On solving weakly singular Volterra equations of the first kind with Galerkin approximations

The basic linear, Volterra integral equation of the first kind with a weakly singular kernel is solved via a Galerkin approximation. It is shown that the approximate solution is a sum with the first term being the solution of Abel's equation and the remaining terms computable as components of the solution of an initialvalue problem. The method represents a significant decrease in the normal number of computations required to solve the integral equation. The principal goal here is to show that under typical assumptions on k and f the first kind integral equation (I) ~f(x) = Ok(x, t) (x t)-ut dt 0, then certain obvious modifications produce the same existence and uniqueness result, the point being that (I) is still converted to a second kind equation to which the usual fixed point methods can be applied. We assume that f(O) = 0 and note here that, unfortunately, the smoothness assumption on f is troublesome for some applications involving discrete data; however, this is the same objection which arises with the original Abel inversion of (I) when k 1. See [3] for a modification of the usual inversion which does not explicitly involve f'. The equivalent second kind equation is as follows. LEMMA. Under the above assumptions on k and f, Eq. (I) is equivalent to the second kind equation Received September 2, 1975; revised October 16, 1975. AMS (MOS) subject classifications (1970). Primary 45L05; Secondary 45L10, 45H05, 45B05, 45D05. Copyright ?3 1976, American Mathematical Society 747 This content downloaded from 157.55.39.104 on Mon, 20 Jun 2016 05:51:18 UTC All use subject to http://about.jstor.org/terms