The solution of a singular integral equation with some applications in potential theory

N T Shawagfeh

doi:10.1155/s0161171298000258

Abstract

An analytical solution is derived for a singular integral equation which governs some twodimensional potential boundary value problems in a region exterior to n‐infinite co‐axial circular strips. An application in electrostatics is discussed.

Highlights

In this paper we derive a solution to the Fredhoim singular integral equation where q and a are constants and I"(0)is a differentiable nction for ]0- (2k# n)l < a,k 0 n-I This integral equation governs the solution of various two-dimensional Difichlit and Neumann potential bounda value problems for the region consisting of the whole (r, O) plane outside the circular strips
The previous investigations in potential problems of circular strips [1,2,3,4,5,6,7,8] were concerned mainly with the case of two strips,where Greens nction approach leads to a singular integral equation with kernel sin(O-) q + log
A different technique has been used by Sampath and Jain [4] based on decoupling the equation into two singular equations which can be solved using eigennctions expansion metod The same technique has been used to solve various bounda value problems involving two circular strips[ 5-8

Summary

INTRODUCTION

In this paper we derive a solution to the Fredhoim singular integral equation where q and a are constants and I"(0)is a differentiable nction for ]0- (2k# n)l < a ,k 0 n-I This integral equation governs the solution of various two-dimensional Difichlit and Neumann potential bounda value problems for the region consisting of the whole (r, O) plane outside the circular strips. The previous investigations in potential problems of circular strips [1,2,3,4,5,6,7,8] were concerned mainly with the case of two strips,where Greens nction approach leads to a singular integral equation with kernel sin(O-) q + log. (2 7b) which transforms the equation into the form j H(y)[R + log(21cosx cosyiy h(x), O< x

SOLUTION OF A SINGULAR INTEGRAL EQUATION

4.1LLUSTTIVE EXAMPLE