A combined BIE-FE method for the Stokes equations

A numerical algorithm for the biharmonic equation in domains with piecewise smooth boundaries is presented. It is intended for problems describing the Stokes flow in the situations where one has corners or cusps formed by parts of the domain boundary and, due to the nature of the boundary conditions on these parts of the boundary, these regions have a global effect on the shape of the whole domain and hence have to be resolved with sufficient accuracy. The algorithm combines the boundary integral equation method for the main part of the flow domain and the finite-element method which is used to resolve the corner/cusp regions. Two parts of the solution are matched along a numerical ‘internal interface’ or, as a variant, two interfaces, and they are determined simultaneously by inverting a combined matrix in the course of iterations. The algorithm is illustrated by considering the flow configuration of ‘curtain coating’, a flow where a sheet of liquid impinges onto a moving solid substrate, which is particularly sensitive to what happens in the corner region formed, physically, by the free surface and the solid boundary. The ‘moving contact line problem’ is addressed in the framework of an earlier developed interface formation model which treats the dynamic contact angle as part of the solution, as opposed to it being a prescribed function of the contact line speed, as in the so-called ‘slip models’. Keywords: Dynamic contact angle; finite elements; free surface flows; hybrid numerical technique; Stokes equations.

Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations $u+Ku = v$ in the space $BC$ of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator $A = I + K$ under consideration, with an emphasis on the function space setting $BC$. Firstly, under which conditions is $A$ a Fredholm operator, and, secondly, when is the finite section method applicable to $A$?

Boundary value problems for the N-wave interaction equations

We consider boundary value problems for the N-wave interaction equations in one and two space dimensions, posed for x [greater-or-equal, slanted] 0 and x,y [greater-or-equal, slanted] 0, respectively. Following the recent work of Fokas, we develop an inverse scattering formalism to solve these problems by considering the simultaneous spectral analysis of the two ordinary differential equations in the associated Lax pair. The solution of the boundary value problems is obtained through the solution of a local Riemann–Hilbert problem in the one-dimensional case, and a nonlocal Riemann–Hilbert problem in the two-dimensional case.

Evaluation of NRC, UC Davis and ADAS approaches to estimate the metabolizable energy values of feeds at maintenance energy intake from equations utilizing chemical assays and in vitro determinations

Feed samples received by commercial analytical laboratories are often undefined or mixed varieties of forages, originate from various agronomic or geographical areas of the world, are mixtures (e.g., total mixed rations) and are often described incompletely or not at all. Six unified single equation approaches to predict the metabolizable energy (ME) value of feeds determined in sheep fed at maintenance ME intake were evaluated utilizing 78 individual feeds representing 17 different forages, grains, protein meals and by-product feedstuffs. The predictive approaches evaluated were two each from National Research Council [National Research Council (NRC), Nutrient Requirements of Dairy Cattle, seventh revised ed. National Academy Press, Washington, DC, USA, 2001], University of California at Davis (UC Davis) and ADAS (Stratford, UK). Slopes and intercepts for the two ADAS approaches that utilized in vitro digestibility of organic matter and either measured gross energy (GE), or a prediction of GE from component assays, and one UC Davis approach, based upon in vitro gas production and some component assays, differed from both unity and zero, respectively, while this was not the case for the two NRC and one UC Davis approach. However, within these latter three approaches, the goodness of fit (r(2)) increased from the NRC approach utilizing lignin (0.61) to the NRC approach utilizing 48 h in vitro digestion of neutral detergent fibre (NDF:0.72) and to the UC Davis approach utilizing a 30 h in vitro digestion of NDF (0.84). The reason for the difference between the precision of the NRC procedures was the failure of assayed lignin values to accurately predict 48 h in vitro digestion of NDF. However, differences among the six predictive approaches in the number of supporting assays, and their costs, as well as that the NRC approach is actually three related equations requiring categorical description of feeds (making them unsuitable for mixed feeds) while the ADAS and UC Davis approaches are single equations, suggests that the procedure of choice will vary dependent Upon local conditions, specific objectives and the feedstuffs to be evaluated. In contrast to the evaluation of the procedures among feedstuffs, no procedure was able to consistently discriminate the ME values of individual feeds within feedstuffs determined in vivo, suggesting that the quest for an accurate and precise ME predictive approach among and within feeds, may remain to be identified. (C) 2004 Elsevier B.V. All rights reserved.

Global asymptotic stability in a class of Putnam-type equations

In this paper, we initiate the study of a class of Putnam-type equation of the form x(n-1) = A(1)x(n) + A(2)x(n-1) + A(3)x(n-2)x(n-3) + A(4)/B(1)x(n)x(n-1) + B(2)x(n-2) + B(3)x(n-3) + B-4 n = 0, 1, 2,..., where A(1), A(2), A(3), A(4), B-1, B-2, B-3, B-4 are positive constants with A(1) + A(2) + A(3) + A(4) = B-1 + B-2 + B-3 + B-4, x(-3), x(-2), x(-1), x(0) are positive numbers. A sufficient condition is given for the global asymptotic stability of the equilibrium point c = 1 of such equations. (c) 2005 Elsevier Ltd. All rights reserved.

On two rational difference equations

In this paper, we study the oscillating property of positive solutions and the global asymptotic stability of the unique equilibrium of the two rational difference equations [GRAPHICS] and [GRAPHICS] where a is a nonnegative constant. (c) 2005 Elsevier Inc. All rights reserved.

On the system of rational difference equations x(n) = A+(1)over-bar y(n-p), y(n) = A+(gamma(n-1))over-bar x(n-r)y(n-5)

In this paper, we study the behavior of the positive solutions of the system of two difference equations [GRAPHICS] where p >= 1, r >= 1, s >= 1, A >= 0, and x(1-r), x(2-r),..., x(0), y(1-max) {p.s},..., y(0) are positive real numbers. (c) 2005 Elsevier Inc. All rights reserved.