The velocities of Rayleigh surface waves and, when they exist, Stoneley interface waves can be obtained as the roots of two irrational functions. Here previous results are extended by using standard operations related to the Wiener-Hopf technique to provide expressions in quadrature for these roots.

The expressions for elliptic integrals, elliptic functions and theta functions given in standard reference books are slowly convergent as the parameter m approaches unity, and in the limit do not converge. In this paper we use Jacobi's imaginary transformation to obtain alternative expressions which converge most rapidly in the limit as m → 1. With the freedom to use the traditional formulae for m ≤ ½ and those obtained here for m ≥ ½, extraordinarily rapidly-convergent methods may be used for all values of m; no more than three terms of any series need be used to ensure eight-figure accuracy.

We investigate the reflection and transmission of SH-waves at a corrugated interface between two different anisotropic, heterogeneous elastic solid half-spaces. Both the media are assumed to be transversely isotropic and vertically heterogeneous. Rayleigh's method is followed and expressions for the reflection and transmission coefficients are obtained in closed form for the first-order approximation of the corrugation. It is found that these coefficients depend on corrugation and are affected by the anisotropy and heterogeneity of the media. Numerical computations for a particular model have been performed.

The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region in R2 together with an impedance condition on its inner boundary and another impedance condition on its outer boundary, from the complete knowledge of the eigenvalues for the two-dimensional Laplacian using the asymptotic expansion of the spectral function for small positive t.

The Kuramoto–Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto–Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan–Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto–Sivashinsky PDE, and the transition to its one-dimensional turbulence.

The free surface due to a submerged source in a fluid of finite depth at infinite Froude number is reconsidered. A conformal transformation technique is used to formulate this problem as an integral equation for the free-surface angle. An elementary solution is found for the equation, which results in a closed form expression for the free-surface elevation. Comparison is made with previous numerical solutions.

Solutions are found to two cusp-like free-surface flow problems involving the steady motion of an ideal fluid under the infinite-Froude-number approximation. The flow in each case is due to a submerged line source or sink, in the presence of a solid horizontal base.

In this paper we prove the existence of solutions for hyperbolic hemivariational inequalities and then investigate optimal control problems for some convex cost functionals.

Fluid withdrawn through a line sink from a layered fluid in a vertically confined porous medium is considered. A hodograph method is used to obtain the shape of the interface for a given sink position at the critical flow rate. The analytical solution is compared with a more general numerical solution developed in earlier work. It was found that the surface profiles obtained by the two methods are in close agreement. However, the present work has the advantage that it gives a fully explicit solution.

Recently, Hanson and Mond formulated a type of generalized convexity and used it to establish duality between the nonlinear programming problem and the Wolfe dual. Elsewhere, Mond and Weir gave an alternate dual, different from the Wolfe dual, that allowed the weakening of the convexity requirements. Here we establish duality between the nonlinear programming problem and the Mond-Weir dual using Hanson-Mond generalized convexity conditions.

We consider the optimal proportional reinsurance problem for an insurer with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Using the technique of stochastic linear–quadratic control theory and the Hamilton–Jacobi–Bellman equation, we derive the explicit expressions for the optimal reinsurance strategies and value function, and present the verification theorem within the framework of the viscosity solution. Furthermore, we extend the results in the linear–quadratic setting to the mean–variance problem, and obtain an efficient strategy and frontier. Some numerical examples are given to show the impact of model parameters on the efficient frontier.

The convergence and stability analysis of a simple explicit finite difference method is studied in this paper. Conditional convergence and stability theorems for this method are given. We have also proved that this scheme is stable in a much stronger sense.

The perturbation of the eigenvalues of a regular Sturm–Liouville problem in normal form which results from a small perturbation of the coefficient function is known to be uniformly bounded. For numerical methods based on approximating the coefficients of the differential equation, this result is used to show that a better bound on the error is obtained when the problem is in normal form. A method having a uniform error bound is presented, and an extension of this method for general Sturm–Liouville problems is proposed and examined.

It is well known that transcritical flow past an obstacle may generate undular bores propagating away from the obstacle. This flow has been successfully modelled in the framework of the forced Korteweg–de Vries equation, where numerical simulations and asymptotic analyses have shown that the unsteady undular bores are connected by a locally steady solution over the obstacle. In this paper we present an overview of the underlying theory, together with some recent work on the case where the obstacle has a large width.

Differentiation of the well-known addition theorem for Legendre polynomials produces results for sums over order m of products of various derivatives of associated Legendre functions. The same method is applied to the corresponding addition theorems for vector and tensor spherical harmonics. Results are also given for Chebyshev polynomials of the second kind, corresponding to ‘spin-weighted’ associated Legendre functions, as used in studies of distributions of rotations.

The complete symmetry group of a forced harmonic oscillator is shown to be Sl(3, R) in the one-dimensional case. Approaching the problem through the Hamiltonian invariants and the method of extended Lie groups, the method used is that of time-dependent point transformations. The result applies equally well to the forced repulsive oscillator and a particle moving under the influence of a coordinate-free force. The generalization to na-dimensional systems is discussed.

We propose and analyze the spectral collocation approximation for the partial integro-differential equations with a weakly singular kernel. The space discretization is based on the pseudo-spectral method, which is a collocation method at the Gauss-Lobatto quadrature points. We prove unconditional stability and obtain the optimal error bounds which depend on the time step, the degree of polynomial and the Sobolev regularity of the solution.

The problem of a source or sink submerged beneath a free surface is investigated in the infinite Froude number limit. Solutions are found for all cases in which the source is situated away from the bottom of the channel. Solutions are also found for the case where the source is situated above the asymptotic level of the free surface, giving fountain type free surface shapes.

The notion of strong system equivalence, which was defined and studied in Anderson, Coppel and Cullen [1], is here given a module-theoretic characterization and a dynamical interpretation.

Optimal control problems with switching costs arise in a number of applications, and are particularly important when standard control theory gives “chattering controls”. A numerical method is given for finding optimal controls for linear problems (linear dynamics, linear plus switching cost). This is used to develop an algorithm for finding sub-optimal control functions for nonlinear problems with switching costs. Numerical results are presented for an implementation of this method.