A generalized jump problem for the Helmholtz equation outside cuts on the plane

Крутицкий, П. А.; Prozorov, K. V.

doi:10.1007/s10625-005-0003-y

Cited by 13 publications

(48 citation statements)

References 3 publications

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“…is a single-layer potential with density function µ 2 , it is continuous in Γ and the jump relation of its normal derivative is standard [17,Theorem 5]. Now let us consider the angular potential v[µ 1 ](x), which is essentially a double-layer potential under the condition (2.4).…”

Section: Statement Of the Problem And Preliminary Resultsmentioning

confidence: 99%

“…Hence, compared with the scattering problems of obstacles with closed smooth boundary, these crack scattering problems are more complicated due to the eventual different surface impedances on both sides as well as the presence of the tips of crack. The inverse scattering problems for arc with Dirichlet or Neumann data on the arc has been considered in [12,14,17,18,22,30,33,34] using iterative methods. Physically, this means that the arc is sound-soft or sound hard.…”

Section: Introductionmentioning

confidence: 99%

“…Since the Cauchy singular operator has additional weak singularities at the ends, then we need to be more careful about the method of computation. On the other hand, the arc Γ is represented in [15][16][17] using the arc length for the parametrization, which is the restriction on this scheme. The novelty of our paper is the new numerical method for computing the integrals with high accuracy and therefore solving this very complicated system of integral equations efficiently.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space

Крутицкий¹,

Liu²,

Sini³

2011

JCM

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The wave scattering problem by a crack Γ in R 2 with impedance type boundary is considered. This problem models the diffraction of waves by thin two-sided cylindrical screens. A numerical method for solving the problem is developed. The solution of the problem is represented in the form of the combined angular potential and single-layer potential. The linear integral equations satisfied by the density functions are derived for general parameterized arcs. The weakly singular integrals and the Cauchy singular integral arising in these equations are computed using a highly accurate scheme with a truncation error analysis. The advantage of the scheme proposed in this paper is, in one hand, the fact that we do not need the analyticity property of the crack and we allow different complex valued surface impedances in both sides of the crack. In the other hand, we avoid the hyper-singular integrals. Numerical implementations showing the validity of the scheme are presented.Mathematics subject classification: 35P25, 35R30, 78A45.

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Section: Statement Of the Problem And Preliminary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space

Крутицкий¹,

Liu²,

Sini³

2011

JCM

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“…where we used the notation from (18). It was shown above that both U (x) and ∇U (x) are continuously extensible from D onto γ(d, /3), whence…”

Section: For F (S) Satisfying Conditions (5) Let Us Check That the Pmentioning

confidence: 99%

“…Assume that u(x) is decomposed according to (18). Using formulae (20) and setting r = |x − x(d)|, ψ = ψ(x, x(d)), we consider the integral over the sector P (d, /3):…”

Section: Nonexistence Of a Weak Solutionmentioning

confidence: 99%

On existence of a classical solution and nonexistence of a weak solution to the Dirichlet problem for the Laplacian with discontinuous boundary data

Крутицкий

2009

Quart. Appl. Math.

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Abstract. The Dirichlet problem for the Laplacian in a planar multiply connected interior domain bounded by smooth closed curves is considered in the case when the boundary data is piecewise continuous; i.e., it may have jumps in certain points of the boundary. It is assumed that the solution to the problem may not be continuous at the same points. The well-posed formulation of the problem is given, theorems on existence and uniqueness of a classical solution are proved, and the integral representation for a classical solution is obtained. The problem is reduced to a uniquely solvable Fredholm integral equation of the second kind and of index zero. It is shown that a weak solution to the problem does not exist typically, though the classical solution exists. Introduction.It is known that if the Dirichlet problem for the Laplacian is considered in a planar domain bounded by sufficiently smooth closed curves, and if the function specified in the boundary condition is smooth enough, then the existence of a classical solution follows from the existence of a weak solution. In the present paper we consider the Dirichlet problem for the Laplacian in a planar multiply connected interior domain bounded by closed curves on the assumption that the boundary data is piecewise continuous; i.e., it may have jumps in certain points of the boundary. We look for a solution to the problem which may be discontinuous at these points. We give the well-posed formulation of the problem. We prove that there exists a unique classical solution to this problem and obtain an integral representation for the classical solution in the form of a double layer potential with some additions. Moreover, we reduce the problem to a uniquely solvable Fredholm integral equation of the second kind and of index zero for the density of a potential.In addition, we prove that a weak solution to this problem may not exist. This result follows from the fact that the square of the gradient of the classical solution, basically, is

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Reconstruction of cracks of different types from far‐field measurements

Liu

Sini

2009

Math Methods in App Sciences

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In this paper, we deal with the acoustic inverse scattering problem for reconstructing cracks of possibly different types from the far-field map. The scattering problem models the diffraction of waves by thin two-sided cylindrical screens. The cracks are characterized by their shapes, the type of boundary conditions and the boundary coefficients (surface impedance). We give explicit formulas of the indicator function of the probe method, which can be used to reconstruct the shape of the cracks, distinguish their types of boundary conditions, the two faces of each of them and reconstruct the possible material coefficients on them by using the far-field map. To test the validity of these formulas, we present some numerical implementations for a single crack, which show the efficiency of the proposed method for suitably distributed surface impedances. The difficulties for numerically recovering the properties of the crack in the concave side as well as near the tips are presented and some explanations are given.

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A generalized jump problem for the Helmholtz equation outside cuts on the plane

Cited by 13 publications

References 3 publications

Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space

Numerical Solution of the Scattering Problem for Acoustic Waves by a Two-Sided Crack in 2-Dimensional Space

On existence of a classical solution and nonexistence of a weak solution to the Dirichlet problem for the Laplacian with discontinuous boundary data

Reconstruction of cracks of different types from far‐field measurements

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