An Integro-Differential Structure for Dirac Distributions

We develop a new algebraic setting for treating piecewise functions and distributions together with suitable differential and Rota–Baxter structures. Our treatment aims to provide the algebraic underpinning for symbolic computation systems handling such objects. In particular, we show that the Green's function of regular boundary problems (for linear ordinary differential equations) can be expressed naturally in the new setting and that it is characterized by the corresponding distributional differential equation known from analysis.