Introduction to Fractional Calculus

So my name is Tigranish Kanyan. Today I will give an introduction to fractional calculus. Okay, so here is plan of my talk. So what is fractional calculus? Then I will give some small historical overview and definitions of different fractional difference integrals. And then I will switch to work from language 13.1 and 13.2 functions, fractional d, capital D and their numerical versions. I will speak about applications, how can we apply this fractional calculus in real world problems? And we'll say a couple of words about the upcoming tutorial that Devendral already mentioned and the whole from your course. And at the end I will give some references and the useful links on the topic. So let's start. We all know that differentiation and integration are two fundamental operations of mathematical analysis. So the first derivative of square function is X, while the result integration is cubic function. And we can generate the nth order derivative of this function. So it's written here including gamma function. And in fact here we can put even negative ns and get the nth or the antiderivative of this square function. However, when we are speaking about derivatives or integrals, we are always assuming that this order N is integer. It can be positive negative, but it's integer. So what if we can we could extend this differentiation integration processes to non integer or even complex orders. So let's make a small like toy exercise. Let's take this formula for the nth order derivative of square function and let's put half into that. So the output will be something like this. And formally this is nothing else than the half order derivative of the square function. So if we will repeat this once more, so. So we'll take the nth order derivative formula of this function and put one half into that. We will get immediately X which is nothing else than the ordinary derivative of square function. But we gained it like using two say half order differentiations. These examples illustrate the basic idea of fractional calculus which generalizes the classical operations to some real or complex number orders. A bit of history so in fact in the Wikipedia article there is a nice historical overview about fractional calculus. I will leave a link here. So fractional calculus was introduced in one of the papers of Abend and he introduced all the like basic basic ideas of that he was solving the mechanical problem. Hubble's mechanical problem, which is to define this equation for this curve from the prescribed transit time will not go into details, but I will say some words about the equations that he obtained. So this is one of the equations from this article from 1823 and in more complex form. You obtained that in the second article. This is it. And in fact, after several simple manipulations, this can be rewritten in the form. In this form, which is nothing else than the Caputo fractional derivatives. We are now calling that Caputo fractional derivative. But everything started from the ABL's mechanical problem. Some details can be found in this article. Link is here, so I will leave it. The file is uploaded into the pattable so you can download this notebook and play with the examples, click on the links and so on. So in fact, in fractional calculus we are working with one unified operator standing for differentiation and integration. And we are calling that fractional difference integral. The fractional difference is written here. Where this alpha is the order of differentiation. It can be real load complex 0 is the lower bound of deeper integral. We'll see where it came. However, this. This is not. This can be anything, not only zero. There are a lot of different definitions for fractional differ integrals. However, we'll talk about three of them. Say main three differ integrals. The first one is called Gruelle Netnikov differ integral that's written here. It is giving some definition to fractional differentiation and integration using some limits and infinite series. And it's extending. It's similar to the ordinary derivative definition by limits. However, this. This one is not very useful for analytic formulas and calculations. For them we are using the next one that is called the Riemann Liu Beijetra integral. It's written here and this is in two words. This is like a derivative of an integral containing the function that we're differentiating or integr multiplied by some some power function. And here we see the zero the lower bond. It can be anything. In fact, for simplicity we are using zero. This one, the Riemann Liu will differ integral is very powerful thing. The theory is rather well developed about that. However, from practical point of view is not very usable for practical problems, real world problems. We are using the third difference that is called Caputo deeper integral. It's written here and this is an integral of derivative of the function. And of course we can see that this Caputo differ integral and Riemann1 they are quite similar and they are connected in fact by this formula. So Caputo different world there is remain minus some finite sum. And in fact we are using this one for real problems. However, we are generating the outputs using the remarry will approach and some minus some sum. And of course this Caputo differ integral coincides with the Riemann label one for negative alphas. Because this Part is missing for the place and couple different integral is the same as the remainder one or negative orders. This means for integration fractional integration. Below I have a animation that is showing the behavior of like ordinary in the remain fractional derivatives of a square function. So this is the function itself, this bold blue line. This is the first derivative X. This is the integral cubic function. And we see that fractional orders, they are like in some sense they are interpolating between these ordinary derivatives and integrals. This is the presentable example. What is fractional calculus? What are the fractional derivatives? Okay, now let's turn to the functions that we have implemented. So in 13.1 we have implemented two functions. First one is called fractional D. The second one is called Caputo D and fractionals D. Fractional D computes the Riemann Lubric fractional derivative. As I said before, it does not have a lot of applications, but it's very important in the theory. And the theory is well developed. There are a lot of books, articles and Professor Oleg Marichev he directed around like 85,000 rules for different input functions. And we are using some of them. The definition is written here once more and let's take a look. How can we use this definition to generate the fractional derivatives? Let's take a cubic function, it's written here and just put it into the. Into the definition of Riemann living. So it will. So here we are calculating say the half order derivative of cubic function and it's written here. And we'll just repeat this operation and we'll immediately get the first derivative cubic function. That's it. Of course, this is very. Yeah, this is a very simple example. But for some, when we're developing this fractionally, we use this approach a lot. Let's take a look on the following example. Oh yeah. So let's calculate the any order fraction of the derivative quadratic function written here. And we can plot that and see that. So this is the function. This is its first derivative. Ordinary is the second derivative. And the fractional derivatives, they are like going from the function to its ordinary derivatives. They're interpolating between them. This animation is about that. And after I'm showing how we can we recover the initial function square function using fractional integration. So here I am using the chain of integrations like this order treat them. So first I'm calculating the first ordinary derivative square function. Then using fractional integration getting factor function. Here it is. After I will. So now I will show some examples on two common mathematical functions. So fractional D applied on X expansion function will generate something like this. And here we see that this part stands for the ordinary differentiation. This is coming from the fractional part. This is quite simple example, let's take a look on Bessel functions. So fractional D applied on Bessel G will give something big. Not very understandable, but it is the fractional derivatives of Bessel functions are calculated using hypergeometric FQ function, which is understandable because Bessel function is a hypergeometric class of functions. From this class, and finally, I would like to show a very, very important case fractional D on major G. The important case, important point here is that when we're applying fractional d on major G, we're getting back another major G with shifted parameters. It's written here. And this is very useful in the theory because we can generate fractional d rules for different inputs using their Mary G representations. Then apply fractional D on that and see, see the output. And here I'm showing a table where we have fractional derivatives of some common functions and ordinary derivatives of them. We see that for X we have one more term, while for sine, for example, the output is between hypergeometric functions and so on. Okay, this was about fractional D. And we are going to the next function that was implemented in 13.1. It's Caputo D. And it's very important because it is the Caputo derivative deeper integral. It's taking also the initial values of functional derivatives at some point at origin for this case. So here's the formula that we are using and this is quite important for practical applications. So I will show some of them also. One more basic property is that the capital D F constant is 0 while the fractional D constant is not 0. So this is tying also capital with the classical theory, classical calculus. So here let's take a look on kaputo d applied on some functions. So for example, exponential, we have this part that coming from the fractional D, the Riemann lubricant part. This is a finite sum that I've showed for base lg, the output is in hypergeometric functions. And now I'm applying kaputo d on a very, very important case, interaction calculus on Metagliffe functions. So a Metagliffe function, Devendra already told about them, but they are very important as they are like analogs of exponential function in the theory of fractional differential equations. Yeah, and so here I'm solving a differential equation. It's a fractional differential equation with constant coefficients and containing the capital d of this order and the output is written in metagliptic functions. If here I will put like the ordinary derivative, then the output will be next functions. So plotting this meta glacier, it has some oscillatory behavior. I think that Aram Mana Selyan will talk a lot about this type of problems. So let's go to the next slide. And in 13.2 we have implemented the numerical versions of fractional div capital D because of course fractional D and coupled up they are important and rather powerful. But as of their definitions, when you are applying them on complex functions, then these calculations might take a lot of time. And also for some cases they can generate very big and not understandable results. And even they can generate some results, including different root objects and so on. So having numerical analogs is profitable. And for example, for integrate we have N integrate for dissolve wave ND solve. And now for fractional D capital D we have N fractional D capital D. So the design of these functions is very simple. It's fractional D function, the variable, the order of differentiation and the point that we are calculating the value numerical fractional derivative. So for the output of these functions will be will always be some number. So here is N fractional D. This is N couple to the applied on metabolic functions. Using this numerical versions of fractional derivatives, we can possibly rapidly plot something. So for example, here I'm taking this case, plotting the plotting the numerical fractional derivative of this order on interval. On this interval. Here is another example. I will not evaluate it because it will take quite more time, for quite a bit time. But however, here I'm plotting the sine function and its derivatives fractional and ordinary. Yeah, and now I will say some words about the third approach, third definition of fractional deeper integrals, the Gruelwald technical one that is not very useful in analytical problems, but it's very, very useful in numerical evaluations. So in N fractional D we have two methods. So the first one is called Riemann Wield. It's understandable. The second one is called it is just simply doing numerical calculations. However, we have implemented Dreamwald Letnikov approach, because for some classes of functions, the Riemann Luvial fractional D is not able to calculate because the definition of it is very, very difficult. However, is doing this. For example, here I took this function and using this Grungalvetnikov method at this point calculated the numerical disorder derivative. If we will apply the remainder one, we'll not get the answer because it's impossible to do. Yeah, so I think that was all about capital D, fractional D and their numerical versions. Now let's talk about applications. So generally speaking fractional calculus is gaining more and more applications in different areas of science. There are a lot of particles on this. Fractional differential equations are used in different problems like fluid dynamics, diffusion problems, control theory, signal processing and others. I will now show only one example. Aram Manacelian will talk about them in his talk. So this is a very nice example. Here we are taking a fractional version of the wave equation. So this fractional version has some capital B's the one capital D in it and we are dissolving that and the output is written in metabolic learn sine functions and we are able to plot it. So this is nice plot and this was quite surprising that dissolve is now able to solve also these type of equations. And this very nice also parallel to implementations we made a big tutorial on fractional calculus. So it will be in 13.2 version. It covers all the topics of this presentations presentation. And also it shows a lot of applications and real world problems. So this is the overview of that page. It will be available in 13.2 and also we made couple of webinars new in 13.1 series. And using that videos we've created a world from view course that is called Introduction to fractional Calculus. It is available via this link. And in that course we discuss the history of fractional calculus, the approaches, different definitions and so on. Plus a lot of examples of fractional differential equations, how we can solve them, plus how Laplace transforms and inverse Laplace transforms are working with these functions and so on. So, so let's I'm going to the end of my presentation and this is the summary. So as I said before, instructional calculus is becoming more and more like used in different areas. And of course when we are turning from classical differential equations to differential equations with fractional derivatives or integrals inside then we we are like extending the class of equations that we can work with solve with. And of course this means that we can describe more physical phenomena or even try to describe more more precisely the already solved ones. And of course understanding this potential of this topic, we have implemented this fractional D, capital D and fractional D and capital T functions and also added a lot of support Dissolve and Laplace transforms. Okay, so now a couple of words about references and useful links. So here I have two books that are giving the basic theory of fractional calculus. These article is quite interesting because it's giving the historical overview of that showing how Abel obtained these equations and how we introduced this fractional calculus. This book is about numerical approaches to fractional deeper integrals, how to calculate them. And we used this book in to develop the functions for 13.2 and fractional d, then capital D. Also this this is a link to Wolfram's blog post where he mentions this fractional D. I wrote a blog post also about fractional calculus. So it's the link is here. This link is about our webinars in July and August, I think. And also I'm giving links to the reference pages, these functions mentioned functions because we try to add a lot of examples into the pages to comprehensively cover all the say the all the aspects of algorithms, the weak points and so on. And also we added a lot of examples of applications. How can we use them? So there will be a tutorial in 13.2 and also the link to the course. I think that's all. Thank you for your time. Presentation.