Mixed boundary value problems of diffraction by a half‐plane with an obstacle perpendicular to the boundary

Abstract

The paper is devoted to the analysis of wave diffraction problems modelled by classes of mixed boundary conditions and the Helmholtz equation, within a half-plane with a crack. Potential theory together with Fredholm theory, and explicit operator relations, are conveniently implemented to perform the analysis of the problems. In particular, an interplay between Wiener-Hopf plus/minus Hankel operators and Wiener-Hopf operators assumes a relevant preponderance in the final results. As main conclusions, this study reveals conditions for the well-posedness of the corresponding boundary value problems in certain Sobolev spaces and equivalent reduction to systems of Wiener-Hopf equations.

New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

“…conversely, for any {a n } n=+∞ n=−∞ ∈ 2 , there exists a function ∈ L 2 ([− , ]) such that the values a n are the Fourier coefficients of . In view of this, the integral transform (11) defines an invertible bounded linear operator between two Hilbert spaces L 2 ([− , ]) and 2 in which the inversion formula is given by (12). Remark 1.…”

“…9 In the present paper, we will propose four new convolutions, which will be written on the basis of two parameters. Such parameters will enable these convolutions with extra levels of flexibility that are useful to enlarge the number of properties of their associated integral operators, as well as the number of possible applications (eg, some classes of wave diffraction problems formulated as boundary value problems [10][11][12][13][14] can be analyzed and solved with the help of convolution operators of Wiener-Hopf plus Hankel type).…”

Odd asymmetric factorization of Wiener-Hopf plus Hankel operators on variable exponent Lebesgue spaces

“…This is the case e.g. in wave diffraction theory [7,8] where Wiener-Hopf plus Hankel operators [2,3,4,9,10,11] are characterizing the equations (or systems of equations) which model some of that problems. Namely, this occurs when the geometry of diffraction objects present certain types of symmetry which give rise to the sum of a Wiener-Hopf and a Hankel operator.…”