FAST COLLOCATION SOLVERS
FOR INTEGRAL EQUATIONS
ON OPEN ARCS

J. SARANEN AND G. VAINIKKO

Abstract:

In this work we develop a unified approach for numerical approximation and fast solution of classical integral equations on open arcs. The approximation is obtained applying the cosine transform and fully discrete trigonometric collocation together with an asymptotic approximation of the operator. The computed approximation is of optimal accuracy order in a large scale of Sobolev norms, and it can be obtained in ${\cal O}(n\log n)$ arithmetical operations. Our results cover logarithmic singular integral equations, Cauchy singular integral equations, as well as hypersingular integral equations.