An integral equation method for a boundary value problem arising in unsteady water wave problems

In this paper we consider the 2D Dirichlet boundary value problem for Laplace’s equation in a non-locally perturbed half-plane, with data in the space of bounded and continuous functions. We show uniqueness of solution, using standard Phragmen-Lindelof arguments. The main result is to propose a boundary integral equation formulation, to prove equivalence with the boundary value problem, and to show that the integral equation is well posed by applying a recent partial generalisation of the Fredholm alternative in Arens et al [J. Int. Equ. Appl. 15 (2003) pp. 1-35]. This then leads to an existence proof for the boundary value problem. Keywords. Boundary integral equation method, Water waves, Laplace’s

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 $A$?

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

A generalized collectively compact operator theory with an application to integral equations on unbounded domains

In this paper a generalization of collectively compact operator theory in Banach spaces is developed. A feature of the new theory is that the operators involved are no longer required to be compact in the norm topology. Instead it is required that the image of a bounded set under the operator family is sequentially compact in a weaker topology. As an application, the theory developed is used to establish solvability results for a class of systems of second kind integral equations on unbounded domains, this class including in particular systems of Wiener-Hopf integral equations with L1 convolutions kernels

A Nyström method for a class of integral equations on the real line with applications to scattering by diffraction gratings and rough surfaces

We propose a Nystr¨om/product integration method for a class of second kind integral equations on the real line which arise in problems of two-dimensional scalar and elastic wave scattering by unbounded surfaces. Stability and convergence of the method is established with convergence rates dependent on the smoothness of components of the kernel. The method is applied to the problem of acoustic scattering by a sound soft one-dimensional surface which is the graph of a function f, and superalgebraic convergence is established in the case when f is infinitely smooth. Numerical results are presented illustrating this behavior for the case when f is periodic (the diffraction grating case). The Nystr¨om method for this problem is stable and convergent uniformly with respect to the period of the grating, in contrast to standard integral equation methods for diffraction gratings which fail at a countable set of grating periods.

Asymptotic behavior at infinity of solutions of multidimensional second kind integral equations

We consider second kind integral equations of the form x(s) - (abbreviated x - K x = y ), in which Ω is some unbounded subset of Rn. Let Xp denote the weighted space of functions x continuous on Ω and satisfying x (s) = O(|s|-p ),s → ∞We show that if the kernel k(s,t) decays like |s — t|-q as |s — t| → ∞ for some sufficiently large q (and some other mild conditions on k are satisfied), then K ∈ B(XP) (the set of bounded linear operators on Xp), for 0 ≤ p ≤ q. If also (I - K)-1 ∈ B(X0) then (I - K)-1 ∈ B(XP) for 0 < p < q, and (I- K)-1∈ B(Xq) if further conditions on k hold. Thus, if k(s, t) = O(|s — t|-q). |s — t| → ∞, and y(s)=O(|s|-p), s → ∞, the asymptotic behaviour of the solution x may be estimated as x (s) = O(|s|-r), |s| → ∞, r := min(p, q). The case when k(s,t) = к(s — t), so that the equation is of Wiener-Hopf type, receives especial attention. Conditions, in terms of the symbol of I — K, for I — K to be invertible or Fredholm on Xp are established for certain cases (Ω a half-space or cone). A boundary integral equation, which models three-dimensional acoustic propaga-tion above flat ground, absorbing apart from an infinite rigid strip, illustrates the practical application and sharpness of the above results. This integral equation mod-els, in particular, road traffic noise propagation along an infinite road surface sur-rounded by absorbing ground. We prove that the sound propagating along the rigid road surface eventually decays with distance at the same rate as sound propagating above the absorbing ground.

On asymptotic behavior at infinity and the finite section method for integral equations on the half-line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

Seasonal pattern of regional carbon balance in the central Rocky Mountains from surface and airborne measurements

[1] High-elevation forests represent a large fraction of potential carbon uptake in North America, but this uptake is not well constrained by observations. Additionally, forests in the Rocky Mountains have recently been severely damaged by drought, fire, and insect outbreaks, which have been quantified at local scales but not assessed in terms of carbon uptake at regional scales. The Airborne Carbon in the Mountains Experiment was carried out in 2007 partly to assess carbon uptake in western U.S. mountain ecosystems. The magnitude and seasonal change of carbon uptake were quantified by (1) paired upwind-downwind airborne CO2 observations applied in a boundary layer budget, (2) a spatially explicit ecosystem model constrained using remote sensing and flux tower observations, and (3) a downscaled global tracer transport inversion. Top-down approaches had mean carbon uptake equivalent to flux tower observations at a subalpine forest, while the ecosystem model showed less. The techniques disagreed on temporal evolution. Regional carbon uptake was greatest in the early summer immediately following snowmelt and tended to lessen as the region experienced dry summer conditions. This reduction was more pronounced in the airborne budget and inversion than in flux tower or upscaling, possibly related to lower snow water availability in forests sampled by the aircraft, which were lower in elevation than the tower site. Changes in vegetative greenness associated with insect outbreaks were detected using satellite reflectance observations, but impacts on regional carbon cycling were unclear, highlighting the need to better quantify this emerging disturbance effect on montane forest carbon cycling.

A geomatics-based approach for the derivation of the spatial distribution of sediment transport processes in periglacial mountain environments

Within this paper modern techniques such as satellite image analysis and tools provided by geographic information systems (GIS.) are exploited in order to extend and improve existing techniques for mapping the spatial distribution of sediment transport processes. The processes of interest comprise mass movements such as solifluction, slope wash, dirty avalanches and rock- and boulder falls. They differ considerably in nature and therefore different approaches for the derivation of their spatial extent are required. A major challenge is addressing the differences between the comparably coarse resolution of the available satellite data (Landsat TM/ETM+, 30 in x 30 m) and the actual scale of sediment transport in this environment. A three-stepped approach has been developed which is based on the concept of Geomorphic Process Units (GPUs): parameterization, process area delineation and combination. Parameters include land cover from satellite data and digital elevation model derivatives. Process areas are identified using a hierarchical classification scheme utilizing thresholds and definition of topology. The approach has been developed for the Karkevagge in Sweden and could be successfully transferred to the Rabotsbekken catchment at Okstindan, Norway using similar input data. Copyright (C) 2008 John Wiley & Sons, Ltd.

Quantifying sediment transport processes in periglacial mountain environments at a catchment scale using geomorphic process units

The research record on the quantification of sediment transport processes in periglacial mountain environments in Scandimvia dates back to the 1950s. A wide range of measurements is. available, especially from the Karkevagge region of northern Sweden. Within this paper satellite image analysis and tools provided by geographic information systems (GIS) are exploited in order to extend and improve this research and to complement geophysical methods. The processes of interest include mass movements such as solifluction, slope wash, dirty avalanches and rock-and boulder falls. Geomorphic process units have been derived in order to allow quantification via GIS techniques at a catchment scale. Mass movement rates based on existing Field measurements are employed in the budget calculation. In the Karkevagge catch ment. 80% of the area can be identified either as a source area for sediments or as a zone where sediments are deposited. The overall budget for the slopes beneath the rockwalls in the Karkevagge is approximately 680 t a(-1) whilst about 150 : a-1 are transported into the fluvial System.

Soils and their distribution on Bambouto volcanic mountain, West Cameroon highland, Central Africa

Morphological, physical and chemical studies were carried out on soils of Mount Bambouto, a volcanic mountain of the West Cameroon highland. These studies show that the soils of this region can be divided into seven groups according to Soils Taxonomy USA [Soil taxonomy: a basic system of soil classification for making and interpreting soils surveys: USDA Agriculture Handbook 436: Washington, DC, US Government Pronting Office, 1975, 754]: lithic dystrandept soils, typical dystrandept soils, oxic dystrandept soils, typical haplohumox soils, typical kandiudox soils, tropopsamment soils and umbriaquox soils. A soils map of this region at scale 1:50,000 has been drawn up, using the seven soils groups above as soil cartography units. These soils are organised into of three main categories: soils with andic characteristics in the upper region of the mountain (lithic dystrandept soils, typical dystrandept soils and oxic dystrandept soils); ferrallitic soils in the lower part of the mountain (typical haplohumox soils and typical kandiudox soils) and imperfectly developed soils (tropopsamment soils and umbraquox soils).