Distributions - Applications

This dossier follows on from the two previous presentations on the subject and to introduce the basic concepts of distribution theory. It presents some fundamental applications of this theory in engineering fields.

We have already seen, in the folder Convolution and Fourier transform how to write the derivation operator as a convolution product, i.e. : $T^{\left(n\right)}={\delta }^{\left(n\right)}\ast T$

allows us to reduce the solution of a differential equation to the search for an inverse of the derivation operator (Green's solution) in a suitable convolution algebra. In a way, we're algebraizing the problem! It's very elegant and clever, but it doesn't solve all the difficulties.

Here, we develop further applications in three directions.

We hope that these three presentations ( as well as the present text), basic knowledge of distributions and an idea of possible applications. Distribution theory is a fine piece of mechanics, based on complex functional spaces. To master the tool, you need to have some idea of its foundations: that's the difficulty of writing on the subject!