonal matrices A and B.

A simple rigorous approach is given to finding boundary conditions for the adjoint differential equation in an optimal control problem. The boundary conditions for a time-optimal problem are calculated from the simpler conditions for a fixed-time problem.

Computation of eigenvalues of regular Sturm-Liouville problems with periodic or semiperiodic boundary conditions is considered. A simple asymptotic correction technique of Paine, de Hoog and Anderssen is shown to reduce the error in the centred finite difference estimate of the kth eigenvalue obtained with uniform step length h from O(k4h2) to O(kh2). Possible extensions of the results are suggested and the relative advantages of the method are discussed.

In this paper consider we optimal control problems with linear state constraints where the states can be discontinuous at the boundary. In fact the state vector models the cause the position and velocity of a particle where the collisions with the boundary that cause the discontinuities are elastic. Necessary conditions are derived by looking at limits of approximate problems that are unconstrained.

A Demianski-type metric is investigated in connection with the Einstein-Maxwell fields. Using complex vectorial formalism, some exact solutions of Einstein-Maxwell field equations for source-free electromagnetic fields plus pure radiation fields are obtained. The radiating Demianski solution, the Debney-Kerr-Schild solution and the Brill solution are derived as particular cases.

In this paper, we consider a class of combined optimal parameter selection and optimal control problems with general constraints. The first aim is to provide a unified approach to the numerical solution of this general class of optimisation problems by using the control parametrisation technique. This approach is supported by some convergence results. The second aim is to show that several different classes of optimal control problems can all be transformed into special cases of the problem considered in this paper. For illustration, four numerical examples are presented.

In modelling phenomena involving diffusion and chemical reactions, coupled systems of linear differential equations are often obtained, which can involve several dependent variables. For two dependent variables, coupled reaction-diffusion systems can be uncoupled, and in principle the original boundary value problem can be reduced to two separate boundary value problems for the classical heat equation. Here we address various aspects of the fundamental unsolved problem of the determination of corresponding uncoupling transformations for systems involving several dependent variables. We present, in an elementary manner, the current state of knowledge relating to this complex problem area. Several new results are obtained here. For example, in reviewing known results two dependent variables we observe that those systems for which uncoupling transformations have been found are essentially those which can be reduced to a coupled system involving a single spatial operator L. In addition, for several dependent variables, the general solution structure for the kernel matrix, involved in the uncoupling transformation, is presented together with some explicit results for values of components of the kernel matrix along characteristics, which are deduced from elementary considerations.