The problem of estimating the limit f∞ of a sequence fn converging as fn − f∞ = O(n−λ) as n → ∞, where λ > 0, is discussed. Using the generalization of the ε-algorithm proposed recently by Vanden Broeck and Schwartz, an acceleration scheme is developed. The method is illustrated on several test sequences and compared to other acceleration procedures.

In the absence of surface tension, the problem of determining a travelling surface pressure distribution that displaces a portion of the free surface in a prescribed manner has been solved by several authors, and this “planing-surface” problem is reasonably well understood. The effect of inclusion of surface tension is to change, in a dramatic way, the singularity in the integral equation that describes the problem. It is now necessary in general to allow for isolated impulsive pressure, as well as a smooth distribution over the wetted length. Such pressure points generate jump discontinuities in free-surface slope. Numerical results are obtained here for a class of problems in which there is a single impulse located at the leading edge of the planing surface and detachment with continuous slope at the trailing edge. These results do not appear to approach the classical results in the limit as the surface tension approaches zero, a paradox that is resolved in Part II, which follows.

In Part I of this series, surface tension was included in the classical two-dimensional planing-surface problem, and the usual smooth-detachment trailing-edge condition enforced. However, the results exhibited a paradox, in that the classical results were not approached in the limit as the surface tension approached zero. This paradox is resolved here by abandoning the smooth-detachment condition, that is, by allowing a jump discontinuity in slope between the planing surface and the free surface at the trailing edge. A unique solution is obtainable for any input planing surface at fixed wetted length if one allows such jumps at both leading and trailing edges. If, as is the case in practice, the wetted length is allowed to vary, uniqueness may be restored by requiring either, but not both, of these slope discontinuities to vanish. The results of Part I correspond to the seemingly more-natural choice of making the trailing-edge detachment continuous, but it appears that the correct choice is to require the leading-edge attachment to be continuous.

Reacting systems with Arrhenius chemistry can exhibit multiple steady states for γ > 4(1 + β)/β where γ is activation energy and β is the heat of reaction. We give a detailed analysis of the extremely rapid jump which takes place as catalyst activity slowly decays through criticality. Previous analyses are asymptotic in the limit γ → ∞; here we relax the large γ assumption and only require that passage through criticality is slow.

The Rapp formula of teletraffic dimensioning is generalized to admit an arbitrary renewal stream of offered traffic. The derivation proceeds from a heavy traffic approximation and provides also an estimate of the order of error involved in the Rapp formula. In principle, the method could be used to seek convenient higher order approximations.

Our equations give an incidental theoretical substantiation of an empirical result relating to marginal occupancy found recently by Potter.

For diffusion problems, the boundary conditions are specified at two distinct points, yielding a two end-point boundary value problem which normally requires iterative techniques. For spherical geometry, it is possible to specify the boundary conditions at the same points, approximately, by using an optimization principle for arbitrary diffusivity. When the diffusivity obeys a power or an exponential law, a first integral exists and iteration can be avoided. For those two exact cases, it is shown that the general optimization result is extremely accurate when diffusivity increases rapidly with concentration.

The partially stiff system of ordinary differential equations

is studied by the methods developed in the earlier papers in this series. Here e is a small positive parameter, x and y are n- and m-vectors respectively, and A is nonsingular. A useful basis for the solution space of the homogeneous system is constructed and the method of variation of parameters is used to obtain useful representations of all solutions. Sufficient conditions are derived under which the formal approximation

is close to the actual solution. it is found that purely imaginary eigenvalues for A require more stringent requirements for the formal technique to be valid. A brief discussion of the case when A is singular shows that there are a great number of possibilities requiring consideration for a general theory. it is suggested that local computation of such cases is likely to be the most effective weapon for any specific system.

This is a discussion of some numerical integration methods for surface integrals over the unit sphere in R3. Product Gaussian quadrature and finite-element type methods are considered. The paper concludes with a discussion of the evaluation of singular double layer integrals arising in potential theory.