Fractional derivatives and exponential derivatives

In schools and universities, when studying mathematics, the operation of differentiation of a function and its derivative are determined. These are fundamental concepts on which the entire apparatus of mathematical analysis is subsequently built.

The usual derivative and its generalizations are used everywhere, for example in machine learning and when training neural networks.

If you differentiate a function, you get its derivative. If you do this twice, you get the second derivative. But what if there is something “in between”? Of course there is, and it is about such objects that this article is written.

What's the point of a derivative of order 1/2?

Fractional derivatives

The concept of a derivative of an integer order is generalized to a fractional order – for example, you can “invent” a derivative of an order . Good intuition about what an order derivative is and how it “acts” on a function gives an operator view of taking the derivative.

Let us consider the derivative as an operator – a mapping from the space of differentiable functions to the space of their derivatives. Let's call the operator the ordinary derivative and the application of the operator to the function is denoted by D^1(f)=df/dx . Then a reasonable task is to find a new operator in the operator space – one that, when applied to the function twice, will give the same result as let's call it $D^{0.5}: (D^{0.5} ◦ D^{0.5})(f) = D^{0.5}(D^{0.5}(f)) = D^1(f)$ . This will be the derivative of the order.

Similarly, you can define an operator for the derivative of any fractional order, and call the corresponding operator . Let me remind you that real numbers are strictly defined as the limits of sequences of rational numbers converging to them, therefore the real order operator can be defined in the same way as the limit of fractional order operators.

There are several different ways to find the exact form (or formula) of a derivative of a fractional order, consistent, firstly, with the property written above, and secondly, giving formulas that coincide in the case of integer orders with classical derivatives. I will highlight four of them and describe them in one or two sentences.

The first method is similar to the definition of an ordinary derivative – through limits. See the formulas in the pictures, the first corresponds to the ordinary derivative, and the second generalizes it for the fractional order alpha, and if you substitute alpha equal to 1, you get the ordinary derivative.
Definition of ordinary derivative
$Determination of the fractional derivative through the limit$
Determination of the fractional derivative through the limit
The second way is to expand the function itself, which we will differentiate by basis, for example into a Taylor or Fourier series, calculate how the operator acts on the basis functions, apply the derivative operator to each term and sum the series back. It's important to note that this approach will work because the operator linear, i.e. the operator applied to the sum will give the same result as the sum of the results of applying the operator to the terms. All that remains to be understood is how the fractional derivative acts on the basis functions. This can be done by looking at the patterns of ordinary differentiation of power functions () and sines-cosines with exponentials (exponent $e^{ax}$ this is an eigenfunction of the differentiation operator). This approach is used in this article: https://habr.com/ru/articles/734000/
The third way is to use Cauchy formula for the iterated integral. The formula makes it possible to express an iterated integral of order K in terms of a single one, and it involves raising the argument of the function to a power and factorial. This formula can be analytically extended to both non-integer and orbital orders of the iterated integral – this is exactly how fractional derivatives are obtained. The formula contains exponentiation – it has no problems with fractional exponents, and factorial – it is generalized using Gamma functions.
The fourth method is integral transformations Fourier or Mellina. Both of these transformations make it very easy to express the image of the derivative of an integer power of a function through a formula into which you can easily substitute the fractional (or even complex) parameter of the power of the derivative. The idea is almost the same as in the previous method – use a ready-made formula or transformation and generalize it to fractional powers of the derivative.

There are a lot of ways to obtain a fractional derivative, for example, on Wikipedia there are more than 15 types of such an operation:

Different types of fractional derivatives

Other types of fractional derivativesOther types of fractional derivatives

Fractional derivatives have many uses, and YouTubers also love to explore this concept, here are a couple of interesting videos about it:

Exponential derivative

The operator approach allows us to calculate other “interesting things”, for example, functions of the operation of taking the derivative itself. The simplest and non-trivial example of a function of the derivative is the exponential: e^{d/dx}. To understand what such an operator does, you need to take the usual Taylor series expansion of the exponential and substitute d/dx instead of x. It will turn out:

That is, this is a series of taking derivatives. What will happen directly to the function if you apply this operator to it? Here are three key examples, with functions :

Exponential derivative for functions x, x^n, sin(x) — Exponential derivative for functions

x, x^n, sin(x) — Exponential derivative for functions

The visible pattern is that the resulting statement acts like this – – works for any sufficiently differentiable functions. This is already clear even from the derivative for and the linearity of the operator – the function on which it acts can be expanded into a Taylor series, and I calculated the derivative of the sine for the sake of an example of direct calculation. If we apply this operator twice, we get . We can generalize this operator to $exp(a\cdot d/dx)$ which shifts the argument by any real number .