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Welcome back to our calls, where we are currently discussing how to solve ordinary differential equations,

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and so far we have programmed ourselves a function that solves these equations, according to the Euler

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methods.

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And we have solved the dysfunction first order and also second order differential equations.

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We've seen that this can be really nice to get the results that we weren't able to get when we would

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just rely on analytical methods.

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Now, of course, this Euler method is just the most simple approach and the most simple way of solving

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these equations.

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We've also seen that when the step size is too small, it can get slightly wrong results.

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And also, you could imagine that when our problem problem becomes more difficult and if we want to

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simulate it over a very long time and we want to still get the accurate solution, then we would really

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have to decrease the step size and then the solution would take forever.

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So it's much better to use a solvent.

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It is provided by Python that is optimized and it uses more improved and advanced methods.

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And to use them, we have to load a module that we have actually loaded in the very beginning of this

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notebook, and this module is called PSI Pi.

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And here we are only going to use the integrated part of place.

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So I wrote from Typekit import integrate.

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But you could also write import side.

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So fine, so pi.

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And then you would just have to write every time, say Pi Dot, integrate and then the solver that we

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are going to use.

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And since I wrote it down like this, we just have to write, integrate dots and then the command that

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we want to use.

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So let me go down to the notebook just where we left after the last lecture, and this was when we have

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simulated this driven and attempt pendulum.

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So as I said now we will discuss improved methods and there are actually two useful solvers in this

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Sci-Fi module.

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There is an old one that is called.

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Yeah, all the story O and the E ins and I was just confused because it says old.

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But it stands, of course, for ordinary linear differential equation and that interface and integration

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methods this you can still use, but it's kind of old and not really profitable anymore.

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Much better is now the new command, which is sci fi not integrate dot solve underscore IVP for now.

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Yeah, and and this is what we're going to use.

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And for this, you can even select different methods and different solvers.

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So it's a nice tool that we will explore in the following lectures.

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And the first thing is we are going to compare our results, for example, to which time and fight decay

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with our old results.

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So here we have the old methods.

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And let me run this.

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We've seen it worked.

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And now we must, of course, verify that our new solver can also restore these results.

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So how does it work?

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Well, first, let's create a new cell and then the the command will be integrate dots.

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So underscore heap, as I wrote above.

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And then we just have to take care of the arguments.

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So if you wouldn't know what the arguments are, how would you find out?

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Well, you have to Google in the internet.

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So here is a link where you can go to or you just Google Python.

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Solve IVP and then you will get probably to this side or to another side.

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And there they will really list all the arguments that you need to provide and we'll explain what they

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mean and what you have to provide.

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So here we first have to provide our function like in our own solution.

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So we need everybody.

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Then we need the time span and this is here given by such a list of two values starting volume and end

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value.

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Then we have to provide the starting value for the value of why.

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And here we use one and we have to give it also as a list.

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The reason for this is that once we try to solve higher order differential equations, then we would

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have to provide here two or more values, as we have seen for the free fall, where we have to provide

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the starting position and the starting velocity.

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So here, from a logical point of view, it wouldn't really be necessary to use this as a list, but

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that's the syntax.

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So we have to provide a list.

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And now it would work already, but fortunately, we can provide the method and we will discuss too

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soon a bit more detail, but a pretty standard method that's always a good first guess is the only a

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quarter four or five methods or short RJ 45.

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So we must now stored the solution and I call this solution or AK 45, and I run this.

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And now we can have a look.

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What is the actual solution?

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So solution underscore our K 45 Bob T would be the syntax for the T list, just an area of some dots.

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And then we can also access the why list.

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Which is, of course, also an array of dots of coordinates, and together they will be dots.

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So let's look at those and let's just copy the plot from here and replace these two commands with our

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new solutions so we can leave everything as it is and just change this one.

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So here we will.

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Right solution RJ 45 T and then Dot.

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Why?

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And if I run this, you will see this is the solution.

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And first of all, you see, OK, the most important thing is that the end value is correct.

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You see, it is correct.

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It's right on the red curve.

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And it's also the same as an hour in a miracle approach.

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But what you can also see is that this could R45.

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The methods used only very, very few points could even see this year only one two three four five 11

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points.

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And therefore, it's very, very fast compared to our 100 points or even 300 that we used here.

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So it's enormous, I think, how comical to solve this differential equation with only such a few points.

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And also what's remarkable is that we never told the wrong a quota algorithm how many points to choose.

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And this is really a nice thing because we have seen previously that once we choose the step with too

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large into all the methods that our solution will become in an inaccurate.

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And so there's always this danger that you don't get the correct solution, but you don't really know

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if that's correct or not.

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However, for the human quota 45, you will not have these problems as the algorithm determines these

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values automatically.

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So the next thing that we are going to do is we, of course, wants to reproduce this graph here.

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So we want to get two more points on the curve.

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So for this, what we can do is we can provide an additional optional argument here.

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So let me copy this.

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And let me also copy the plot.

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And to do this, we just have to provide an additional argument here, which is called to evaluate,

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to evaluate, and then we have to provide here some.

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And for example, we could provide a very similar array to be we right and peddled within space.

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And then we go from, I don't really remember from zero to let's see zero to 30 and the step number

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would be three hundreds.

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Yes.

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So this would be age times and max, cola and max that I run this around this.

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And then it tells R.K. 45 explicitly to use these points.

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And so at least these points as the T points for which it will evaluate the solution.

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So I think in general, it's really much better to have only a few points here because then the calculations

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will be faster and in the end, you will only be interested in the end value.

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But since for some reason you want to have a dense list of points here, and then you can use this by

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using such an optional argument, and then you will get such a nice solution, which is exactly what

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we got before with our Euler methods.