Dual integral equations -- revisited
type: Item
instance of: scholarly article
title: Dual integral equations -- revisited
zbMATH Open document ID: 1021.45003
DOI: 10.1155/S0161171203007488
author: Sudeshna Banerjea
author: C. C. Kar
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2003-06-18T00:00:00Z
full work available at URL: https://eudml.org/doc/50325
review text: The authors consider the following dual integral equations: \[ \left\{\begin{aligned} \int_0^\infty A(k)L(k,y) dk &=-R\exp(-Ky),\quad y\in G,\\ \int_0^\infty kA(k)L(k,y) dk &=iK(1-R)\exp(-Ky),\quad y\in B, \end{aligned}\right.\tag{1} \] where \(L(k,y) = k\cos ky-K\sin ky,\quad G=(0,\infty)-B\), \(A(k)\) is an unknown function, and \(R\) is an unknown constant. This integral equation arises in the well-known problem of scattering water waves in deep water by a vertical barrier under the assumption of linearized theory. The solution of (1) is obtained here by noting the behaviour of the second equation of (1) at the end points of the interval \(G\), which can be determined from physical consideration. Equation (1) is then reduced to a singular integral equation whose kernel involves Cauchy and logarithmic type singularities, the solution of which is known. Knowing the solution of (1), the solution of the corresponding scattering problems is obtained in a closed form.
reviewed by: Nikolay Yakovlevich Tikhonenko
Mathematics Subject Classification ID: 45F10
Mathematics Subject Classification ID: 76B15
Mathematics Subject Classification ID: 45E05
Mathematics Subject Classification ID: 45E10
zbMATH DE Number: 1930013
zbMATH Keywords: dual integral equations
zbMATH Keywords: scattering of water waves by a vertical barrier
zbMATH Keywords: singular integral equation with logarithmic and Cauchy kernel
MaRDI profile type: Publication
OpenAlex ID: W2129572863
description: scientific article; zbMATH DE number 1930013