AN APPLICATION OF APPROXIMATION-THEORY TO NUMERICAL-SOLUTIONS FOR FREDHOLM INTEGRAL-EQUATIONS OF THE 2ND KIND

We present a numerical scheme for approximating the solution x = x(t) of the Fredholm equation of the second kind x(t) - integral-b(a) k(s, t)x(s) ds = f(t) a less-than-or-equal-to t less-than-or-equal-to b (F) The function k = k(t, s) may have a singularity in its derivatives at t = a or at s = a. Although the solution may have singularities in its derivatives at the left endpoint of the interval [a, b], we achieve a rate of convergence that is fast by using spline approximation results of Rice, and results of de Boor describing approximation by splines with variable knots.