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One day he had a question is a good start for us to solve it numerically.

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And later what we will do is we're going to solve it using, of course, neural networks.

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So what is what does what does it mean?

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That heat equation.

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So the heat equation is a partial differential equation that describes the or how the temperature of

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a physical system changes over time.

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And we can see how it's written.

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It's quite short, like d, u, d, t equals K, and the second derivative of u over d x.

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So the change of temperature or u, whatever the value we are talking about with time.

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Will depend on the second derivative of temperature with space.

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This is how we solve it.

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And the K is a diffusion factor or a constant that.

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Govern how?

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How how fast the temperature with time will change based on the spatial coordinate.

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So this one we call it spatial parameter or spatial part of the equation.

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As it changes the the temperature will change with space.

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With X, of course, this is 1D and we have X, but if it's 2D, you will have two dimensions.

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That is, you will have the change of temperature with x and temperature with Y.

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Now the other parameter or component of the equation is the temporal component of the equation, and

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temporal is just time based change of temperature with time.

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This is what we mean about temporal.

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A part of the equation.

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And the problem, what we are going to solve is we will have a rod.

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This rod will be frozen.

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Well, it's just zero degrees.

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It will have zero degrees.

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Was not, say, frozen because it will be made from aluminum.

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And then what we will have is two heat sources.

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Now, as the time goes, we will have these two heat sources.

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And as the time goes, this rod will change its temperature from zero to a higher value.

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So the temperature will start to rise in the tips and of course, also in the middle of the rod.

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Little by little, until, of course, it will reach to depending on the heat sources temperature.

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So.

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How we are going to solve it.

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First, in order to solve it, we we will need, of course, to to discretize or basically simplify

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how this equation is written.

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At the end, we are solving this equation numerically.

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And this is how we need to take into consideration the change of like change of temperature with space

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and change of temperature with time.

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And of course, we need to care about.

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So let's take or let's start from the easy part, which is the Kay.

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And the Kay.

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Here is the in regards to heat equation for an actual heat problem, which is we're trying to solve

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how the temperature will change with time.

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And space.

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Of course, we we will need a heat capacity.

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Property and heat capacity is a property of a substance that that quantifies the ability to store thermal

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energy.

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And we can see different values of like depending on the on the on the metal or on the material itself.

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So we can see here 0.45 for iron.

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We can see here aluminum, which we will use today like we will use for aluminum or in our numerical

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solution.

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And of course we have different quantities here.

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We have joule for gram and Celsius and here we have joule for mole Celsius.

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So we need to also be careful about this.

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So.

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Okay, now we're.

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We're ready.

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So what we should do, let's start solving it.

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So here we can see first the in order to care about or in order to solve the issue of a change of temperature

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with X and change of temperature with T.

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What we need to solve is we need to do a numerical derivation.

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A numerical derivation is basically we.

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Or the first derivative we need to.

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Calculate that it change the location where where the curve will will change basically the slope at

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that location.

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And in order to do that, of course, we have many formulas.

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For example, a central derivative method, we will use this one and H is the difference between this

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point and this point, which we will calculate around it, which is in this case, for example, we

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try to derivate this point.

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We have to calculate the function value at X plus h and minus the x minus H, this is X.

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So here is the lower value, here is the higher value.

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So we need to minus both values and then divide by the distance.

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This is how we will calculate or this is the general idea of a numerical derivation is very simple and

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of course we don't really care about the to describe the actual curve equation.

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So for example.

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If you have a curve that is X squared.

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So we will have it looks like this.

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And when we do our first derivative, we will have two X which will look like this.

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And if we have the second derivative, so deriving the already derived curve, we will have the slope

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of the whole curve will be, of course, for every point it will be two.

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And if we continue and two the third derivative, we will reach to zero.

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So this is how this like numerical derivation works.

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And actually there is many ways or sorry, there are many ways to calculate the numerical derivations

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in things we call usually in numerical computation, we call it schemes.

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So we have, for example, first order derivative schemes, we have second order derivative schemes

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and it is it will provide us with.

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A simple formula that we will use actually the numerical value of every point to calculate the change

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at this point.

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So in here we have a forward difference scheme and there is of course the backward difference scheme.

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So it will take, it will it depends on the, the point of interest and you consider that point in front

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of it which is called the forward difference and backward difference is the point behind it.

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And central difference is we will consider both points.

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And of course, for second derivatives, usually you need or not, usually you need more points.

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In this case, you will have, for example, three point central difference for second derivative.

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And we can see here, of course, this is we're not talking about deriving a curve and then derive it

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again.

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So it's not applying first derivative two times, which will work.

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But in what we do is we, we, we, we don't, we're not really interested in the first derivative curve

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and then we derive it again.

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But we will do is we will take the equation and we discretize it means we make it written in a, in

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a simple um, uh, in a simple values and or numerically possible to calculate values.

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That is, we will take this curve and then what we will describe it in, in, in terms of the function

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values of every point.

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So this is what we will do here.

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And by this we will be able to calculate the second derivative directly, the value of the second derivative

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from the function value itself.

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And I'm not talking about the first derivative, so it is like applying it first derivative, and then

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again another third, another derivatives to get the second derivative.

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But rather than going for this two step thing, we can go from one step which is using these three points.

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Or in this case also this scheme, what we call the scheme is the way we calculate this derivative.

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We will need to calculate this point.

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So, so basically this is how we can actually, uh, well, uh, translate a heat equation or any equation

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from a formula that looks like this to an actual values that can be calculated at every point.

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So.

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Now we understand how we can get care, which is a constant.

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So depending on the material, we simply get it and we know how to change the first derivative into,

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for example, a scheme.

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Of course we can pick any scheme we want, although a side note is of course to choose.

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The more accurate the derivative, the more accurate the scheme, and this is the inherent numerical

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property of the scheme.

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So you have the some schemes will have a first order accurate what we call and the second order accurate.

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And it is just the how numerically accurate the scheme itself.

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So in my experience most of the problems we don't really need second order or most of the time first

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order schemes will work.

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But the idea is some schemes will provide us inherently numerically feature that it will actually be

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more accurate.

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So it matters the choose the choosing of scheme.

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But however we will choose a very simple schemes in this problem to solve this problem.

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So for for our problem, what we will choose is the forward looking derivative for for space and time

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difference.

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And this this scheme.

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And for the second derivative we will choose the this three point scheme.

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And when we apply this to these two schemes on the equation, what we have it, we will end up with

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something that looks like this.

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So you at the time we were calculating plus the x minus 2UTX plus u x minus d x.

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So this is the three points we're calculating, using three points.

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Basically this scheme and the X square is this one.

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Of course, this is dy dx means the difference x the difference of x because this is there is no D here,

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so it's not like derivative.

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So just be careful.

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It's a little bit sometimes get confusing.

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So here for the first derivative is quite also simple.

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You take u t plus x minus u t x over t dt.

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This is how we discretize the heat equation from its original form, which is this one into a simple

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way, which is our simple form.

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Which is this one.

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So the this is, of course, the the theoretical way to describe it.

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And what we will do is we are going to actually apply it in a python code in in the next lessons.