Gq. Han et Lq. Zhang, ASYMPTOTIC ERROR EXPANSION OF A COLLOCATION-TYPE METHOD FOR HAMMERSTEIN EQUATIONS, Applied mathematics and computation, 72(1), 1995, pp. 1-19
In recent papers Kumar and Sloan have considered the numerical solutio
n of the Hammerstein equation y(x) = f(x) + integral(a)(b) k(x, t)g(t,
y(t)) dt, x is an element of [a, b] by a method that first applied th
e standard collocation procedure to an equivalent equation for z(t) =
g(t, y(t)) and then obtained an approximation to y by use of the equat
ion y(x)=f(x) + integral(a)(b) k(x,t)z(t)dt, x is an element of [a, b]
. In this paper, the asymptotic error expansion of this method is obta
ined. We show that when piecewise polynomial of Pi(p-1) are used, the
approximation solution admits an error expansion in even powers of the
step-size h, beginning with a term h(2p). Thus, Richardson's extrapol
ation can be performed on the solution and this will increase the accu
racy of numerical solution greatly.