A polynomial collocation method for the numerical solution of weakly singular and singular integral equations on non-smooth boundaries

The aim of this paper is to show the e ciency of the use of smoothing changes of variable in the numerical treatment of 1D and 2D weakly singular and singular integral equations. The introduction of a smoothing transformation, besides smoothing the solution, allows also the use of a very simple and e cient collocation method based on Chebyshev polynomials of the rst kind and their zeros. Further, we propose proper smoothing changes of variable also for the numerical approximation of those collocation matrix elements, which are given by weakly singular, singular or nearly singular integrals. Several numerical tests are given to point out the e ciency of the numerical approach we propose