Numerical analysis of graded mesh methods for a class of second kind integral equations on the real line

Abstract

In this paper we are concerned with the numerical analysis of the collocation method based on graded meshes of second kind integral equations on the real line of the form φ(s)=ψ(s)+ ∫ R κ(s−t)z(t)φ(t) dt, s∈ R , where κ∈L 1 ( R ) , z∈L ∞ ( R ) , and ψ∈BC( R ) , the space of bounded continuous complex-valued functions on R , are assumed known and the function φ∈BC( R ) is to be determined. We introduce some new graded meshes for the collocation method of the integral equation, which are different from those used previously for the Wiener–Hopf integral equation in the case when the solution decays exponentially at infinity, and establish optimal local and global L ∞ -norm error estimates under the condition that the solution decays only polynomially at infinity.