Superconvergence of a collocation-type method for simple turning points of Hammerstein equations

Abstract:

In this paper a simple turning point ($y = {y^c}$, $\lambda = {\lambda ^c}$) of the parameter-dependent Hammerstein equation \[ y(t) = f(t) + \lambda \int _a^b {k(t,s)g(s,y(s))\;ds,\quad t \in [a,b],}\] is approximated numerically in the following way. A simple turning point ($z = {z^c}$, $\lambda = {\lambda ^c}$) of an equivalent equation for $z(t):=\lambda g(t,y(t))$ is computed first. This is done by solving a discretized version of a certain system of equations which has (${z^c}$, ${\lambda ^c}$) as part of an isolated solution. The particular discretization used here is standard piecewise polynomial collocation. Finally, an approximation to ${y^c}$ is obtained by use of the (exact) equation \[ y(t) = f(t) + \int _a^b {k(t,s)z(s)\;ds,\quad t \in [a,b].}\] The main result of the paper is that, under suitable conditions, the approximations to ${y^c}$ and ${\lambda ^c}$ are both superconvergent, that is, they both converge to their respective exact values at a faster rate than the collocation approximation (of ${z^c}$) does to ${z^c}$.