A new collocation-type method for Hammerstein integral equations

Abstract:

We consider Hammerstein equations of the form \[ y(t) = f(t) + \int _a^b {k(t,s)g(s,y(s)) ds,\quad t \in [a,b],} \] and present a new method for solving them numerically. The method is a collocation method applied not to the equation in its original form, but rather to an equivalent equation for $z(t): = g(t,y(t))$. The desired approximation to y is then obtained by use of the (exact) equation \[ y(t) = f(t) + \int _a^b {k(t,s)z(s) ds,\quad t \in [a,b].} \] Advantages of this method, compared with the direct collocation approximation for y, are discussed. The main result in the paper is that, under suitable conditions, the resulting approximation to y converges to the exact solution at a rate at least equal to that of the best approximation to z from the space in which the collocation solution is sought.