Distributions - Operations and derivatives

A wide variety of physical, chemical, biological, and even economic phenomena can be modeled using differential equations or partial differential equations. The solution to a differential equation is a function that is n-times continuously differentiable. However, it became apparent in the early 20th century that these differentiability constraints were too restrictive and that – for certain phenomena – it might be worthwhile to introduce discontinuous functions as solutions. In the 1930s, Jean Leray introduced the concept of weak solutions for hydrodynamic equations (turbulent solutions of the Navier-Stokes equations). Shortly thereafter, Leonid Sobolev applied this concept to the theory of potential. Building on this work and seeking to provide a coherent framework for it, Laurent Schwartz developed (1945–1950) a general and rigorous theory known as the “theory of distributions.”

At the same time, since the late 19th century, Heaviside’s symbolic calculus had been very popular among engineers because, although it often defied standard mathematical conventions, it had the merit of yielding exact results.

Then, in 1926, Dirac introduced his famous function $\{\begin{array}{cc}\xi \left(x\right)={\mathrm{e}}^{-\frac{1}{1-x^2}}& \mathrm{si}\mathrm{\hspace{0.17em}}\left|x\right|<1\\ \xi \left(x\right)=0& \mathrm{si}\mathrm{\hspace{0.17em}}\left|x\right|\ge 1\end{array}$ zero outside the origin, equal to +∞ at the origin, and with a definite integral of 1, to model a unit impulse at time t = 0 with no effect outside t = 0. Even more difficult for mathematicians to accept, this δ function was also introduced as a derivative of the Heaviside function H, namely the function that equals 1 for x > 0 and 0 for x < 0—a function that is precisely not differentiable at 0! The Dirac delta function was used in integration by parts; it was differentiated (referred to as δ′), assigned a Fourier transform equal to 1, and used in convolution products.

Once again, these various operations make perfect sense within the framework of distribution theory.

In this study, we present the entire "toolkit" that this theory provides. But before we get to the purely operational aspects,...