Product Integration for the Generalized Abel Equation

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.