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

Authors: Giuliana Criscuolo, Giuseppe Mastroianni and Giovanni Monegato
Journal: Math. Comp. 55 (1990), 213-230
MSC: Primary 65R20; Secondary 45L05
MathSciNet review: 1023045
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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

$\displaystyle \int_{ - 1}^1 {K(x,y)f(x)\,dx,} \quad - 1 \leq y \leq 1,$

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 \vert x - y\vert$, $ K(x,y) = \vert x - y{\vert^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.

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Article copyright: © Copyright 1990 American Mathematical Society