Convergence properties of a class of product formulas for weakly singular integral equations

Abstract:

We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type \[ \int _{ - 1}^1 {K(x,y)f(x) dx,} \quad - 1 \leq y \leq 1,\] where the kernel $K(x,y)$ is weakly singular and the function $f(x)$ has singularities only at the endpoints $\pm 1$. In particular, when $K(x,y) = \log |x - y|$, $K(x,y) = |x - y{|^v}$, $v > - 1$, and $f(x)$ has algebraic singularities of the form ${(1 \pm x)^\sigma }$, $\sigma > - 1$, we prove that the uniform rate of convergence of the rules is $O({m^{ - 2 - 2\sigma }}{\log ^2}m)$ in the case of the first kernel, and $O({m^{ - 2 - 2\sigma - 2v}}\log m)$ if $v \leq 0$, or $O({m^{ - 2 - 2\sigma }}\log m)$ if $v > 0$, for the second, where m is the number of points in the quadrature rule.