We used function theoretic method to solve a singular integral equation with logarithmic kernel in two disjoint finite intervals where the unknown function satisfying the integral equation may be bounded or unbounded at the nonzero finite endpoints of the interval concerned. An appropriate solution of this integral equation is then applied to solve the problem of scattering of time harmonic sur...

In this paper, we establish that the Lagrangian-type material differentiation formulas, that allow to express the first-order derivative of a (regular) surface integral with respect to a geometrical domain perturbation, still hold true for the strongly singular and hypersingular syrface integrals usually encountered in boundary integral formulations. As a consequence, this work supports previou...

This paper is concerned with the construction and study of specific finite and infinite integral transforms arising from a singular Sturm-Liouville problem on a half line, which is connected to the hydrogen atom equation. The paper goes further to describe fully the image of some spaces of square integrable functions with respect to some measure under the finite and infinite integral transforms...

We study singularity formation of three-dimensional ~3-D! vortex sheets without surface tension using a new approach. First, we derive a leading order approximation to the boundary integral equation governing the 3-D vortex sheet. This leading order equation captures the most singular contributions of the integral equation. By introducing an appropriate change of variables, we show that the lea...

This paper is concerned with the problem of scattering of time-harmonic electromagnetic waves from penetrable diffraction gratings in the 2D polarization case. We propose a new, weakly singular, integral equation formulation for the scattering problem which is proved to be uniquely solvable. A main feature of the new integral equation formulation is that it avoids the computation of the normal ...

A nonlinear boundary value problem for Laplace's equation is solved numerically by using a reformulation as a nonlinear boundary integral equation. Two numerical methods are proposed and analyzed for discretizing the integral equation, both using product integration to approximate the singular integrals in the equation. The first method uses the product Simpson's rule, and the second is based o...

The weak-form of Helmholtz differential equation, in conjunction with vector test-functions (which are gradients of the fundamental solutions to the Helmholtz differential equation in free space) is utilized as the basis in order to directly derive non-hyper-singular boundary integral equations for the velocity potential, as well as its gradients. Thereby, the presently proposed boundary integr...

A direct function theoretic method is applied to solve a weakly singular integral equation whose kernel involves logarithmic singularity. This method avoids the occurrence of strong singularity. The solution of this integral equation is then applied to re-investigate the well known problem of water wave scattering by a partially immersed vertical barrier. c © 2008 Published by Elsevier Ltd