Moving boundary value problems for the wave equation

We study certain boundary value problems for the one-dimensional wave equation posed in a time-dependent domain. The approach we propose is based on a general transform method for solving boundary value problems for integrable nonlinear PDE in two variables, that has been applied extensively to the study of linear parabolic and elliptic equations. Here we analyse the wave equation as a simple illustrative example to discuss the particular features of this method in the context of linear hyperbolic PDEs, which have not been studied before in this framework.

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.

Well-posed boundary value problems for integrable evolution equations on a finite interval

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

A Nyström method for a boundary value problem arising in unsteady water wave problems

This paper is concerned with solving numerically the Dirichlet boundary value problem for Laplace’s equation in a nonlocally perturbed half-plane. This problem arises in the simulation of classical unsteady water wave problems. The starting point for the numerical scheme is the boundary integral equation reformulation of this problem as an integral equation of the second kind on the real line in Preston et al. (2008, J. Int. Equ. Appl., 20, 121–152). We present a Nystr¨om method for numerical solution of this integral equation and show stability and convergence, and we present and analyse a numerical scheme for computing the Dirichlet-to-Neumann map, i.e., for deducing the instantaneous fluid surface velocity from the velocity potential on the surface, a key computational step in unsteady water wave simulations. In particular, we show that our numerical schemes are superalgebraically convergent if the fluid surface is infinitely smooth. The theoretical results are illustrated by numerical experiments.

The value of badger populations and control of tuberculosis in cattle in England and Wales: A note

This note reports on the results of a choice experiment survey of 400 people in England and Wales, conducted to estimate the value that society places on changes to the size of the badger population. The study was undertaken in the context of the possible need to reduce the badger population by culling to help control bovine tuberculosis in cattle. The study found that people were concerned about the problem of bovine tuberculosis in cattle, which was reflected in their willingness to pay to control the disease, and gave a relatively low value to changes in the size of the badger population (within limits). However, people did not like the idea of a policy that intentionally killed large numbers of badgers and had a relatively very high willingness to pay not to have such a policy.

Use of fibrolytic enzymes to improve the nutritive value of ruminant diets - A biochemical and in vitro rumen degradation assessment

Two commercial enzyme products, Depol 40 (D) and Liquicell 2500 (L), were characterised from a biochemical standpoint and their potential to improve rumen degradation of forages was evaluated in vitro. Enzyme activities were determined at pH 5.5 and 39 degreesC. Analysis of the enzyme activities indicated that L contained higher xylanase and endoglucanase, but lower exoglucanase, pectinase and alpha-amylase activities than D. The Reading Pressure Technique (RPT) was used to investigate the effect of enzyme addition on the in vitro gas production (GP) and organic matter degradation (OMD) of alfalfa (Medicago sativa L.) stems and leaves. A completely randomised design with factorial arrangement of treatments was used. Both alfalfa fractions were untreated or treated with each enzyme at four levels, 20 h before incubation with rumen fluid. Each level of enzyme provided similar amounts of filter paper (D1, L1), endoglucanase (D2, L2), alpha-L-arabinofuranosidase (D3, L3) and xylanase units (D4, L4) per gram forage DM. Enzymes increased the initial OMD in both fractions, with improvements of up to 15% in leaves (D4) and 8% in stems (L2) after 12 h incubation. All enzyme treatments increased the extent of degradation (96 h incubation) in the leaf fractions, but only L2 increased final OMD in the stems. Direct hydrolysis of forage fractions during the pre-treatment period did not fully account for the magnitude of the increases in OMD, suggesting that the increase in rate of degradation was achieved through a combined effect of direct enzyme hydrolysis and synergistic action between the exogenous (applied) and endogenous (rumen) enzymes. (C) 2003 Elsevier Science B.V. All rights reserved.

Whole crop cereals 2. Prediction of apparent digestibility and energy value from in vitro digestion techniques and near infrared reflectance spectroscopy and of chemical composition by near infrared reflectance spectroscopy

Samples of whole crop wheat (WCW, n = 134) and whole crop barley (WCB, n = 16) were collected from commercial farms in the UK over a 2-year period (2003/2004 and 2004/2005). Near infrared reflectance spectroscopy (NIRS) was compared with laboratory and in vitro digestibility measures to predict digestible organic matter in the dry matter (DOMD) and metabolisable energy (ME) contents measured in vivo using sheep. Spectral models using the mean spectra of two scans were compared with those using individual spectra (duplicate spectra). Overall NIRS accurately predicted the concentration of chemical components in whole crop cereals apart from crude protein. ammonia-nitrogen, water-soluble carbohydrates, fermentation acids and solubility values. In addition. the spectral models had higher prediction power for in vivo DOMD and ME than chemical components or in vitro digestion methods. Overall there Was a benefit from the use of duplicate spectra rather than mean spectra and this was especially so for predicting in vivo DOMD and ME where the sample population size was smaller. The spectral models derived deal equally well with WCW and WCB and Would he of considerable practical value allowing rapid determination of nutritive value of these forages before their use in diets of productive animals. (C) 2008 Elsevier B.V. All rights reserved.

The role of animal nutrition in improving the nutritive value of animal-derived foods in relation to chronic disease

Foods derived from animals are an important source of nutrients in the diet; for example, milk and meat together provide about 60 and 55% of the dietary intake of Ca and protein respectively in the UK. However, certain aspects of some animal-derived foods, particularly their fat and saturated fatty acid (SFA) contents, have led to concerns that these foods substantially contribute to the risk of CVD, the metabolic syndrome and other chronic diseases. In most parts of Europe dairy products are the greatest single dietary source of SFA. The fatty acid composition of various animal-derived foods is, however, not constant and can, in many cases, be enhanced by animal nutrition. In particular, milk fat with reduced concentrations of the C12-16 SFA and an increased concentration of 18:1 MUFA is achievable, although enrichment with very-long-chain n-3 PUFA is much less efficient. However, there is now evidence that some animal-derived foods (notably milk products) contain compounds that may actively promote long-term health, and research is urgently required to fully characterise the benefits associated with the consumption of these compounds and to understand how the levels in natural foods can be enhanced. It is also vital that the beneficial effects are not inadvertently destroyed in the process of reducing the concentrations of SFA. In the future the role of animal nutrition in creating foods closer to the optimum composition for long-term human health is likely to become increasingly important, but production of such foods on a scale that will substantially affect national diets will require political and financial incentives and great changes in the animal production industry.