Heat equation and first-half integration
Introduction to the fractional derivative

In this first paragraph, we show that the half-order derivative is naturally introduced when we try to calculate a heat flow using Fourier's law.

We give ourselves the interval [0, + ∞[ where the space variable y lives. Note that this variable y is typically a direction orthogonal to a principal direction x of a fluid flow. The time variable t is assumed to be positive. We give ourselves a strictly positive diffusivity constant µ and a function f(•) which, for this example, depends only on time. A function u(y, t) is sought as a solution to the heat equation