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.

A benchmark method for global optimization problems in structure determination from powder diffraction data

Quasi-Newton-Raphson minimization and conjugate gradient minimization have been used to solve the crystal structures of famotidine form B and capsaicin from X-ray powder diffraction data and characterize the chi(2) agreement surfaces. One million quasi-Newton-Raphson minimizations found the famotidine global minimum with a frequency of ca 1 in 5000 and the capsaicin global minimum with a frequency of ca 1 in 10 000. These results, which are corroborated by conjugate gradient minimization, demonstrate the existence of numerous pathways from some of the highest points on these chi(2) agreement surfaces to the respective global minima, which are passable using only downhill moves. This important observation has significant ramifications for the development of improved structure determination algorithms.

Will rising household incomes solve China's micronutrient deficiency problems?

Rapid economic growth in China has resulted in substantially improved household incomes. Diets have also changed, with a movement away from traditional foods and towards animal products and processed foods. Yet micronutrient deficiencies, particularly for calcium and vitamin A, are still widespread in China. In this research we model the determinants of the intakes of these micronutrients using household panel data, asking particularly whether continuing income increases are likely to cause the deficiencies to be overcome. Nonparametric kernel regressions and random effects panel regression models are employed. The results show a statistically significant but relatively small positive income effect on both nutrient intakes. The local availability of milk is seen to have a strong positive effect on intakes of both micronutrients. Thus, rather than relying on increasing incomes to overcome deficiencies, supplementary government policies, such as school milk programmes, may be warranted.

Mediterranean ecosystems: problems and tools for conservation

Mediterranean ecosystems rival tropical ecosystems in terms of plant biodiversity. The Mediterranean Basin (MB) itself hosts 25 000 plant species, half of which are endemic. This rich biodiversity and the complex biogeographical and political issues make conservation a difficult task in the region. Species, habitat, ecosystem and landscape approaches have been used to identify conservation targets at various scales: ie, European, national, regional and local. Conservation decisions require adequate information at the species, community and habitat level. Nevertheless and despite recent improvements/efforts, this information is still incomplete, fragmented and varies from one country to another. This paper reviews the biogeographic data, the problems arising from current conservation efforts and methods for the conservation assessment and prioritization using GIS. GIS has an important role to play for managing spatial and attribute information on the ecosystems of the MB and to facilitate interactions with existing databases. Where limited information is available it can be used for prediction when directly or indirectly linked to externally built models. As well as being a predictive tool today GIS incorporate spatial techniques which can improve the level of information such as fuzzy logic, geostatistics, or provide insight about landscape changes such as 3D visualization. Where there are limited resources it can assist with identifying sites of conservation priority or the resolution of environmental conflicts (scenario building). Although not a panacea, GIS is an invaluable tool for improving the understanding of Mediterranean ecosystems and their dynamics and for practical management in a region that is under increasing pressure from human impact.