Convolution Equations

Abstract

The objective of this chapter is to show that the solution of ordinary differential equations, if based on distributions as opposed to functions, can be obtained by (mostly) algebraic methods. These methods are rigorous forms of the so-called Heaviside’s operational or symbolic calculus. The close relationship to the integral transforms that convert convolution into the ordinary multiplication is also shown. With this chapter we stop using uppercase letters such as T to denote distributions. Instead, we start using lowercase letters such as the ones typically used to denote functions, for example f. We also adopt the convention of denoting the Laplace transform of a distribution, say f, with the same letter, but changed to uppercase, e.g., $$F = {\mathcal {L}}\{f\}$$. When we need to distinguish between the ordinary and the distributional differential operator, we will in general denote the former by $$\frac{\textrm{d}^{}}{\textrm{d}t^{}}$$ and continue to denote the latter by $$D$$.