Superconvergence of a Collocation-Type Method for Simple Turning Points of Hammerstein Equations

In this paper a simple turning point (y = yc, A = Ac) of the parameterdependent Hammerstein equation y(t) = f(t) + X k(t,s)g(s,y(s))ds, te[a,b], is approximated numerically in the following way. A simple turning point (z = zc, X = Xe) of an equivalent equation for z(t) := Xg(t, y(t)) is computed first. This is done by solving a discretized version of a certain system of equations which has (2e, Ac) as part of an isolated solution. The particular discretization used here is standard piecewise polynomial collocation. Finally, an approximation to yc is obtained by use of the (exact) equation rb y(t) = f(t)+l k(t,s)z(s)ds, te[a,b]. IJ a The main result of the paper is that, under suitable conditions, the approximations to yc and Ac are both superconvergent, that is, they both converge to their respective exact values at a faster rate than the collocation approximation (of zc) does to zc.