Wave propagation from lateral Cauchy data using a boundary element method

Chapko, Roman; Johansson, Björn; Muzychuk, Yuriy; Hlova, A.

doi:10.1016/j.wavemoti.2019.102385

Cited by 7 publications

(6 citation statements)

References 18 publications

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“…As far as wave propagation phenomena are concerned, research aiming towards solving Cauchy problems for hyperbolic equations was carried out in the past in [8,9,[11][12][13][14]. The socalled quasi-reversibility method has been applied to solve the Cauchy problem for the wave equation [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

“…Other related research was based on the use of mixed formulations of such an underlining method, see Bécache et al [12] for details. Numerical methods have also been devised and applied [8,9,13]. For instance, the authors of [13] applied a hybrid boundary integral approach incorporating the Galerkin boundary element method to reconstruct an unknown boundary condition from lateral Cauchy data in the wave equation on a 3-D annulus.…”

Section: Introductionmentioning

confidence: 99%

“…Numerical methods have also been devised and applied [8,9,13]. For instance, the authors of [13] applied a hybrid boundary integral approach incorporating the Galerkin boundary element method to reconstruct an unknown boundary condition from lateral Cauchy data in the wave equation on a 3-D annulus. Worth mentioning here is also the analysis and solution of Weber [17] for the ill-posed inverse heat conduction problem obtained by approximating the parabolic heat equation with a hyperbolic damped wave equation with small parameter.…”

Section: Introductionmentioning

confidence: 99%

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Solution of the Cauchy problem for the wave equation using iterative regularization

Alosaimi

Lesnic

Johansson³

2021

Inverse Problems in Science and Engineering

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solution of the Cauchy problem for the wave equation using iterative regularization

Alosaimi

Lesnic

Johansson³

2021

Inverse Problems in Science and Engineering

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“…This single-layer method has been implemented and tested in various planar domains both doubly and simply connected as well as unbounded, for details and references see the overview in [3]. The method is not restricted to the Laplace equation but can also be applied to, for example, the similar ill-posed problem in elasticity and for other stationary and non-stationary elds of various physical processes, see [4,5].…”

Section: Introductionmentioning

confidence: 99%

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2020

VAM

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We consider the Cauchy boundary value problem for the Laplace equation in a planar bounded doubly-connected domain. Our goal is to reconstruct the Cauchy data on the interior boundary from given Cauchy data on the outer boundary. It is an example of the linear ill-posed inverse problem. Using the indirect integral equation method based on a double-layer representation for the Cauchy problem we receive a system of integral equations to be solved for two unknown densities. It is shown that the system has a unique solution for a dense set of data. Next we parametrize this system to periodical integral equations and separate the given hypersingularity in some kernels as a special weight function. The numerical solution of integral equations is realized by Nystroem method based on trigonometrical quadrature rules. As result the full discrete system of linear equations with respect to approximation values of unknown densities in the quadrature points is received. Since this system is ill-posed it is solved by Tikhonov regularization. The value of the regularization parameter is chosen by trial and error. It gives us the possibility to receive the stable solution. The inuence of the discretization is also included in a brief error analysis. Results of numerical experiments for the reconstruction of the function and its normal derivative on the interior boundary show that accurate approximations can be obtained with the double-layer method also for the noisy Cauchy data. In the case of noisy data, random pointwise errors are added to the function values on the outer boundary with the percentage given in terms of the least squares norm.

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“…We also note here other applications of the Laguerre transform that are not related to the direct use of the convolution product of functions. In particular, it was used to reduce evolution problems to boundary value problems for infinite systems of elliptic equations (for details, see [2,6,7,10,15,22] and references therein).…”

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confidence: 99%

The Laguerre transform of a convolution product of vector-valued functions.

Muzychuk

2021

Mat. Stud.

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The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions. The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences. The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

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Wave propagation from lateral Cauchy data using a boundary element method

Cited by 7 publications

References 18 publications

Solution of the Cauchy problem for the wave equation using iterative regularization

Solution of the Cauchy problem for the wave equation using iterative regularization

�� ֲ�� ²�� ز �� В��

The Laguerre transform of a convolution product of vector-valued functions.

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Wave propagation from lateral Cauchy data using a boundary element method

Cited by 7 publications

References 18 publications

Solution of the Cauchy problem for the wave equation using iterative regularization

Solution of the Cauchy problem for the wave equation using iterative regularization

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The Laguerre transform of a convolution product of vector-valued functions.

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