Revisiting the fundamental equations of the mathematical physics in the space of generalised functions

Abstract

A broad class of physical processes is described by second-order differential equations. The equation of waves, the diffusion equation and the stationary equation, which are found both in classical and quantum physics, belong to this class. In this article, these equations are reformulated under the assumption that the source determining the physical phenomenon acts instantaneously, using the formalism of generalised functions. Some particular cases of these equations are precisely solved by applying the Fourier transform. The obtained solutions form a subspace of the singular generalised functions’ space and are mentioned by the product between a step function and a function with at most a finite number of discontinuity points.