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 / the first kind integral equation (0 /(x) =/*/c(x, t)(x-tyau(t)dt, 0 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 /(0) = 0 and note here that, unfortunately, the smoothness assumption on / 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 /'. 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 © 1976, American Mathematical Society 747 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use