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All right, so let us now come to another example, and this is called the heat equation and the heat

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equation describes how heat or how temperature changes in a spatial region.

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For example, we could use a single baard and it would be a one dimensional system as here where we

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make it hot on the left hand side and called on the right hand side.

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And then we see how over time the temperature changes inside the bar.

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So this is how you write it down in two, three or more dimensions.

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What you have here in the left hand side of time derivative of the you, which is in the array that

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has different values for different times.

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So you see here we have a dependence on our.

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So basically, we have here such an array.

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You want you two until you end and then we have also a time dependence.

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So for every time step, these values here will change and devalue.

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You want to you and will describe the temperature at this particular point in space.

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And so we have you on the left hand side the time derivative.

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So this is how it changes over time.

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On the right hand side, we have a constant and then the the plus operator.

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Which is basically the second order derivative in space.

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And we add of the different components.

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So as I wrote down here in one dimension, this is just eight times to second the derivative with respect

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to X of you x A..

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So this is our differential equation.

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And so maybe you realize this now that this is fundamentally a very different a system that we are solving

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here.

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So before we had the harmonic oscillator and we have the loren's system, which in both cases was just

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a single particle, it was in one dimension and two dimensional in three dimensions where we had X,

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Y and Z coordinates, and we only had a single set of these coordinates.

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So we had just had one particle that we had to describe.

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And here it's different.

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We have a sample in space, for example, in one dimension, but could also be in two or more dimensions.

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And this is disc criticised by individual values.

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Each of these values is just a temperature, so it's just a scalar.

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But in terms of the mathematics, this is you could see an end dimensional system.

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So since we have full values, you want to you and we will have and different coupled differential equations.

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So from the from the methodology, it's pretty similar to the Lawrence system where we had three coupled

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equations.

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But now we have and coupled equations and the end can be pretty large.

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It can be several thousands of equations that are coupled.

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And here we will see how we can solve it.

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So first of all, to show you why this is such a system, it is described by so many differential equations.

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We must, of course, this ties the right hand side.

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So this is the left hand side.

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It's exactly in the shape that we want to have.

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It is basically you dot.

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And so we must describe ties only the right insight.

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And so we must just criticize the second derivative.

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So we have learned that there are different methods to do this, and you can increase, of course,

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the accuracy by using certain methods for the derivative.

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But you will use just a very simple approach.

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We will just disparities by using the central differences methods.

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So basically, the value of the second derivative of you at the position J or with the Index J is given

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by the some of the neighbors minus two times devalue at the actual position.

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And then we divide by Delta X, which is the spatial distance between two neighboring cells here and

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we square it.

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And this is pretty straightforward.

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We can do this for every value of J, except for the element that is at the end.

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So basically, you want and you n or when you use Python indices, you zero and you and minus one.

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So we will see that we can even leave these two values out for the derivative.

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But we could also just say, OK, in this case, we don't use the central methods we use to double the

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forward or to double back what methods.

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And so then this is what we get.

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This is, of course, not such a good approximation.

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But yeah, there's nothing else we can do because these are at the edges and there are missing neighbors,

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so we have to determine somehow the second or the spatial derivative.

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So this means we can now write down this equation here for every individual element of this array with

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the disparities.

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Second Order derivative.

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So we're right, for example, for the derivative of UJA is equal to eight times and then this democratize

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derivative.

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So eight times u j minus one minus two times U J plus a huge plus one.

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If we divide by Delta X squared and we do this for every single cell and for every index, so we get

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such a very large set of differential equations.

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And just as I mentioned, we don't really have to consider the left and the right differential equation

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in a different way, we can just say here the temperature is supposed to be constant.

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And this would correspond to a so-called heat bath in in physics.

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So sometimes you consider, for example, an infinitely large water pool with the constant temperature

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and the temperature is not effected by the object that we are actually simulating.

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And so you so you see the left and the right and correspond to such a heat bath and the temperature

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is then constant.

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This makes it even more simple.

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So then you could just neglect these two equations here and say the temperature change is just zero.

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All right.

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And so now we begin and we implement these equations and solve them once again with our integrate module

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from Spain.

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So let's run the cell and then let's go ahead and implement everything.