Kress Smoothing Transformation for Weakly Singular Fredholm Integral Equations of Second Kind

Year 2013, Volume: 10 Issue: 1, - , 01.05.2013

Hassan M. S. Bawazir Ali Abd Rahman Nor’aini binti Aris

Abstract

The paper investigates a numerical method for the second kind Fredholm
integral equation with weakly singular kernel k(x, y), in particular, when k(x, y) = ln |x−y|,
and k(x, y) = |x − y|
−α, −1 ≤ x, y ≤ 1, 0 < α < 1. The solutions of such equations may
exhibit a singular behaviour in the neighbourhood of the endpoints x = ±1. We introduce
a new smoothing transformation based on the Kress transformation for solving weakly
singular Fredholm integral equations of the second kind, and then using the Hermite
smoothing transformation as a standard. With the transformation an equation which
is still weakly singular is obtained, but whose solution is smoother. The transformed
equation is then solved numerically by product integration methods with interpolating
polynomials. Two types of interpolating polynomials, namely the Gauss-Legendre and
Chebyshev polynomials, have been used. Numerical examples are presented to investigate
the performance of the former.

Keywords

Integral equation, weakly singular integral equation, smoothing transformation

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