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In this course.

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We will learn, of course, about pens.

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And the structure of this course is going to be first start with the finite difference methods, the

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usual way of solving pdes or partial differential equations to basically model a system.

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And later, what we are going to do is to actually go to pens.

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We will start with a little bit hard part of pens, which is well, basically solving, solving them

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using PyTorch, a little bit more programming needed in here.

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And after that, the course will become much easier as we are going to use deep library to solve pens.

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It will make us it will make our life much easier.

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And actually we can model a complex systems in a very short time.

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So finite difference methods are usually used for solving partial differential equations.

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Of course, it's what we call usually a numerical solution.

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A numerical is just a way to solve equations.

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Not, of course, analytical, not using well math laws by our hand.

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What we try to use is numerical solutions that give us a more flexibility in solving these partial differential

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equations.

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So the first part of this course, we are going to first understand how to solve partial differential

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equations.

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Numerically, the old way of doing it, let's say using finite difference method.

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And to get this to do this, we will just take the heat equation that describes how the heat or the

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temperature or basically the heat propagates and change the temperature into in our in a system.

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So we will have the rod and we will have two heat sources and we will see how this this rod will increase

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in temperature.

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So it's it's it's quite interesting and we will see the how we will code it all by ourselves from zero

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using of course Python.

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Now after that we will step our game a little bit.

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The heat equation is kind of diffusion equation and it is more or less always like it's much more robust

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as it's going to.

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It has this diffusion well factor that kind of will make it kind of converge most of the time or many

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times much easier than the convection equation or the conviction that doesn't have convection term like

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over D y.

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So we have this.

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We will.

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Well, now we want to step it a little bit.

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We have this diffusion term and we have this convection term and in which we will try to solve an equation

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that has both, which is the burgers equations.

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And these burgers equations are well, more complicated than one D heat equation.

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We are going to solve it using also a finite difference method.

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We'll go step by step and we will this time solve these two beautiful equations and get hopefully the

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answer.

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Now we become a familiar with finite difference methods and how to solve the problems with that.

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We step it now.

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Well, now we have to work with pens.

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But of course we have to work kind of cautiously.

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So we will start by solving 1D burgers equations because why?

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Why burgers?

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Well, first 1d is of course is easier and 1d burgers has has many things we need to learn.

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The first one is, as you can see this the second derivative of u over the first derivative of u over

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the x and derivative of u over time, which is interesting to see how the solution will progress also

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in time.

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So this is going to be actually our solution and this is going to be solved using pens.

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This is going to be the first pens.

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Well, problem we we solve now, after that, what we need to do is how about go to solve a 2D problem?

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The the the first problem we solved is 1D problem and now we need to go and solve a 2D problem.

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To do that, we took the 2D heat equation and we well write our code to to solve it with of course pens.

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Now, solving with PyTorch is, of course, important.

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And and most likely, if you are most likely if you want to write your own software, it's better to

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write.

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With Python.

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You have a control on everything, more or less, much more higher control than if you use a library.

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However, there are libraries that actually make pens solutions easier.

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For example, the NXT is one of them and this library we are going to use it to first solve a 1D heat

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equation just to know the library.

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How can we achieve it?

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Achieve a solution using this library and we get a little bit familiar and we will see how our coding

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will become easier and our solution will we can get to a solution faster compared with, you know,

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writing our torch problem.

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Now we get familiar and we get stronger with pens.

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Why don't we solve Navier-Stokes equations?

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I'm sure many of you like.

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Navier-Stokes equations, which is equations that is used to model fluid dynamics.

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We take these equations which are, in our case, what we are going to solve is a steady state.

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Navier-Stokes equations.

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However, in these equations I'm putting on this slide, we have time.

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Just you put it zero because du over t is going to be zero in our case.

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And as you can see, there are or as you know, there are many forms of Navier-Stokes equations.

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This one we are going to solve two momentum equations, one in your direction.

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One is in the direction in the horizontal axis and the vertical axis.

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And the third one is the continuity equation, which is this one.

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Of course Navier-Stokes equations might have other equations, for example, the energy equation and

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in Navier-Stokes you can even expand through other things like reactions and even electrohydrodynamics.

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It's it can get quite complicated.

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However, let's take the simple 2D steady state equation and or we model it into a steady state using

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these equations and solve a 2D problem.

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And with that, hopefully we get already familiar enough with solving.

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Well equations using pens.

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And by that our course is over and hopefully it will be a very important step and try to reduce your

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time learning pens or physics informed neural networks.