Boundary value problems for the Helmholtz equation in a half-plane

The Dirichlet and impedance boundary value problems for the Helmholtz equation in a half-plane with bounded continuous boundary data are studied. For the Dirichlet problem the solution can be constructed explicitly. We point out that, for wavenumbers k > 0, the solution, although it satisfies a limiting absorption principle, may increase in magnitude with distance from the boundary. Using the explicit solution we propose a novel radiation condition which we utilise in formulating the impedance boundary value problem. By reformulating this problem as a boundary integral equation we prove uniqueness and existence of solution for a certain range of admissable impedance boundary data

Approximate solution of second kind integral equations on infinite cylindrical surfaces

The paper considers second kind integral equations of the form (abbreviated)φφK+=g, in which S is an infinite cylindrical surface of arbitrary smooth cross-section. The “truncated equation” (abbreviated )aaaaKEφφ+=g, obtained by replacing S by Sa, a closed bounded surface of class C2, the boundary of a section of the interior of S of length 2a, is also discussed. Conditions on k are obtained (in particular, implying that K commutes with the operation of translation in the direction of the cylinder axis) which ensure that I-K is invertible, that I - Ka is invertible and (I — Ka)-1 uniformly bounded for all sufficiently large a, and that aφ converges to φ in an appropriate sense as ∞→a. Uniform stability and convergence results for a piecewise constant boundary element collocation method for the truncated equations are also obtained. A boundary integral equation, which models three-dimensional acoustic scattering from an infi- nite rigid cylinder, illustrates the application of the above results to prove existence of solution (of the integral equation and the corresponding boundary value problem) and convergence of a particular collocation method

Asymptotic behaviour at infinity of solutions of second kind integral equations on unbounded regions of Rn

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

Scattering by one-dimensional rough surfaces

We consider the two-dimensional Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane. We look for a solution in the form of a double-layer potential using, as fundamental solution, the Green's function for the impedance half-plane. This leads to a boundary integral equation which can be solved for any bounded and continuous boundary data provided the boundary itself does not differ too much from the flat boundary {(x1,h) ∈ R2 : x1 ∈ R} (h > 0). We show this by calculating the symbol of the integral operator in the integral equa- tion in the flat boundary case, and then using standard operator perturbation results. Continuous dependence of the solution on the shape of the boundary is shown

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

We 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

A high frequency boundary element method for scattering by a class of nonconvex obstacles

This is a post-peer-review, pre-copyedit version of an article published in Numerische Mathematik. The final authenticated version is available online at: https://doi.org/10.1007/s00211-014-0648-7EPSR

Sobolev spaces on non-Lipschitz subsets of Rn with application to boundary integral equations on fractal screens

We study properties of the classical fractional Sobolev spaces on non-Lipschitz subsets of Rn. We investigate the extent to which the properties of these spaces, and the relations between them, that hold in the well-studied case of a Lipschitz open set, generalise to non-Lipschitz cases. Our motivation is to develop the functional analytic framework in which to formulate and analyse integral equations on non-Lipschitz sets. In particular we consider an application to boundary integral equations for wave scattering by planar screens that are non-Lipschitz, including cases where the screen is fractal or has fractal boundary

Computing Fresnel integrals via modified trapezium rules

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement

On stability of discretizations of the Helmholtz equation (extended version)

We review the stability properties of several discretizations of the Helmholtz equation at large wavenumbers. For a model problem in a polygon, a complete $k$-explicit stability (including $k$-explicit stability of the continuous problem) and convergence theory for high order finite element methods is developed. In particular, quasi-optimality is shown for a fixed number of degrees of freedom per wavelength if the mesh size $h$ and the approximation order $p$ are selected such that $kh/p$ is sufficiently small and $p = O(\log k)$, and, additionally, appropriate mesh refinement is used near the vertices. We also review the stability properties of two classes of numerical schemes that use piecewise solutions of the homogeneous Helmholtz equation, namely, Least Squares methods and Discontinuous Galerkin (DG) methods. The latter includes the Ultra Weak Variational Formulation

Multiplexed identification, quantification and genotyping of infectious agents using a semiconductor biochip

The emergence of pathogens resistant to existing antimicrobial drugs is a growing worldwide health crisis that threatens a return to the pre-antibiotic era. To decrease the overuse of antibiotics, molecular diagnostics systems are needed that can rapidly identify pathogens in a clinical sample and determine the presence of mutations that confer drug resistance at the point of care. We developed a fully integrated, miniaturized semiconductor biochip and closed-tube detection chemistry that performs multiplex nucleic acid amplification and sequence analysis. The approach had a high dynamic range of quantification of microbial load and was able to perform comprehensive mutation analysis on up to 1,000 sequences or strands simultaneously in <2 h. We detected and quantified multiple DNA and RNA respiratory viruses in clinical samples with complete concordance to a commercially available test. We also identified 54 drug-resistance-associated mutations that were present in six genes of Mycobacterium tuberculosis, all of which were confirmed by next-generation sequencing