The Numerical Solution of an Abel Integral Equation by a Product Trapezoidal Method

An approximate solution to the Abel integral equation ${{\int_0^t {f(s)ds} } / {\sqrt {t^2 - s^2 } = g(t)}}$, $t \geqq 0$, may be obtained by computing the values $f_h (jh)$ of a continuous function $f_h (s)$ which is linear in each interval $[ih,(i + 1)h],h > 0,i = 0,1,2, \cdots $, and which satisfies the equations \[ {{\int_0^{kh} {f_h (s)ds} } / {\sqrt {(kh)^2 - s^2 } = g(kh),\quad k > 0}},\]\[ f_h (0) = f(0) = {{2g(0)} / {\pi .}}\] When $g(t)$ is Lipschitz continuous with Lipschitz constant B, then it is shown that \[ \mathop {\sup }\limits_{0 \leqq t \leqq T} \left| {f_h (s)} \right| \leqq BT + {{2\left| {g(0)} \right|} / {\pi .}}\] Furthermore, if f has a bounded third derivative on $[0,T]$, then there is a constant C such that \[ \mathop {\sup }\limits_{0 \leqq t \leqq T} \left| {f(s) - f_h (s)} \right| \leqq Ch^2 \log ({T / h}).\]