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1

Metamorphism - an integral transform reducing the order of a differential equation

Kisil Vladimir V.

Journal of Applied Analysis

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2023

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Vol. 29, nr 2

219--227

EN

We propose an integral transform, called metamorphism, which allows us to reduce the order of a differential equation. For example, the second-order Helmholtz equation is transformed into a first-order equation, which can be solved by the method of characteristics.

2

Topological sensitivity analysis for a coupled nonlinear problem with an obstacle

Abdelbari M., Nachi K., Sokolowski J.

Control and Cybernetics

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2017

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Vol. 46, no. 1

5--25

EN

The Topological Derivative has been recognized as a powerful tool in obtaining the optimal topology for several kinds of engineering problems. This derivative provides the sensitivity of the cost functional for a boundary value problem for nucleation of a small hole or a small inclusion at a given point of the domain of integration. In this paper, we present a topological asymptotic analysis with respect to the size of singular domain perturbation for a coupled nonlinear PDEs system with an obstacle on the boundary. The domain decomposition method, referring to the SteklovPoincar´epseudo-diﬀerential operator, is employed for the asymptotic study of boundary value problem with respect to the size of singular domain perturbation. The method is based on the observation that the known expansion of the energy functional in the ring coincides with the expansion of the Steklov-Poincar´e operator on the boundary of the truncated domain with respekt to the small parameter, which measures the size of perturbation. In this way, the singular perturbation of the domain is reduced to the regular perturbation of the Steklov-Poincar´e map ping for the ring. The topological derivative for a tracking type shape functional is evaluated so as to obtain the useful formula for application in the numerical methods of shape and topology optimization.

3

Ritz solution of the Helmholtz equation in a stadium-shaped domain with zero normal derivatives – applications to fluid sloshing and thermo-convective stability

Wang C. Y.

Archives of Mechanics

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2017

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Vol. 69, nr 6

471--483

EN

The eigenvalues and eigenfunctions of the Helmholtz equation with Neumann conditions are obtained for the stadium-shaped domain. The variational Ritz method is found to be accurate and efficient in determining these eigenvalues and eigenfunctions. The eigenfunctions show the evolution and switching of mode shapes from a long rectangular strip to a circle. These new results are applied to the sloshing of a liquid in a tank, and to the onset of thermo-convective stability in a confined porous layer.

4

Acoustic Scattering of 3D Complex Systems Having Random Rough Surfaces by Scalar Integral Equations

Guel-Tapia J. A., Villa-Villa F., Mendoza-Suárez A., Pérez-Aguilar H.

Archives of Acoustics

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2016

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Vol. 41, No. 3

461--472

EN

We propose a numerical surface integral method to study complex acoustic systems, for interior and exterior problems. The method is based on a parametric representation in terms of the arc’s lengths in curvilinear orthogonal coordinates. With this method, any geometry that involves quadric or higher order surfaces, irregular objects or even randomly rough surfaces can be considered. In order to validate the method, the modes in cubic, spherical and cylindrical cavities are calculated and compared to analytical results, which produced very good agreement. In addition, as examples, we calculated the scattering in the far field and the near field by an acoustic sphere and a cylindrical structure with a rough cross-section.

5

Sound Radiation from a Surface Source Located at the Bottom of the Wedge Region

Szemela K.

Archives of Acoustics

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2015

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Vol. 40, No. 2

223--234

EN

Applying rigorous analytical methods, formulas describing the sound radiation have been obtained for the wedge region bounded by two transverse baffles with a common edge and bottom. It has been assumed that the surface sound source is located at the bottom. The presented formulas can be used to calculate the sound pressure and power inside the wedge region. They are valid for any value of the wedge angle and represent a generalization of the formulas describing the sound radiation inside the two and three-wall corner region. Moreover, the presented formulas can be easily adapted for any case when more than one sound source is located at the bottom. To demonstrate their practical application, the distribution of the sound pressure modulus and the sound power have been analyzed in the case of a rectangular piston located at the wedge’s bottom. The influence of the transverse baffle on the sound power has been investigated. Based on the obtained formulas, the behaviour of acoustic fields inside a wedge can be predicted.

6

Growth and Lδ-approximation of solutions of the Helmholtz equation in a finite disk

Kumar D, Basu A.

Journal of Applied Analysis

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2014

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Vol. 20, nr 2

119--128

EN

In this paper we study the growth and Lδ-approximation, 1 ≤ δ ≤ ∞, of solutions (not necessarily entire) of Helmholtz-type equations. Moreover, we obtain the characterization of order and type of H ∈ HR, 0 < R < ∞, in terms of decay of approximation errors En(H,R0) and Ein,δ(H,R0), i = 1,2. Our results extend and improve the results obtained by McCoy [J. Approx. Theory 25 (1979), 153–168].

7

Green's functions for interior and exterior Helmholtz problems

Kukla S., Siedlecka U., Zamorska I.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

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2012

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Vol. 11, nr 1

53-62

EN

In the paper, the derivation of Green's functions for Helmholtz equation in circular, annular and exterior circular domains is presented. The Green's functions are assumed of the form a cosine series. An example of application of the Green's functions in frequency analysis of a composite membrane is presented.

8

Approximation of entire function solutions of the Helmholtz equation having slow growth

Kumar D.

Journal of Applied Analysis

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2012

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Vol. 18, nr 2

179-196

EN

In this paper, we study the Chebyshev polynomial approximation of entire solutions of Helmholtz equations in R2 in Banach spaces (B(p, q, m) space, Hardy space and Bergman space). Some bounds on generalized order of entire solutions of Helmholtz equations of slow growth have been obtained in terms of the coefficients and approximation errors using function theoretic methods.

9

Approximate BEM analysis of thin electromagnetic shield

Jabłoński P.

Prace Naukowe Politechniki Śląskiej. Elektryka

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2011

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z. 4

55--70

EN

A method of approximate analysis of a thin electromagnetic shield is considered and proposed in the paper. Due to presumably small thickness of the shield, its numerical analysis is troublesome. Applying the Boundary Element Method (BEM) to solve equations for a thin shield creates two difficulties: significant increase of the number of algebraic equations, and the presence of nearly singular integrals. The proposed model avoids them both by using an approximate analytical solution for the shield. Numerical examples confirm its usability.

PL

W pracy zaproponowano przybliżoną metodę analizy pola elektromagnetycznego w otoczeniu cienkościennego ekranu elektromagnetycznego z zastosowaniem metody elementów brzegowych (MEB). Z powodu założonej niewielkiej grubości ekranu analiza numeryczna napotyka na problemy. Zastosowanie MEB niesie ze sobą dwie trudności: znaczny wzrost liczby równań algebraicznych oraz obecność całek prawieosobliwych. Przedstawiona metoda unika obydwu trudności poprzez zastosowanie przybliżonego analitycznego rozwiązania w obszarze ekranu. Zaprezentowane przykłady numeryczne potwierdzają jej użyteczność w rozpatrywanej klasie zagadnień.

10

Investigation of Novel method of spherical aberration correction for enhanced spherical hybrid reflector antennas

Ponomarev O. P.

Zeszyty Naukowe Wydziału Elektroniki i Informatyki Politechniki Koszalińskiej

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2010

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Nr 2

77--92

11

Triangular Bézier patches in modelling smooth boundary surface in exterior Helmholtz problems solved by the PIES

Zieniuk E., Szerszeń K.

Archives of Acoustics

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2009

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Vol. 34, No. 1

51-61

EN

The paper proposes the use of triangular parametric Bézier patches as a new and effective way to generate three-dimensional boundaries in acoustics problems. The boundary geometry composed of triangular Bézier patches has been directly linked to the parametric integral equation system (PIES) to numerical solving exterior Helmholtz problems. A primary advantage of the proposed approach is to avoid the necessity of conventional domain or boundary discretization. The obtained numerical solutions compared with literature exacts results are characterized by high accuracy and convergence.

12

Dirichlet/Dirichlet and Dirichlet/Dirichlet-Neumann/Neumann non-overlapping iterative domain decomposition methods

Kubacki S., Bogusławski A.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2008

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Vol. 12, No 1-2

85-104

EN

A new iterative non-overlapping domain decomposition method is proposed for solving the one- and two-dimensional Helmholtz equation on parallel computers. The spectral collocation method is applied to solve the Helmholtz equation in each subdomain based on the Chebyshev approximation, while the patching conditions are imposed at the interfaces between subdomains through a correction, being a linear function of the space coordinates. Convergence analysis is performed for two applications of the proposed method (DDLC and DDNNLC algorithms - the meaning of these abbreviations is explained below) based on the works of Zanolli and Funaro et al. Numerical tests have been performed and results obtained using the proposed method and other iterative algorithms have been compared. Parallel performance of the multi-domain algorithms has been analyzed by decomposing the two-dimensional domain into a number of subdomains in one spatial direction. For the one-dimensional problem, convergence of the iteration process was quickly obtained using the proposed method, setting a small value of the ? constant in the Helmholtz equation. Another application of the proposed method may be an alternative to other iterative schemes when solving the two-dimensional Helmholtz equation.

13

Transmission problems for the Helmholtz equation for a rectilinear-circular lune

Denysenko V.

Opuscula Mathematica

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2007

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Vol. 27, no. 2

197-203

EN

The question related to the construction of the solution of plane transmission problem for the Helmholtz equation in a rectilinear-circular lune is considered. An approach is proposed based on the method of partial domains and the principle of reflection for the solutions of the Helmholtz equation through the segment.

14

The reactance wave diffraction problem by a strip in a scale of Bessel potential spaces

Castro L. P., Natroshvili D.

Opuscula Mathematica

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2006

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Vol. 26, no. 2

289-303

EN

We consider a boundary-transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain reactance conditions are assumed on it. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well-posedness of the problem is shown for a range of regularity orders of the Bessel potential spaces, and for a set of possible reactance numbers (dependent on the wave number).

15

A solution of 3D Helmholtz equation for boundary geometry modeled by Coons patches using the Parametric Integral Equation System

Zieniuk E., Szerszeń K.

Archives of Acoustics

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2006

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Vol. 31, No. 1

99-111

EN

In this paper, the authors propose an algorithm for numerical solution of the 3D Helmholtz equation using the Parametric Integral Equation System (PIES). The PIES, unlike the traditional Boundary Integral Equation (BIE), is characterized by the fact that the boundary geometry has been considered in its mathematical formalism. Polygonal Coons surfaces have been used to describe the 3D domain. This makes it possible to obtain continuous solutions without any discretization of the 3D domain.

16

Ocena wpływu perturbacji parametrów zadania brzegowego na rozwiązanie równania Helmholtza

Winkler A.

Zeszyty Naukowe. Budownictwo / Politechnika Śląska

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2005

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z. 104

395-402

PL

Niniejszy artykuł jest prezentacją nowego systemu algebraicznego oraz jego aplikacji do teorii perturbacji cząstkowych równań różniczkowych. Nowa metoda zastosowania została do rozwiązania zadania brzegowego dla równania Helmholtza. Wprowadzone zostały zaburzenia parametrów równania oraz warunków początkowych i brzegowych. W nowym systemie klasyczne problemy perturbacyjne opisywane równaniami różniczkowymi mogą być rozwiązane w sposób tak prosty jak zwykłe równania różniczkowe stosowane w akustyce, matematyce czy technice. Nie są konieczne dodatkowe przekształcenia analityczne.

EN

This paper presents applications of the new algebraic system in the theory of perturbed partial differential equations. New method is applied to solve boundary problems for Helmholtz equation. Perturbation of equation parameters as well as for initial and boundary conditions are considered. Classical perturbation problems described by differential equations are solved in the new algebraic system as easy as usual differential problems of applied acoustics, mathematics and techniques. Additional analytical transformations are not applied acoustics, mathematics and techniques. Additional analytical transformations are not required.

17

Mesh-free approach to Helmholtz equation based on radial basis functions

Kowalczyk P., Mrozowski M.

Journal of Telecommunications and Information Technology

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2005

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nr 2

71-74

EN

Recently, a radial basis functions (RBFs) method, which was originally proposed for interpolation problems, has been developed and applied to solve partial differential equations and eigenproblems. Properties of that method (meshfree algorithm) allows one to use it in many areas, including electromagnetics. In this paper the mesh-free RBF method for solving Helmholtz equation was applied and a new adaptive algorithm for defining the set of interpolation centers was proposed. Using the proposed approach the cutoff wavelengths and the field distribution in cylindrical waveguides of arbitrary cross-section were calculated with a high accuracy.

18

A Matrix Decomposition MFS Algorithm For Helmholtz Problems

Karageorghis A., Smyrlis Y. S.

Vibrations in Physical Systems

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2004

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Vol. 21

47--54

EN

The Method of Fundamental Solutions (MFS) is a boundary-type method for the solution of certain elliptic boundary value problems. In this work, we develop an efficient matrix decomposition MFS algorithm for solution of exterior Helmholtz problems in domains surrounding circular domains in two dimensions. These ideas are extended for the solution of exterior Helmholtz problems in domains surrounding axisymmetric domains.

19

"Bottom crystal" and possibility of water wave attenuation

Popov I. Yu

Archives of Mechanics

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2003

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Vol. 55, nr 2

221-232

EN

The influence of periodic bottom structure ("bottom crystal'') on surface water waves is considered. The problem reduces to a two-dimensional Helmholtz operator with periodic potential. Zero-range potential method based on the theory of self-adjoint extensions of symmetric operators is used. It is shown that there is a gap in the spectrum. An application of this spectral property to the problem of wave attenuation is discussed.