# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On the numerical solution of the direct scattering problem for an open sound-hard arc. (English) Zbl 0854.65106
A boundary integral equation approach is used to solve the (Neumann) boundary value problem for the Helmholtz equation in ${ℝ}^{2}\setminus {\Gamma }$ modelling the scattering phenomena for time-harmonic acoustic waves by a sound-hard open arc ${\Gamma }$. (Such problems arise also in the analysis of cracks.) Using the so-called cosine substitution the integral equation is found to be essentially the same as that for a closed boundary, considered e.g. by R. Kress [J. Comput. Appl. Math. 61, No. 3, 345-360 (1995; Zbl 0839.65119)]. Hence, the integral equation is solved approximately by a quadrature formula method, and error estimates in Hölder norms are found by standard techniques, cf. S. G. Michlin, S. Prößdorf [Singuläre Integraloperatoren, Akademie-Verlag Berlin (1980; Zbl 0442.47027)]. A numerical example (bowl-shaped open arc) shows exponential convergence.
##### MSC:
 65N38 Boundary element methods (BVP of PDE) 76Q05 Hydro- and aero-acoustics 76M15 Boundary element methods (fluid mechanics) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation