Geodesic
Dirichlet and Neumann boundary value problems for Helmholtz equation in unbounded domains with piecewise smooth part of boundary
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 148 (2006) no. 3, pp. 94-108

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In this paper we study the boundary value problems modelling scattering waves by a domain with the rough boundary. We assume that a domain is the half plane and a finite part of boundary is characterized by a piecewise smooth function. We also assumed that singularities of boundaries are the edges. We prove the theorems of existence and uniqueness of solution of the boundary value problems. We find the integral equations of second kind and we show that these equations are equivalent to the boundary value problems. We propose the numerical algorithms for scattering problems. They are based on the spline-subdomains method for integral equations. We establish the convergence of this numerical algorithm. 
@article{UZKU_2006_148_3_a7,
     author = {E. K. Lipachev},
     title = {Dirichlet and {Neumann} boundary value problems for {Helmholtz} equation in unbounded domains with piecewise smooth part of boundary},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {94--108},
     publisher = {mathdoc},
     volume = {148},
     number = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2006_148_3_a7/}
}
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E. K. Lipachev. Dirichlet and Neumann boundary value problems for Helmholtz equation in unbounded domains with piecewise smooth part of boundary. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 148 (2006) no. 3, pp. 94-108. http://geodesic.mathdoc.fr/item/UZKU_2006_148_3_a7/