JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS Volume 24, Number 4, Winter 2012 SUPERCONVERGENT NYSTR ¨ OM AND DEGENERATE KERNEL METHODS FOR INTEGRAL EQUATIONS OF THE SECOND KIND C. ALLOUCH, P. SABLONNI ` ERE, D. SBIBIH AND M. TAHRICHI Communicated by Giovanni Monegato ABSTRACT. We propose, in this paper, new methods for approximating the solution of a second kind integral equa- tion with a smooth kernel or kernel having a discontinuity along the diagonal. By using an interpolatory projection at Gaussian points onto the space of (discontinuous) piecewise polynomials of degree ≤ r - 1, we prove that the proposed methods exhibit convergence orders 3r and 4r for the iterated version. In comparison with Kulkarni’s of the same conver- gence order, we show that our methods are faster and simpler to implement. The theoretical results obtained are illustrated by some numerical examples. 1. Introduction. Let us consider the linear integral equation of the second kind (1) u -Ku = f, where K is the compact linear operator defined on the space C [0, 1] by Ku(s)=  1 0 k(s, t)u(t) dt, s ∈ [0, 1] with k(·, ·) ∈C ([0, 1] × [0, 1]) and f ∈C [0, 1]. Assume that the homogenous integral equation u -Ku = 0 has, in C([0, 1]), only the trivial solution; then the operator (I-K) is invertible and (I-K) -1 = ∞  n=0 K n . Keywords and phrases. Collocation method, Kulkarni’s method, integral equa- tion, superconvergence. Research supported by URAC-05. Received by the editors on June 29, 2011, and in revised form on February 28, 2012. DOI:10.1216/JIE-2012-24-4-463 Copyright c 2012 Rocky Mountain Mathematics Consortium 463