Generalized solution to the Volterra equations with piecewise continuous kernels (Q400785)

The Volterra equation \(\int_0^tk(t,s)u(s)ds=f(t)\) of the first kind is studied on a compact interval in the space of Schwartz distributions. Here, \(k\) is piecewise continuous/piecewise smooth, and \(f(0)\neq0\). Existence and uniqueness is shown by obtaining a regular Volterra equation (of ``almost'' second kind) after removing the ``singularity'' \(f(0)\). Moreover, by considering Taylor expansions of the given data, a corresponding asymptotic expansion of the solution of this regular equation is obtained (with qualitative error estimates) in the form of certain linear combinations of expressions of the type \((\ln t)^jt^k\).