Asymptotic error expansion of a collocation-type method for Hammerstein equations

摘要

In recent papers Kumar and Sloan have considered the numerical solution of the Hammerstein equation y(x)=f(x)+∫abk(x,t)g(t,y(t))dt, xϵ[a,b] by a method that first applied the 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 equation y(x)=f(x)+∫abk(x,t)z(t)dt, xϵ[a,b] In this paper, the asymptotic error expansion of this method is obtained. We show that when piecewise polynomial of /GVp−1 are used, the approximation solution admits an error expansion in even powers of the step-size h, beginning with a term h2p. Thus, Richardson's extrapolation can be performed on the solution and this will increase the accuracy of numerical solution greatly.