Convergence properties of a class of product formulas for weakly singular integral equations. (English) Zbl 0705.65017
The authors study product quadrature rules \(\int^{1}_{- 1}K(x,y)f(x)dx\simeq \sum^{m}_{i=1}w_{m,i}(y)f(x_{m,i}),\) with \(K(x,y)=| x-y|^{\nu}\), \(\nu >-1\), or \(K(x,y)=\log | x- y|\), of interpolatory type, based on the zeros of some classes of generalized Jacobi orthogonal polynomials. If \(f(x)=(1\pm x)^{\sigma}\), \(\sigma >-1\), then they obtain uniform rate of convergence of the quadrature rules. Their results generalize some of those presented earlier in the literature.
Reviewer: D.Acu
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 65R20 | Numerical methods for integral equations |
| 41A55 | Approximate quadratures |
| 30E20 | Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
Keywords:
weakly singular integral equations; product quadrature rules; Jacobi orthogonal polynomials; uniform rate of convergenceReferences:
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