Fast numerical schemes for Fredholm integral equations of the second kind

Fast numerical schemes for Fredholm integral equations of the second kind have been developed. The integral equations are firstly discretized by the open Clenshaw-Curtis quadrature rule on a nodal point set $\Xi\sb{n}$. Generally n has to be chosen fairly large in order to obtain a certain accuracy. When the kernels are sufficiently smooth, we have shown that the linear systems of equations can be approximated well by their low rank approximations on $\Xi\sb{m}$ with $m \ll n$ by the eigenvalue expansions or the singular value decompositions of the integral operators. Most computations are now accomplished on $\Xi\sb{m}$. The Chebyshev expansions are used to define the interpolation formulas. We have shown that, if the kernel $\kappa\ \in\ C\sp{p}$ and the right hand side function $y\ \in\ C\sp{q}$ for some integers $p,q \u3e 0$, the schemes converge at the rate of $o(1/m\sp{p-l}) + o (1/n\sp{\nu -1}$), where the integer $\nu \ge$ min(p,q).^ When the kernels $\kappa(s,t$) are non-smooth along the line $s=t$, we have firstly described a high order quadrature rule. Then, we proposed two iteration methods to efficiently solve the corresponding equations.