Explicit solutions of Volterra integro-differential convolution equations (Q2034036)

The following convolution type Volterra integro-differential equation of the first order is considered: \[ \frac{\partial Q(t,z)}{\partial z} = \alpha(t,z) - \int_0^t \beta(t-s)Q(t,s)\, ds, \quad t,z\in (0,T)\times (0,T), \] where where \(T>0\), and \(\alpha, \beta\) are two given locally integrable functions and \(Q\) satisfies a boundary condition \(Q(t,s) = \gamma(t), \) where \(\gamma(t)\) is locally integrable. An explicit solution of this equation is derived. The novelty of this paper is the use of a biconvolution algebra.