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On Galerkin methods for Abel-type integral equations. (English) Zbl 0665.65096
Integral equations of the form $\int\sp{x}\sb{0}(x-y)\sp{\alpha - 1}k(x,y)f(y)dy=\Gamma (\alpha)g(x)$, $0<x<1$, where $0<\alpha <1$ and $k(x,x)=1$, are treated by an approximation ansatz with piecewise polynomials of specified maximal degree, not required to be continuous across the nodes. The questions of robustness and stability are thoroughly discussed, and optimal rates of convergence in the supremum norm are derived. The effect of approximating the inner products of the Galerkin scheme by quadrature methods (in particular by a “natural” quadrature and by a “cheap” one) are investigated. Finally, results of numerical case studies are displayed, also for use of a particular collocation method.
Reviewer: R.Gorenflo

65R20Integral equations (numerical methods)
45E10Integral equations of the convolution type
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