Search results for "quadrature domain"

showing 3 items of 3 documents

Free boundary methods and non-scattering phenomena

2021

We study a question arising in inverse scattering theory: given a penetrable obstacle, does there exist an incident wave that does not scatter? We show that every penetrable obstacle with real-analytic boundary admits such an incident wave. At zero frequency, we use quadrature domains to show that there are also obstacles with inward cusps having this property. In the converse direction, under a nonvanishing condition for the incident wave, we show that there is a dichotomy for boundary points of any penetrable obstacle having this property: either the boundary is regular, or the complement of the obstacle has to be very thin near the point. These facts are proved by invoking results from t…

FOS: Physical sciencesBoundary (topology)01 natural sciencesinversio-ongelmatTheoretical Computer ScienceMathematics - Analysis of PDEsMathematics (miscellaneous)ConverseFOS: MathematicsPoint (geometry)0101 mathematicsMathematical PhysicsComplement (set theory)MathematicsosittaisdifferentiaaliyhtälötQuadrature domainsScatteringApplied MathematicsResearch010102 general mathematicsMathematical analysisMathematical Physics (math-ph)010101 applied mathematicsComputational MathematicsObstacleInverse scattering problemAnalysis of PDEs (math.AP)Research in the Mathematical Sciences
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Quadrature Formula Based on Interpolating Polynomials: Algorithmic and Computational Aspects

2007

The aim of this article is to obtain a quadrature formula for functions in several variables and to analyze the algorithmic and computational aspects of this formula. The known information about the integrand is {λi(f)}i=1n, where λi are linearly independent linear functionals. We find a form of the coefficients of the quadrature formula which can be easy used in numerical calculations. The main algorithm we use in order to obtain the coefficients and the remainder of the quadrature formula is based on the Gauss elimination by segments method. We obtain an expression for the exactness degree of the quadrature formula. Finally, we analyze some computational aspects of the algorithm in the pa…

Quadrature domainsMathematical analysisGauss–Laguerre quadratureTanh-sinh quadratureGauss–Kronrod quadrature formulaMathematics::Numerical Analysissymbols.namesakesymbolsGauss–Jacobi quadratureGaussian quadratureApplied mathematicsGauss–Hermite quadratureClenshaw–Curtis quadratureMathematics
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Quadrature domains for the Helmholtz equation with applications to non-scattering phenomena

2022

In this paper, we introduce quadrature domains for the Helmholtz equation. We show existence results for such domains and implement the so-called partial balayage procedure. We also give an application to inverse scattering problems, and show that there are non-scattering domains for the Helmholtz equation at any positive frequency that have inward cusps.

metaharmonic functionsmatematiikkapartial balayageyhtälötmean value theoremMathematics::Numerical Analysis35J05 35J15 35J20 35R30 35R35quadrature domainnon-scattering phenomenaMathematics - Analysis of PDEsFOS: MathematicsHelmholtz equationacoustic equationAnalysisAnalysis of PDEs (math.AP)
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