A fast two-grid and finite section method for a class of integral equations on the real line with application to an acoustic scattering problem in the half-plane

We consider the numerical treatment of second kind integral equations on the real line of the form ∅(s) = ∫_(-∞)^(+∞)▒〖κ(s-t)z(t)ϕ(t)dt,s=R〗 (abbreviated ϕ= ψ+K_z ϕ) in which K ϵ L_1 (R), z ϵ L_∞ (R) and ψ ϵ BC(R), the space of bounded continuous functions on R, are assumed known and ϕ ϵ BC(R) is to be determined. We first derive sharp error estimates for the finite section approximation (reducing the range of integration to [-A, A]) via bounds on (1-K_z )^(-1)as an operator on spaces of weighted continuous functions. Numerical solution by a simple discrete collocation method on a uniform grid on R is then analysed: in the case when z is compactly supported this leads to a coefficient matrix which allows a rapid matrix-vector multiply via the FFT. To utilise this possibility we propose a modified two-grid iteration, a feature of which is that the coarse grid matrix is approximated by a banded matrix, and analyse convergence and computational cost. In cases where z is not compactly supported a combined finite section and two-grid algorithm can be applied and we extend the analysis to this case. As an application we consider acoustic scattering in the half-plane with a Robin or impedance boundary condition which we formulate as a boundary integral equation of the class studied. Our final result is that if z (related to the boundary impedance in the application) takes values in an appropriate compact subset Q of the complex plane, then the difference between ϕ(s)and its finite section approximation computed numerically using the iterative scheme proposed is ≤C_1 [kh log⁡〖(1⁄kh)+(1-Θ)^((-1)⁄2) (kA)^((-1)⁄2) 〗 ] in the interval [-ΘA,ΘA](Θ<1) for kh sufficiently small, where k is the wavenumber and h the grid spacing. Moreover this numerical approximation can be computed in ≤C_2 N log⁡N operations, where N = 2A/h is the number of degrees of freedom. The values of the constants C1 and C2 depend only on the set Q and not on the wavenumber k or the support of z.

The impedance boundary value problem for the Helmholtz equation in a half-plane

We prove unique existence of solution for the impedance (or third) boundary value problem for the Helmholtz equation in a half-plane with arbitrary L∞ boundary data. This problem is of interest as a model of outdoor sound propagation over inhomogeneous flat terrain and as a model of rough surface scattering. To formulate the problem and prove uniqueness of solution we introduce a novel radiation condition, a generalization of that used in plane wave scattering by one-dimensional diffraction gratings. To prove existence of solution and a limiting absorption principle we first reformulate the problem as an equivalent second kind boundary integral equation to which we apply a form of Fredholm alternative, utilizing recent results on the solvability of integral equations on the real line in [5].

Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane

We consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane, this problem arising in electromagnetic scattering by one-dimensional rough, perfectly conducting surfaces. We propose a new boundary integral equation formulation for this problem, utilizing the Green's function for an impedance half-plane in place of the standard fundamental solution. We show, at least for surfaces not differing too much from the flat boundary, that the integral equation is uniquely solvable in the space of bounded and continuous functions, and hence that, for a variety of incident fields including an incident plane wave, the boundary value problem for the scattered field has a unique solution satisfying the limiting absorption principle. Finally, a result of continuous dependence of the solution on the boundary shape is obtained.

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.

Decreased monsoon precipitation in the Northern Hemisphere due to anthropogenic aerosols

The Northern Hemisphere monsoons are an integral component of Earth's hydrological cycle and affect the lives of billions of people. Observed precipitation in the monsoon regions underwent substantial changes during the second half of the 20th century, with drying from the 1950s to mid-1980s and increasing precipitation in recent decades. Modeling studies suggest anthropogenic aerosols has been a key factor driving changes in tropical and monsoon precipitation. Here we apply detection and attribution methods to determine whether observed changes are driven by human influences using fingerprints of individual forcings (i.e. greenhouse gas, anthropogenic aerosol and natural) derived from climate models. The results show that the observed changes can only be explained when including the influence of anthropogenic aerosols, even after accounting for internal climate variability. Anthropogenic aerosol, not greenhouse gas or natural forcing, has been the dominant influence on Northern Hemisphere monsoon precipitation over the second half of the 20th century.

From services to civilian: the geographies of veterans’ post-military lives

This paper explores the spatiality of (post)military identities, demonstrating the continuing impact of having been part of the military community despite the passage of time. Our tri-service respondents highlighted the challenges faced even by those deemed to have ‘successfully’ transitioned to ‘Civvy Street’, articulating discourses of loss and separation. While some had achieved closure with their past military selves, others struggled and became stuck in a liminal space between civilian and military lives that perpetuated feelings of isolation. Our work contributes to understandings of military geographies and highlights the importance of conceptualising post-institutional transitions as a process in order to understand how individuals negotiate their identities in changing spatial circumstances.

Children’s perspectives on their working lives and on public action against child labour in Burkina Faso

The majority of children in Burkina Faso undertake work and, compared to other West African countries, it has the highest rate of children involved in hazardous work (Diallo 2008; NISD 2008), which is one working child out of two. Based on a qualitative survey, this chapter presents the perspectives of Burkinabe children working in two of the hazardous sectors: a quarry and an artisanal mine. The findings show that children acknowledge very difficult working conditions in these sites. However a variety of reasons maintain them in work, which they perceive as a solution and thus challenging action against child labour in Burkina Faso.

Recovering lost lives: researching women in legal history

Drawing on the research I undertook into the life of Gwyneth Bebb, who in 1913 challenged the Law Society of England and Wales for their refusal to admit women to the solicitors’ profession, this article focuses on the range of sources one might use to explore the lives of women in law, about whom there might be a few public records but little else, and on the ways in which sources, even official ones, might be imaginatively used. It traces the research process from the case that inspired the research (Bebb v The Law Society [1914] 1 Ch 286) through to the creation of an entry in the Oxford Dictionary of National Biography and what this means for women’s history, emphasising the importance of asking the ‘woman question’ and seeking out the broader significance of a woman’s life in the context of her times.

Half-remembered quotations from mostly forgotten speeches: the limits of Labour’s European policy discourse

This article uses discourse analysis to study the continuities in British foreign policy thinking within the Labour party from the 1960s to the present day. Using representative extracts from speeches by Hugh Gaitskell, Harold Wilson, Tony Blair and Gordon Brown, it identifies the ideational consis- tencies in the leaders’ attitudes to: Empire; federalism in the EEC/EU; and laying down conditions that have to be met before any constructive engagement with ‘Europe’ can be countenanced. We argue that these consistencies, spanning a 50-year period, exemplify a certain stagnation both within Labour’s European discourses and within British foreign policy thinking more widely. We develop the idea that Labour party thinking has been crucially framed by both small ‘c’ conser- vative and upper-case Conservative ideology, popularised by Winston Churchill in his ‘three circles’ model of British foreign policy.

Caring responsibilities, change and transitions in young people's family lives in Zambia

Despite diversity in family dynamics within and between societies, globally, it is adults who are usually relied upon to care for family members who are sick, disabled or have other care needs. Young people in Zambia and other African countries affected by the HIV epidemic are under increasing pressure to obtain a good education and employment to support their families, whilst some also have to cope with the loss of parents and care for relatives with little external support. This article discusses the findings of qualitative research that explored the experiences of young people (aged 14-30) who had significant family caring responsibilities and those without such responsibilities in Zambia. Interviews and life-mapping methods were conducted with a total of 35 young people living in rural and urban areas, 12 parents and relatives and 12 professionals. We analyse young people's experiences and perceptions of socially expected transitions, such as completing education and earning an income to support themselves and their families, in addition to more unpredictable changes in young people's family lives. ‘Critical moments’ (Thomson et al, 2002), such as bereavement and loss of parents and other family members, disinheritance of assets and property grabbing, migration and mobility between different relatives homes, parental divorce and separation, often had significant impacts on young people’s ability to navigate their pathways to adulthood according to wider social norms and expectations. A more relational conceptualisation of youth transitions is needed that takes account of young people's caring responsibilities and changing family dynamics.

Those Essex girls: the lives and letters of Lettice Knollys, Penelope Rich, Dorothy Perrott Percy, and Frances Walsingham

This essay examines the lives and letters of four of the most influential women at the court of Queen Elizabeth I: Lettice Knollys her daughters Penelope Rich and Dorothy Perrott Percy and her daughter-in-law Frances Walsingham, all of whom lived scandalous but powerful lives.

Evolution PDEs and augmented eigenfunctions. half line.

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.