MATHEMATICS OF COMPUTATION Volume 72, Number 242, Pages 729–756 S 0025-5718(02)01431-X Article electronically published on March 8, 2002 NYSTR ¨ OM-CLENSHAW-CURTIS QUADRATURE FOR INTEGRAL EQUATIONS WITH DISCONTINUOUS KERNELS SHEON-YOUNG KANG, ISRAEL KOLTRACHT, AND GEORGE RAWITSCHER Abstract. A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis quadrature for Fredholm integral equations of the second kind x(t)+  b a k(t, s)x(s)ds = y(t), whose kernel k(t, s) is either discontinuous or not smooth along the main diag- onal, is presented. This scheme is of spectral accuracy when k(t, s) is infinitely differentiable away from the diagonal t = s. Relation to the singular value de- composition is indicated. Application to integro-differential Schr¨odinger equa- tions with nonlocal potentials is given. 1. Introduction Let the integral operator, (Kx)(t)=  b a k(t,s)x(s)ds, a ≤ t ≤ b, map C q [a,b] ,q> 1, into itself. In the present paper, we consider the numerical solution of the corresponding Fredholm integral equation of the second kind, x(t)+  b a k(t,s)x(s)ds = y(t),y ∈ C q ,a ≤ t ≤ b. (1) When the kernel k(t,s) has a discontinuity either by itself or in its partial deriva- tives along the main diagonal t = s, one cannot expect a high accuracy Nystr¨ om quadrature based on Newton-Cotes or Gaussian integration rules, see, e.g., Figure 2 of Section 5, since, except for the trapezium rule, the standard error bounds for these rules are not applicable. If the function x(t) were known, then the discretiza- tion of the integral operator in (1) would be straightforward: for any fixed t the interval [a,b] can be partitioned so that in each subinterval the integrand is smooth. When x(t) is unknown it is generally not possible to get an accurate discretization of (1) with x(t) and x(s) sampled at the same support points, without using some Received by the editor March 29, 2001 and, in revised form, July 9, 2001. 2000 Mathematics Subject Classification. Primary 45B05, 45J05, 65Rxx, 65R20, 81U10. Key words and phrases. Discontinuous kernels, fast algorithms, nonlocal potentials. The work of the first author is partially supported by a fellowship from alumni of Mathematics Department, Chungnam National University, Korea. c 2002 American Mathematical Society 729