Two sums given by J. B. Miller are evaluated in terms of classical hypergeometric results.

A method for solving quasilinear parabolic equations of the types

that differs radically from previously known methods is proposed. For each initial-boundary-value problem of one of these types that has boundary conditions of the first kind (second kind), a conjugate initial-boundary-value problem of the other type that has boundary conditions of the second kind (first kind) is defined. Based on the relations connecting the solutions of a pair of conjugate problems, a series of parabolic equations with constant coefficients that do not change step to step is constructed. The method proposed consists in calculating the solutions of the equations of this series. It is shown to have linear convergence. Results of a series of numerical experiments in a finite-difference setting show that one particular implementation of the proposed method has a smaller domain of convergence than Newton's method but that it sometimes converges faster within that domain.

The existence of stable periodic oscillatory solutions in a two species competition model with time delays is established using a combination of Hopf-bifurcation theory and the asymptotic method of Krylov, Bogoliuboff and Mitropoisky.

In this paper we consider a projective connection as defined by the nth-order Adler-Gelfand-Dikii (AGD) operator on the circle. It is well-known that the Korteweg-de Vries (KdV) equation is the archetypal example of a scalar Lax equation defined by a Lax pair of scalar nth-order differential (AGD) operators. In this paper we derive (formally) the KdV equation as an evolution equation of the AGD operator (at least for n ≤ 4) under the action of Vect(S1). The solutions of the AGD operator define an immersion R → RPn−1 in homogeneous coordinates. In this paper we derive the Schwarzian KdV equation as an evolution of the solution curve associated with Δ(n), for n ≤ 4.

A system of new integral equations is presented. They are derived from Maxwell's equations and describe radio-frequency (RF) current densities on a two-dimensional flat plate. The equations are generalisations of Pocklington's integral equation showing phase-retardation in two dimensions. These singular equations are solved, numerically, for the case of one-dimensional geometry. The solutions are shown to display effects which correspond to damped resonance when the wavelength of the current matches aspects of the geometry of the conductor.

A class of linear systems subject to sudden jumps in parameter values is considered. To solve this class of stochastic control problem, we try to seek the best feedback control law depending only on the measurable output. Based on this idea, we convert the original problem into an approximate constrained deterministic optimization problem, which can be easily solved by any existing nonlinear programming technique. An example is solved to illustrate the efficiency of the method.

Two-dimensional free-surface flows produced by a submerged source in a fluid of infinite depth are considered. It is assumed that the point on the free surface just above the source is a stagnation point and that the fluid outside two shear layers is at rest. The free-surface profile and the shape of the shear layers are determined numerically by using a series-truncation method. It is shown that there is a solution for each value of the Froude number F > 0. When F tends to infinity, the flow also describes a thin jet impinging in a fluid at rest.

Singular perturbation methods are applied to an analysis of the operation of an isothermal gas step slider bearing of narrow geometry and operating at moderate bearing numbers. Approximate expressions are obtained for the pressure field in the lubricating gap, as well as the load-carrying capacity of the bearing; and the influence of the nature of the bearing step on those quantities is investigated. Comparisons are made with results obtained using a standard numerical package.

A calculation of the electromagnetic response of a thin conductor in the presence of an exciting primary magnetic field has been attempted by various authors. Analytic solutions are obtainable when either the conductor is of infinite extent or when the problem possesses some symmetry. The loss of symmetry makes the problem difficult to solve except for the simplest shape – that of a circular conductor. A numerical method has been used for the rectangular conductor by other authors. In this paper we consider the response due to a thin plane conductor of arbitrary shape. The method involves the numerical generation of a set of body-fitted orthogonal curvilinear coordinates which maps the conductor onto a unit square. Good orthogonal grids can be generated for shapes that do not deviate too far from the rectangular. In terms of these curvilinear coordinates the vector potential for the area current density satisfies an integro-differential equation which is solved numerically.

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.

It is known that many optimization problems can be reformulated as composite optimization problems. In this paper error analyses are provided for two kinds of smoothing approximation methods of a unconstrained composite nondifferentiable optimization problem. Computational results are presented for nondifferentiable optimization problems by using these smoothing approximation methods. Comparisons are made among these methods.

This paper examines the control of an interface between a suspension of sedimenting particles in liquid and a bed of dense-packed particles at the bottom of the suspension. The problem arises in the operation of continuous thickeners (e.g. in mineral processing) and is here mathematically described by a first order inhomogeneous partial differential equation for the concentration C(x, t) of particles. The controlled variable is the height H* of the bed, and the control variables are the volume fluxes injected at the feed level and removed at the bed. A strategy to control the interface is devised, and control is confirmed and demonstrated by a series of numerical experiments.

Connections between a linear partial difference equation with constant coefficients and a nonlinear partial difference equation are established by means of a comparison theorem and a continuous dependence of parameters theorem. A linearized oscillation theorem is also established as an application.

We wish to correct a minor error in the recent paper [2]. That paper was concerned with an integral equation defined on a closed polygon Γ with r corners at the points x0, x2, …, x2r = x0. We parameterized Γ using a mapping γ:[−π,π] → Γ defined as follows. For each l, introduce the mid-point x2l−1 of the side joining x2l—2 to x2l. Then introduce 2r + 1 points in parameter space

with the property that for each j = 1, …, 2r

where mj are integers and . Then γ(s) is defined by

for j = 1, …, 2r. The {Sj} are then the preimages of the {xj} under γ. Moreover, in view of (1), a family of uniform meshes can be constructed on [−π, π] which include {Sj} as the break-points. Then γ maps these to meshes which are uniform on each segment joining xj−1 to xj (which we denote Γj). These meshes are used to discretize the integral equation.

The linear stability properties are examined of long wavelength vortex modes in two time-periodic flows. These flows are the motion which is induced by a torsionally oscillating cylinder within a viscous fluid and, second, the flow which results from the sinusoidal heating of an infinite layer of fluid. Previous studies concerning these particular configurations have shown that they are susceptible to vortex motions and linear neutral curves have been computed for wavenumbers near their critical value. These computations become increasingly difficult for long wavelength motions and here we consider such modes using asymptotic methods. These yield simple results which are formally valid for small wavenumbers and we show that the agreement between these asymptotes and numerical solutions is good for surprisingly large wavenumbers. The two problems studied share a number of common features but also have important differences and, between them, our methods and results provide a basis which can be extended for use with other time-periodic flows.

In this note we present a variable order continuation method for the solution of nonlinear equations when only a poor estimate of a solution is known. The method changes continuously from one which improves the global convergence characteristics to one which attains rapid convergence to a solution and proves to be more efficient than methods previously presented in [2].