Wave diffraction by a half-plane with an obstacle perpendicular to the boundary

Castro, L. P.; Kapanadze, David

doi:10.1016/j.jde.2012.08.030

Cited by 9 publications

(6 citation statements)

References 25 publications

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“…conversely, for any {a n } n=+∞ n=−∞ ∈ 2 , there exists a function ∈ L 2 ([− , ]) such that the values a n are the Fourier coefficients of . In view of this, the integral transform (11) defines an invertible bounded linear operator between two Hilbert spaces L 2 ([− , ]) and 2 in which the inversion formula is given by (12). Remark 1.…”

Section: Definition 2 (Finite Two-parameter Fourier-type Transformatimentioning

confidence: 99%

“…9 In the present paper, we will propose four new convolutions, which will be written on the basis of two parameters. Such parameters will enable these convolutions with extra levels of flexibility that are useful to enlarge the number of properties of their associated integral operators, as well as the number of possible applications (eg, some classes of wave diffraction problems formulated as boundary value problems [10][11][12][13][14] can be analyzed and solved with the help of convolution operators of Wiener-Hopf plus Hankel type).…”

Section: Introductionmentioning

confidence: 99%

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New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

Castro

Guerra

Tuan³

2020

Math Meth Appl Sci

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We propose four new convolutions exhibiting convenient factorization properties associated with two finite interval integral transformations of Fourier‐type together with their norm inequalities. Moreover, we study the solvability of a class of integral equations of Wiener‐Hopf plus Hankel type (on finite intervals) with the help of the factorization identities of such convolutions. Fourier‐type series are used to produce the solution formula of such equations, and a Shannon‐type sampling formula is also obtained.

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Section: Definition 2 (Finite Two-parameter Fourier-type Transformatimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

Castro

Guerra

Tuan³

2020

Math Meth Appl Sci

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“…But the setting is classical only (i.e., in Hilbert spaces) and the approach applies only to Dirichlet and Neumann but not to impedance conditions. Other known results are either limited to special situations such as the case of rectangles and depend strongly on the geometry of the domain [4,5,6,7,9,10,12,13,14,38] or rather sophisticated analytical methods are applied [29,50], or precise settings of appropriate function spaces are missing (see, e.g., [33,47]). For a historical survey and for further references we recommend [8,48,50].…”

Section: Below)mentioning

confidence: 99%

Mixed impedance boundary value problems for the Laplace–Beltrami equation

Castro¹,

Duduchava²,

Speck³

2020

J. Integral Equations Applications

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This work is devoted to the analysis of the mixed impedance-Neumann-Dirichlet boundary value problem (MIND BVP) for the Laplace-Beltrami equation on a compact smooth surface C with smooth boundary. We prove, using the Lax-Milgram Lemma, that this MIND BVP has a unique solution in the classical weak setting H 1 (C) when considering positive constants in the impedance condition. The main purpose is to consider the MIND BVP in a nonclassical setting of the Bessel potential space H s p (C), for s > 1/p, 1 < p < ∞. We apply a quasilocalization technique to the MIND BVP and obtain model Dirichlet-Neumann, Dirichletimpedance and Neumann-impedance BVPs for the Laplacian in the half-plane. The model mixed Dirichlet-Neumann BVP was investigated by R. Duduchava and M. Tsaava (2018).The other two are investigated in the present paper. This allows to write a necessary and sufficient condition for the Fredholmness of the MIND BVP and to indicate a large set of the space parameters s > 1/p and 1 < p < ∞ for which the initial BVP is uniquely solvable in the nonclassical setting. As a consequence, we prove that the MIND BVP has a unique solution in the classical weak setting H 1 (C) for arbitrary complex values of the nonzero constant in the impedance condition.

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“…It is also clear that convolution type equations are very often used in the modelling of a broad range of different problems (cf. [9,10]), and so additional knowledge on their solvability is very welcome.…”

Section: Introductionmentioning

confidence: 99%

New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

2018

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We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadraticphase Fourier integral operators are also studied (including a Riemann-Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity).As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.

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Wave diffraction by a half-plane with an obstacle perpendicular to the boundary

Cited by 9 publications

References 25 publications

New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

New convolutions and their applicability to integral equations of Wiener‐Hopf plus Hankel type

Mixed impedance boundary value problems for the Laplace–Beltrami equation

New Convolutions for Quadratic-Phase Fourier Integral Operators and their Applications

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