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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Product integration for the generalized Abel equation
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by Richard Weiss PDF
Math. Comp. 26 (1972), 177-190 Request permission


The solution of the generalized Abel integral equation \[ g(t) = \int _0^t {\{ k(t,s)/{{(t - s)}^\alpha }\} f(s)} ds,\;0 < \alpha < 1,\] 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.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Math. Comp. 26 (1972), 177-190
  • MSC: Primary 65P05
  • DOI:
  • MathSciNet review: 0299001