Applications of Convolution

Abstract

As a first application, the convolution algebra \(\mathcal{D}_+^{\prime}(\mathbb{R})\) is used to introduce symbolic calculus that allows to solve inhomogeneous ordinary differential equations with constant coefficients where the inhomogeneity is a given element in \(\mathcal{D}_+^{\prime}(\mathbb{R})\). In particular, Volterra’s integral equation is solved by using the convolution algebra \(\mathcal{D}_+^{\prime}(\mathbb{R})\). Next we address the problem of solving inhomogeneous partial differential equations with constant coefficients. An important step for a solution is based on the concept of an elementary solution and to determine these. Without proof we mention the important result of Malgrange and Ehrenpreis, which states that every constant coefficient partial differential operator has at least one elementary solution. The concept of an hypoelliptic partial differential operator with constant coefficients is defined and characterized through regularity properties of one of its elementary solutions. Elementary solutions of the following partial differential operators are calculated: the Laplace operator in \(\mathbb{R}^n\), the operator associated to the heat equation in \(\mathbb{R}^{n+1}\), and the wave operator in \(\mathbb{R}^4\). Knowing an elementary solution allows to solve the inhomogeneous equation by calculating the convolution product of the elementary solution with the inhomogeneity (when the convolution product is defined). As an example we solve Maxwell’s equation in vacuum in the case where the densities of electric charge and of electric current are distributions with natural support properties.