Numerical solution of Urysohn integral equations using the iterated collocation method

  • Authors:

  • Khosrow Maleknejad;Hesamoddin Derili;Saeed Sohrabi

  • Affiliations:

  • School of Mathematics, Iran University of Science and Technology, Tehran, Iran;Department of Mathematics, Faculty of Science, Islamic Azad University (Karaj Unit), Karaj, Iran;School of Mathematics, Iran University of Science and Technology, Tehran, Iran

  • Venue:
  • International Journal of Computer Mathematics
  • Year:
  • 2008

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Abstract

In this paper, we analyse the iterated collocation method for the nonlinear operator equation x = y+K(x) with K a smooth kernel. The paper expands the study begun by H. Kaneko and Y. Xu concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. Let x* denote an isolated fixed point of K. Let Xn, n≥1, denote a sequence of finite-dimensional approximating subspaces, and let Pn be a projection of X onto Xn. The projection method for solving x = y+K(x) is given by xn = Pny+PnK(xn), and the iterated projection solution is defined as [image omitted] . We analyse the convergence of {xn} and {[image omitted] } to x*, giving a general analysis that includes the collocation method. A detailed analysis is then given for a large class of Urysohn integral operators in one variable, showing the superconvergence of {[image omitted] } to x*.