scholarly journals The Potential Method for the Reactance Wave Diffraction Problem in a Scale of Spaces

2006 ◽  
Vol 13 (2) ◽  
pp. 251-260
Author(s):  
Luis P. Castro ◽  
David Natroshvili
Keyword(s):  
Boundary Value Problem ◽  
Wave Diffraction ◽  
Boundary Value ◽  
Diffraction Problem ◽  
Potential Method ◽  
Wave Number ◽  
Dimensional Case ◽  
Bessel Potential ◽  
Regularity Properties ◽  
Bessel Potential Spaces

Abstract This paper is concerned with a screen type boundary value problem arising from the wave diffraction problem with a reactance condition. We consider the problem in a weak formulation within Bessel potential spaces, and where both cases of a complex and a pure real wave number are analyzed. Using the potential method, the boundary value problem is converted into a system of integral equations. The invertibility of the corresponding matrix pseudodifferential operator is shown in appropriate function spaces which allows the conclusion about the existence and uniqueness of a weak solution to the original problem. Higher regularity properties of solutions are also proved to exist in some scale of Bessel potential spaces, upon the corresponding smoothness improvement of given data. In particular, the 𝐶 α -smoothness of solutions in a neighbourhood of the screen edge is established with arbitrary α < 1 in the two-dimensional case and α < 1/2 in the three-dimensional case.

scholarly journals DIFFRACTION BY A UNION OF STRIPS WITH IMPEDANCE CONDITIONS IN BESOV AND BESSEL POTENTIAL SPACES

2008 ◽  
Vol 13 (2) ◽  
pp. 183-194
Author(s):  
Luis P. Castro ◽  
David Kapanadze
Keyword(s):  
Wave Diffraction ◽  
Besov Spaces ◽  
Differential Operators ◽  
Diffraction Problem ◽  
Bessel Potential ◽  
Impedance Boundary ◽  
Pseudo Differential Operators ◽  
Impedance Parameters ◽  
Impedance Conditions ◽  
Bessel Potential Spaces

We consider an impedance boundary‐value problem for the Helmholtz equation which models a wave diffraction problem with imperfect conductivity on a union of strips. Pseudo‐differential operators acting between Bessel potential spaces and Besov spaces are used to deal with this wave diffraction problem. In particular, these operators allow a reformulation of the problem into a system of integral equations. The main result presents impedance parameters which ensure the well-posedness of the problem in scales of Bessel potential spaces and Besov spaces.

Mixed type boundary value problems for Laplace–Beltrami equation on a surface with the Lipschitz boundary

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Roland Duduchava
Keyword(s):  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Beltrami Equation ◽  
Bessel Potential ◽  
Convolution Operators ◽  
Classical Setting ◽  
Type Boundary ◽  
Mellin Convolution ◽  
Bessel Potential Spaces

AbstractThe purpose of the present research is to investigate a general mixed type boundary value problem for the Laplace–Beltrami equation on a surface with the Lipschitz boundary 𝒞 in the non-classical setting when solutions are sought in the Bessel potential spaces \mathbb{H}^{s}_{p}(\mathcal{C}), \frac{1}{p}<s<1+\frac{1}{p}, 1<p<\infty. Fredholm criteria and unique solvability criteria are found. By the localization, the problem is reduced to the investigation of model Dirichlet, Neumann and mixed boundary value problems for the Laplace equation in a planar angular domain \Omega_{\alpha}\subset\mathbb{R}^{2} of magnitude 𝛼. The model mixed BVP is investigated in the earlier paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain, Georgian Math. J.27 (2020), 2, 211–231], and the model Dirichlet and Neumann boundary value problems are studied in the non-classical setting. The problems are investigated by the potential method and reduction to locally equivalent 2\times 2 systems of Mellin convolution equations with meromorphic kernels on the semi-infinite axes \mathbb{R}^{+} in the Bessel potential spaces. Such equations were recently studied by R. Duduchava [Mellin convolution operators in Bessel potential spaces with admissible meromorphic kernels, Mem. Differ. Equ. Math. Phys.60 (2013), 135–177] and V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl.443 (2016), 2, 707–731].

scholarly journals Mixed boundary value problems for the Helmholtz equation in a model 2D angular domain

2020 ◽  
Vol 27 (2) ◽  
pp. 211-231
Author(s):  
Roland Duduchava ◽  
Medea Tsaava
Keyword(s):  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Mixed Boundary ◽  
Boundary Value ◽  
Bessel Potential ◽  
Angular Domain ◽  
Mixed Boundary Value Problems ◽  
Classical Setting ◽  
Mellin Convolution ◽  
Bessel Potential Spaces

AbstractThe purpose of the present research is to investigate model mixed boundary value problems (BVPs) for the Helmholtz equation in a planar angular domain {\Omega_{\alpha}\subset\mathbb{R}^{2}} of magnitude α. These problems are considered in a non-classical setting when a solution is sought in the Bessel potential spaces {\mathbb{H}^{s}_{p}(\Omega_{\alpha})}, {s>\frac{1}{p}}, {1<p<\infty}. The investigation is carried out using the potential method by reducing the problems to an equivalent boundary integral equation (BIE) in the Sobolev–Slobodečkii space on a semi-infinite axis {\mathbb{W}^{{s-1/p}}_{p}(\mathbb{R}^{+})}, which is of Mellin convolution type. Applying the recent results on Mellin convolution equations in the Bessel potential spaces obtained by V. Didenko and R. Duduchava [Mellin convolution operators in Bessel potential spaces, J. Math. Anal. Appl. 443 2016, 2, 707–731], explicit conditions of the unique solvability of this BIE in the Sobolev–Slobodečkii {\mathbb{W}^{r}_{p}(\mathbb{R}^{+})} and Bessel potential {\mathbb{H}^{r}_{p}(\mathbb{R}^{+})} spaces for arbitrary r are found and used to write explicit conditions for the Fredholm property and unique solvability of the initial model BVPs for the Helmholtz equation in the non-classical setting. The same problem was investigated in a previous paper [R. Duduchava and M. Tsaava, Mixed boundary value problems for the Helmholtz equation in arbitrary 2D-sectors, Georgian Math. J. 20 2013, 3, 439–467], but there were made fatal errors. In the present paper, we correct these results.

Diffraction by a half plane perpendicular to the distinguished axis of a gyrotropic medium (oblique incidence)

1981 ◽  
Vol 59 (3) ◽  
pp. 403-424 ◽  
Author(s):  
S. Przeździecki ◽  
R. A. Hurd
Keyword(s):  
Half Plane ◽  
Fourier Transforms ◽  
Boundary Value ◽  
Diffraction Problem ◽  
Closed Form Solution ◽  
Second Order ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
Electromagnetic Plane Wave

An exact, closed form solution is found for the following half plane diffraction problem. (I) The medium surrounding the half plane is gyrotropic. (II) The scattering half plane is perfectly conducting and oriented perpendicular to the distinguished axis of the medium. (III) The direction of propagation of the incident electromagnetic plane wave is arbitrary (skew) with respect to the edge of the half plane. The result presented is a generalization of a solution for the same problem with incidence normal to the edge of the half plane (two-dimensional case).The fundamental, distinctive feature of the problem is that it constitutes a boundary value problem for a system of two coupled second order partial differential equations. All previously solved electromagnetic diffraction problems reduced to boundary value problems for either one or two uncoupled second order equations. (Exception: the two-dimensional case of the present problem.) The problem is formulated in terms of the (generalized) scalar Hertz potentials and leads to a set of two coupled Wiener–Hopf equations. This set, previously thought insoluble by quadratures, yields to the Wiener–Hopf–Hilbert method.The three-dimensional solution is synthesized from appropriate solutions to two-dimensional problems. Peculiar waves of ghost potentials, which correspond to zero electromagnetic fields play an essential role in this synthesis. The problem is two-moded: that is, superpositions of both ordinary and extraordinary waves are necessary for the spectral representation of the solution. An important simplifying feature of the problem is that the coupling of the modes is purely due to edge diffraction, there being no reflection coupling. The solution is simple in that the Fourier transforms of the potentials are just algebraic functions. Basic properties of the solution are briefly discussed.

The Three-Dimensional Problem of Statics of the Elastic Mixture Theory with Displacements Given on the Boundary

1999 ◽  
Vol 6 (6) ◽  
pp. 517-524
Author(s):  
M. Basheleishvili
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Three Dimensional ◽  
Mixture Theory ◽  
Potential Method ◽  
Fredholm Integral Equations ◽  
Infinite Domains ◽  
Elastic Mixture ◽  
Uniqueness Of The Solution ◽  
Dimensional Boundary

Abstract The first three-dimensional boundary value problem is considered for the basic equations of statics of the elastic mixture theory in the finite and infinite domains bounded by the closed surfaces. It is proved that this problem splits into two problems whose investigation is reduced to the first boundary value problem for an elliptic equation which structurally coincides with an equation of statics of an isotropic elastic body. Using the potential method and the theory of Fredholm integral equations of second kind, the existence and uniqueness of the solution of the first boundary value problem is proved for the split equation.

Stability in the Special Saturated Liquid Film along an Inclined Heated Plate

2012 ◽  
Vol 516-517 ◽  
pp. 202-207
Author(s):  
Xiao Chao Fan ◽  
Rui Jing Shi ◽  
Bo Wei
Keyword(s):  
Reynolds Number ◽  
Boundary Value Problem ◽  
Liquid Film ◽  
Boiling Point ◽  
Boundary Value ◽  
Physical Property ◽  
Wave Number ◽  
Heated Plate ◽  
Saturated Liquid ◽  
Inclined Angle

Stable analysis of flow and heat transfer in the saturated liquid film of liquid low boiling point gases falling down an inclined heated plate is investigated. Firstly, the boundary value problem of linear stability differential equation (Orr–Sommerfeld equation) on small perturbation is derived representing surface tension by nonlinear relationship on temperature. Then, the expression of the wave velocity is got by solving the boundary value problem of O–S equation using the perturbation method. The effects of the inclined angle and some other factors, such as Reynolds number, wave number, temperature of the plate and the parameter for the physical property, on stability in the saturated liquid film of liquid low boiling point gas N2 are numerically analyzed by MATLAB software. Finally, it is shown and analyzed a new critical Reynolds number which is actually the extension of Yih’s.

The Diffraction Forces for a Ship Moving in Oblique Seas

1979 ◽  
Vol 23 (02) ◽  
pp. 127-139
Author(s):  
Armin Walter Troesch
Keyword(s):  
Boundary Value Problem ◽  
Surface Condition ◽  
Boundary Value ◽  
Diffraction Problem ◽  
Second Order ◽  
Exciting Force ◽  
Forward Speed ◽  
Order Problem ◽  
Incident Waves ◽  
The Waves

The diffraction problem of a fixed slender ship moving in incident waves is formulated. The waves are assumed to be of the same order as the beam of the ship and are from an oblique heading. The boundary value problem is linearized with respect to wave amplitude and solved by the method of matched asymptotic expansions. The oscillating forward-speed potential is solved to two orders of magnitude. The first order is just the zero-speed case while the second-order problem involves solving a boundary-value problem with a nonhomogeneous free-surface condition. The solution to this second-order problem is given in terms of three auxiliary potentials, each satisfying a separate part of the boundary conditions. For zero forward speed, the sectional exciting force is calculated and compared with the commonly used integrand of the Khaskind relations. The two give different values, but when integrated over the hull both show the same total exciting force. The pressure distribution on an ore carrier for both zero forward speed and an abbreviated form of the forward-speed case is given and compared with experiments.

Sign in / Sign up

Export Citation Format

Share Document