[Fractional Calculus Demystified: Part 1] Some Intuition Explained

Disclaimer: side effects of studying fractional calculus include wanting to apply it to everything and anything and developing a love for analysis.

Haven’t we have had enough to talk about with regular calculus!? The short answer is no, but we like to studying fun looking things anyway. Have you heard of fractional calculus? Who even knew that these things existed, let alone mattered?

Let’s look back a bit to the origin of the ideas and some motivation from physics for studying such objects that appear in the study of motion, fluids, voting patterns, material science, machine learning, finance, and countless many others. My aim is to give you a short and sweet introduction to the fundamental ideas behind fractional calculus and its application.

Fractional calculus is a term coined to describe certain operators that certainly look like derivatives and integrals but are a bit more funky and nuanced. Its origins date back to discussions between l’Hopital and Leibniz, names we are familiarized with in a first semester Calculus class. They were but a few among the many people who helped solidify the larger study of mathematical Analysis as we know it today. Making use of Leibniz’s notation, l’Hopital posed the question of whether a “half” derivative made sense. In this sense, the term “fractional derivative” was considered despite the order of such a derivative not having to be rational number (fraction).

The way they thought about this came from a back and forth series of letters. The trouble was agreeing upon a proper definition. In math, definitions must be attempts at describing important observed objects in such a way that meaningful results may be concluded about such objects. Crafting definitions, in many ways, is an exercise in precision and forward thinking. They must seem to agree so much with known and future results that they almost appear to be natural, or even obvious, but by no means are they trivial. As such, one natural place to look to define this fractional integration object is to go back to basics — i.e., the Fundamental Theorem of Calculus.

The Fundamental Theorem

This theorem is at the heart of calculus, hence its name. Proven in this form by Newton in the early 18th century, the theorem describes, under appropriate assumptions, how derivatives and integrals behave with one another. In such cases the following holds:

Notice, when both a derivative and integral are applied to a function, we recover the original function back! Suppose that the value of our function at x = a is 0 so that we can ignore the f(a) for now. Then, such a term would disappear. An important distinction, however, is the placing of the derivative inside or outside the integral. We’ll keep that in mind in later discussions.

It turns out that this theorem will somehow lead us towards a good notion of a fractional calculus! Before we jump to our fractional calculus “wishlist,” let’s look at some motivating examples. Remember, our definition should be motivated by observed phenomena and be somehow forward-looking. We’ll be motivated by fluids! Well, sort of.

A Motivating Example

One example is when looking at the influence of external forces upon a material. Denote by σ(t) the stress on a material while denoting ε(t) the strain on the material. For example, suppose you really want ice cream just as I do at the time of writing this article. Your material would be the ice cream. In order to have the yummy treat, you must attack the scoop of ice cream with your spoon. When scooping up the ice cream, the spoon(you) exerts a force on the ice cream which we call stress. At the point of contact, the ice cream gets strained and begins to deform which is why you are able to scoop up a piece of something which previously was a complete solid. I’m sure there is a life lesson there, now that I think about it.

Anyway, there are two idealized examples. One is the perfectly elastic solid (think of a rubber material). Hooke’s Law states that, for an elastic solid,

That is, the stress on a material is directly proportional (no derivative here) to the strain on the material and that relationship is captured by modulus of elasticity E. On the opposite side of the spectrum, we can consider viscous liquids (think water, honey) where Newton’s Law states

That is, the stress on a viscous liquid is proportional to the (remember derivative is tracking some sort of rate of change) rate at which you apply strain. This proportion is captured in the viscosity constant η. This relationship has to do with the fact that ceasing to apply force to a liquid does not necessarily immediately stop its motion. A key distinction between a solid and a liquid is that perfect liquids will take up as much space as their container allows while a solid retains its volume whenever it can.

We refer to this article for further discussion on elasticity vs. viscosity. A key takeaway is that these are idealized scenarios. In other words, there are very few materials that are perfectly elastic or perfectly viscous, i.e. perfectly solid or perfectly liquid. Most things are somewhere in between!

Take a look at Hooke’s Law and Newton’s Law side-by-side. The only real difference is in the derivative term. We can ask ourselves the same question that L’Hospital asked himself in his letter to Leibniz! If most materials are in the in-between land of viscoelastic materials, would they possibly be better modeled under a relationship that is also in between Hooke and Newton? In other words, can we consider, for example,

for a material that is “half” viscous “half” elastic, whatever that may be interpreted physically to mean. Therefore, a central question is about capturing an “in-between” regime of behavior that classical objects don’t let us consider so nicely. Another observed physical phenomenon that we will tackle is that of anomalous diffusion. More on that later.

Enjoyed this story? Here are parts 2 and 3!

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There are many great sources out there. A link to the details of Kai Diethelm’s wonderful book on fractional calculus which is a major reference for this series of articles: https://link.springer.com/book/10.1007/978-3-642-14574-2