Introduction to the fractional derivative

When we introduce the notion of derivative, we quickly realize that we can apply the concept of derivative to the derivative function itself, and thereby introduce the second derivative, then successive derivatives of whole order. Integration, the inverse operation of the derivative, can also be considered as a derivative of order "minus one". We might also ask whether these successive-order derivatives have a fractional-order equivalent. According to a recent history of mathematics thesis, numerical derivation of fractional order can be traced back to various correspondences between Gottfried Leibniz, Guillaume de L'Hôspital and Johann Bernoulli at the end of the XVII ^e century. But these great pioneers came up against a paradox.

One might think that this search for fractional derivation is a matter of "pure" mathematics of no interest to the engineer. However, a simple example from fluid mechanics shows how the half-order derivative appears quite naturally when we want to explain a heat flow exiting laterally from a fluid flow as a function of the time evolution of the internal source. Now that the half-order derivative has been introduced, we need to be careful about its precise definition in the most general situations. The same applies to the definition of the fractional-order derivative α, where α is typically a real number between zero and one. For a long time, following the work of Joseph Liouville and Bernhard Riemann in the mid-19 ^e century, several definitions coexisted without perfect compatibility between them. In this article, we show that with distribution theory, all ambiguities have been removed.

A particular interest in fractional derivation lies in the mechanical modeling of rubber, in short, all kinds of materials that retain the memory of past deformations and whose behavior is said to be viscoelastic. This is where fractional derivation comes in naturally. In the last paragraph, we offer a brief introduction to this difficult subject.