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Product integration for the generalized Abel equation


Author: Richard Weiss
Journal: Math. Comp. 26 (1972), 177-190
MSC: Primary 65P05
DOI: https://doi.org/10.1090/S0025-5718-1972-0299001-7
MathSciNet review: 0299001
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Abstract: The solution of the generalized Abel integral equation

$\displaystyle g(t) = \int_0^t {\{ k(t,s)/{{(t - s)}^\alpha }\} f(s)} ds,\;0 < \alpha < 1,$

where $ k(t,s)$ is continuous, by the product integration analogue of the trapezoidal method is examined. It is shown that this method has order two convergence for $ \alpha \in [{\alpha _1},1)$ with $ {\alpha _1} \doteqdot 0.2117$. This interval contains the important case $ \alpha = \tfrac{1}{2}$. Convergence of order two for $ \alpha \in (0,{\alpha _1})$ is discussed and illustrated numerically. The possibility of constructing higher order methods is illustrated with an example.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1972-0299001-7
Keywords: Generalized Abel equation, numerical solution, product integration methods
Article copyright: © Copyright 1972 American Mathematical Society

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