The oblique derivative problem for the Helmholtz equation and scattering tidal waves

Крутицкий, П. А.

doi:10.1098/rspa.2001.0787

Cited by 12 publications

(13 citation statements)

References 17 publications

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“…To satisfy the boundary conditions, we put (10) in (3), use limit formulas for potentials [1,2,8], and arrive at the system of the integral equations for the densities mðsÞ; nðsÞ…”

Section: Integral Equations At the Boundarymentioning

confidence: 99%

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The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

Крутицкий

2004

Journal of Differential Equations

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The mixed Dirichlet-Neumann problem for the Laplace equation in a bounded connected plane domain with cuts (cracks) is studied. The Neumann condition is given on closed curves making up the boundary of a domain, while the Dirichlet condition is specified on the cuts. The existence of a classical solution is proved by potential theory and boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density in potentials satisfies the uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of cuts are investigated. r

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“…To satisfy the boundary conditions, we put (10) in (3), use limit formulas for potentials [1,2,8], and arrive at the system of the integral equations for the densities mðsÞ; nðsÞ…”

Section: Integral Equations At the Boundarymentioning

confidence: 99%

“…Dirichlet and Neumann problems for propagative Helmholtz equation in unbounded domains with cracks have been studied and reduced to the uniquely solvable integral equations in [4,7]. The oblique derivative problems in the exterior of cuts in a plane for Laplace and Helmholtz equations were treated in [3,8].…”

Section: Introductionmentioning

confidence: 99%

The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

Крутицкий

2004

Journal of Differential Equations

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“…As shown in [1], [5], [12,Appendix], for such µ(σ) the angular potential w 1 [µ](x) belongs to the class K. In particular, the inequality (1) holds with = −q if q ∈ (0, 1). Moreover, integrating w 1 [µ](x) by parts and using (5), we express the angular potential in terms of a double layer potential…”

Section: ) and Satisfies Conditions (5)mentioning

confidence: 99%

“…It follows from [1,5,17], [12,Appendix] that for such µ(s), ν(s) the function (6) belongs to the class K and satisfies all the conditions of the problem U except the boundary conditions (2a).…”

Section: Is Given By (4) By H[ν µ](X)mentioning

confidence: 99%

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On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

Крутицкий

2007

Quart. Appl. Math.

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Abstract. The mixed Dirichlet-Neumann problem for the Laplace equation in an unbounded plane domain with cuts (cracks) is studied. The Dirichlet condition is given on closed curves making up the boundary of the domain, while the Neumann condition is specified on the cuts. The existence of a classical solution is proved by potential theory and a boundary integral equation method. The integral representation for a solution is obtained in the form of potentials. The density of the potentials satisfies a uniquely solvable Fredholm integral equation of the second kind and index zero. Singularities of the gradient of the solution at the tips of the cuts are investigated. Introduction. The boundary of a 2-D cracked domain consists of both closed curves and open arcs (cuts).Open arcs model cracks in solids and screens or wings in fluids. Different physical processes in cracked domains can be described by boundary value problems for the Laplace equation, for example, distribution of stationary heat and electric fields in cracked solids, electric flow in cracked semiconductors, flow of an ideal fluid over several obstacles and wings, etc. Appropriate boundary conditions must be specified on the total boundary, i.e., on both closed curves and open arcs (cracks). The Neumann boundary condition reflects the nonflow (of fluid, electric current, etc.) through the boundary. The Dirichlet boundary condition corresponds to the given temperature in heat theory, fluid pressure in hydrodynamics, electric potential in electrostatics, etc.Boundary value problems with mixed boundary conditions were not treated in cracked domains by rigorous mathematical methods before. Even in the case of Laplace and Helmholtz equations the problems in domains bounded by closed curves [2], [13]- [17] and problems in the exterior of cuts (cracks) [14,16], [18]-[20] were treated separately, because different methods were used in their analysis. Previously the Neumann problem in the exterior of a cut was reduced to a hypersingular integral equation [14,16,18,19] or

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On the diffraction of Poincaré waves

Martin

2001

Math Methods in App Sciences

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The diffraction of tidal waves (Poincaré waves) by islands and barriers on water of constant finite depth is governed by the two‐dimensional Helmholtz equation. One effect of the Earth's rotation is to complicate the boundary condition on rigid boundaries: a linear combination of the normal and tangential derivatives is prescribed. (This would be an oblique derivative if the coefficients were real.) Corresponding boundary‐value problems are treated here using layer potentials, generalizing the usual approach for the standard exterior boundary‐value problems of acoustics. Singular integral equations are obtained for islands (scatterers with non‐empty interiors) whereas hypersingular integral equations are obtained for thin barriers. Copyright © 2001 John Wiley & Sons, Ltd.

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The oblique derivative problem for the Helmholtz equation and scattering tidal waves

Abstract: The oblique derivative problem for the two-dimensional Helmholtz equation in the exterior of cuts in the plane is reduced to a Fredholm integral equation of the second kind, which is uniquely solvable. The problem describes scattering tidal waves by reefs and spits.

Cited by 12 publications

References 17 publications

The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks

On the mixed problem for harmonic functions in a 2-D exterior cracked domain with Neumann condition on cracks

On the diffraction of Poincaré waves

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