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And the previous lecture, we have seen that there exist many methods that are much more accurate compared

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to the conventional Euler method for solving differential equations.

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And one of those methods was the quotes methods.

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And there is actually a whole family of running Qatar methods.

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And one of them was the longer cutoff for five methods, which we are going to implement throughout

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the following two lectures.

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So the reason why you're able to quit our method is used so often throughout physics is because it is

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not that time consuming.

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It's not that difficult to implement, but still, it gives very accurate results.

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And to explain to you why when we put our method is so much better than the Euler method is pretty difficult.

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So we can have a look at this link here, and you can see here's a derivation, and I really don't want

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to go through this whole derivation with you together because you see it is quite demanding.

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And this is not even going to put up four or five methods, but only the a quarter four methods.

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So the simple or simplified and a simplified way, you could say that the reason why the woman quit

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on method is much better than the Euler method is that it combines several terms so that arrows cancel

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each other.

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That's a bit similar to the way that we have to find the more accurate versions for the derivatives

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in the earlier section of this course.

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So what we are going to implement here is the home equity method called Arc four, which will be the

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first step, and then we will take it to even further.

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And then we will implement the R.K. four five method, which is the same that is implemented previously

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in the problem solver that we have compared to here.

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So to get started, let me go back and let me copy the codes for the implementation of our oil methods,

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because here we will define a function called R.K. four.

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And of course, in general, it will be pretty similar to the old method, and we will just change the

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way the the way we'll use updated.

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So let me scroll up.

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Let me search for our Arla methods.

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So not the Euler second order method, but the First Order methods.

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I think here it should be in a second or the pendulum.

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We have solved quite a few examples.

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Yeah, here it is.

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The Euler.

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So you go to the cell, it's should be cell number seven and we copy this and go back to the bottom

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of our notebook and paste it here.

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And now what we are going to change is, of course, the name.

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We're going to name it, R.K. four and four.

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The arguments.

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We still need the same things and also the beginning will be pretty much the same at first.

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We use our starting value for Y and yes, added to Y and the same for T zero.

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Then we start with these lists where we will store our values T and Y, and then there will be the loop

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over the next steps.

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And now this will be what we are going to calculate next.

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And to understand what we are going to do, let me open up this Wikipedia link in a new browser.

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And there I will explain to you how we update the value for white in the form of code for methods.

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So the trick of avoiding a quota method is that we do not simply go from the old y well you to the new

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Y value, but we have intermediate positions over which we will average in a certain way.

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So let me write down how I mean this.

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So first of all, we are going to calculate some intermediate positions.

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So these will be K1 and K2, OK, three and four in this case, because it is a fourth order when we

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put our methods and then we will calculate the value K, which will be some weighted average.

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In this case, it will be one over six times K one plus one over three times K two plus one over three

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times K three and then one over six times K4.

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And this K value will be the value that we will add to our value y.

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So basically, instead of this derivative four, we go straight from the old value to the new value.

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We go from the old value to the new one by adding the value K and K consists out of a weighted average

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of four different points that we are going to construct next.

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So where can you find these values?

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You can find them in any mathematical equation collection for numerical mathematics, for example,

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or also on Wikipedia.

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So here's the classic fourth.

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All the methods the of the hungry put our methods.

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And this is a bit of a cryptic way of writing it down in such a tableau, and the values for the key

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values are always here at the bottom.

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So this would mean one of a six times K1, one over three times K2 and so on.

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And Cassie, there exist many different methods and there exist many different values for these key

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values and also for the other values.

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So these are the ones that we are going to use next.

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I would say, let's first use this one.

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These are the arguments or the coefficients for the H value.

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So for the step with or yeah, step size, we called it here.

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So this will always be calculated in the following way.

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So step size times F and then T come our way, but we will now modify these arguments here.

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So we will change the time argument and we will also change to y arguments.

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So basically, we are taking positions that are in between the old and the new Y value.

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So let me copy this and then we are going to modify it in all four cases.

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So in the first case, or I would say, let's first consider the time argument which is described by

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these coefficients.

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And this means for the first time argument, we will add zero.

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Here we will add one half of h or sorry, one half times h.

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You're also one half times h and then one times h.

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So we right plus major page of a two plus page of a two plus page times one.

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So just h and then we also have to modify the Y arguments.

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And here we have to add a linear combination of the K one K to three K four values.

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And of course, for K one, we cannot add any other K value, but four K two, we can add some multiple

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of the K one four K three.

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We can add a weighted some of the K1 and K2, and for K four, we can add a weighted sum of K one,

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two and three.

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And this is described by these coefficients here.

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This means for the K2 value, we add one half of K one for K three, we add one half of K two and four

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K four.

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We add one times K three.

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And you see in general, this can be much more complicated.

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For example, here we will add for K for a weighted sum of K one, two and three weighted by these coefficients.

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And these coefficients are always calculated in the way so that certain arrows cancel out so that the

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solution will then be more accurate.

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So let's go ahead.

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Let's remember one half on half and one.

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So we have here plus one half times K1, so we can also write K one over two here.

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We had also one half point and a K two column K two of a two.

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And here we just had one in the K three column.

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So like this and this is actually it.

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So we have constructed four new points that are pretty similar to that from the syntax is pretty similar

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to the change for the Y value that we have before the Euler method.

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So it's h times some value of our function, which was the derivative.

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And the only thing that we have to change here is the argument, the time argument and the Y arguments,

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and we have constructed four different K points then have calculated the weighted average over these

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points.

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And then we have updated our value y with the new value of K for the value of each year.

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For the time update, we didn't change anything because we still go in steps of h.

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So of course, we have seen that in our example previously, there is some time adaptation.

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So when I scroll up to here, for example, we have seen that we do not take equidistant time steps.

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So here there's also another trick implemented that's intelligently choses to step with parameter so

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that the error is still not too large, but so that we need less points.

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But this is a whole nother thing, and this is even more difficult to implement.

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So I think we should not bother with it because in the end, we are often often interested in these

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equidistant lists of points.

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And here we are, even asked the solver to create such lists.

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So I think it's OK if we just take an equidistant method here.

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So we have now implemented the art form methods, and I would say we do not need to test it here right

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away.

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Let's just run it and see that there is no error.

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OK?

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This worked fine.

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And before we test it, I would say, let's go right to the next step and implement the arc four five

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methods, and then we will compare both of these methods with our own Euler methods.