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So before we get started with solving our first example of a differential equation, let me tell you

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a bit about the background.

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So the first method that we will discuss here and that we will program ourself is called the Arla Method.

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And it's basically the most simple approach on numerically solving a first order differential equation.

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So what's the first order differential equation?

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Well, we have a function Y, for example, which depends on T, and the derivative of this function

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is equal to another function, which has terms depending on T and terms, depending on the function

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itself.

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So basically, we have T, we have Y, and we have a first derivative of Y and no ion derivatives.

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This characterizes the first or that differential equation.

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And what's the idea behind it to solve it numerically?

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Well, we can just think about how would we express the derivative of Y with respect to T.

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So of course, there are different possibilities that we can write down now.

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But one of them is that we take the value of Y at the position T and add a small value h, then subtract

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the value at the position T and divide by H.

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So this is if if H is small, if so, basically here we would have to write the limit of h goes to zero,

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then this would also be equal to y of T minus y of T minus H or also y of T plus minus y of T minus

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h divided by two h and the limit of h to zero.

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This all gives the same value.

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So here this is only approximately true.

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However, if we say OK, we consider that h a small and we say this is true, then we can solve this

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equation for this term here y at the time, T plus h.

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So we write times h and then plus y of T.

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So we get this expression here, which means when we know the value of Y at the time T, we also know

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the value of Y at the time T plus h because we can just calculate the derivative at the time T and then

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multiply with h and added to the initial value.

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And this is the whole idea.

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We will repeat this step here over and over again to propagate in time zero two, we can repetitively

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iterate this propagation process.

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So if we want to write this down a bit more in terms of programming, then we should write this down

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with indices and not with time arguments here.

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So we know that at the end plus one time step, which basically corresponds to the value Y and plus

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one and at time T plus h, we can express this value as the value before, which is why n at this step

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and corresponding to a time T and then plus the derivative, which is actually the function here.

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So you see from this expression and times h.

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So this is the whole idea.

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We will repeat this step over and over again.

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So maybe you see, now that this is not so difficult, it's difficult at all because this is just the

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function that we have provided that we will program.

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And then we just have to iterate this process where we will add to the initial value, the value of

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the function times, some small step size.

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And then we repeat this and we will look at then y and plus one y and plus two and so on.

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So in the next lecture, we are going to solve a first example, which will be the radioactive decay.

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And you see, this is the differential equation and we will use this method and programme it ourselves.