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.

Scattering by infinite one-dimensional rough surfaces

We consider the Dirichlet boundary-value problem for the Helmholtz equation in a non-locally perturbed half-plane. This problem models time-harmonic electromagnetic scattering by a one-dimensional, infinite, rough, perfectly conducting surface; the same problem arises in acoustic scattering by a sound-soft surface. ChandlerWilde & Zhang have suggested a radiation condition for this problem, a generalization of the Rayleigh expansion condition for diffraction gratings, and uniqueness of solution has been established. Recently, an integral equation formulation of the problem has also been proposed and, in the special case when the whole boundary is both Lyapunov and a small perturbation of a flat boundary, the unique solvability of this integral equation has been shown by Chandler-Wilde & Ross by operator perturbation arguments. In this paper we study the general case, with no limit on surface amplitudes or slopes, and show that the same integral equation has exactly one solution in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including the incident plane wave, the Dirichlet boundary-value problem for the scattered field has a unique solution.

Salt and ionic cocrystalline forms of amides: protonation of carbamazepine in aqueous media

The products of reactions of the pharmaceutical amide carbamazepine (CBZ) with strong acids under aqueous conditions were investigated by both powder and single crystal X-ray diffraction. Despite previous claims to the contrary, it was found that salt forms with CBZ protonated at the amide O atom could be isolated from reactions with both HCl and HBr. These forms include the newly identified hydrate phase [CBZ(H)][Cl]·H O. Reactions with other mineral acids (HI and HBF ) gave ionic cocrystalline (ICC) forms (CBZ· [acridinium][I ]·2.5I and CBZ·[H O ] [BF ] ·H O) as well as the salt form CBZ·[CBZ(H)][BF ]·0.5H O. Reaction 2 4 3 2 5 2 0.25 4 0.25 2 4 2 of CBZ with a series of sulfonic acids also gave salt forms, namely, [CBZ(H)][O SC H ], [CBZ(H)][O SC H (OH)]· 3 6 5 3 6 4 0.5H O, [CBZ(H)] [O SCH CH SO ], and [CBZ(H)][O SC H (OH) (COOH)]·H O. CBZ and protonated CBZ(H) 2 2 3 2 2 3 3 6 3 2 moieties can be differentiated in the solid state both by changes to molecular geometry and by differing packing preferences