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So we start our numerical solution for the heat equation by actually this.

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Well, Jupyter Notebook.

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Will I like Jupyter Notebook?

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Of course, because it's easy to see how the numbers are changing.

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And so we start by just writing the name of the of the file.

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And in our case, we can call it, well, heat.

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It's Maine.

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And now we have the file and we can press save just to make sure that the file is actually saved.

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So starting with, well, our solution, it's all usual libraries because, uh, because we will write

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everything by most of the things by ourselves.

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So we first need to import.

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We make it a little bit like zoom in, import numpy.

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And from.

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Of course, Matlab, the numpy is for math and from matplotlib or matrices to be more accurate.

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Regarding the numpy.

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NumPy arrays by plot.

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To plot things.

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And of course shift and enter.

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So this means it got imported into the our system.

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Now, the first thing is we need to set some variables that in the well, in the like describing our

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system kind of or it is like what we call post-processing.

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So the first thing is we need the length of the rod, let's say, to be ten.

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And the reason I put a ten because I tested one meter, it's just it will diffuse too fast.

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So we'll just put it ten and K equals 0.89 and.

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Temp.

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Or temperature left equals.

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We can consider 100 degrees and temperature.

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At.

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Right?

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Right.

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Will be, let's say 200.

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So you will have in balance one is or in the left, you have 100, and on the right you have 200.

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So.

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The total simulation time.

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Total.

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All time.

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Our simulation time well equals ten and shifted.

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So now we decide or we considered a rod one temperature, one part or one end of it will have 100 degrees,

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the other end will have 200 degrees.

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And um, we will later define the initial condition, which is the rod itself will have a temperature

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of zero.

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So what does it mean?

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This is what we call boundary condition.

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The boundary condition is.

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At the right, it will be 200 degrees and the left will be 100 degrees.

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So the X equals 0.1.

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And of course, this is the value for aluminum and you can, of course, change it as you like.

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I think it's a little bit high.

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And yeah, just just put it wherever like for this thing you can put it any value one.

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But of course we if you're simulating an actual material now took the number is just for aluminum but

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if you simulate any an actual material you have to check the actual the material value itself.

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So we consider that we have a ten meter rod or like, Yeah, a rod and we have 0.1, which is the so

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you will have a oh point like ten centimeter basically.

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Yeah.

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Ten centimeter for every A this is the mesh you will calculate.

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So.

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What we will have is we will have the actual.

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Yeah, this one.

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We will have the actual road goes like this and here we will have 200 degrees.

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And here you will have um, just let me look at the left.

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100.

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Yeah, it's correct.

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100 is left and.

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And this is basically our road.

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And then what we will do is here we have a discretized.

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So every we will cut this road into small, small, small parts.

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And this part will be a 0.1 meter, of course.

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And of course the road is ten meter.

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Of course, this is just dummy value, just whatever it is.

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But and if you want to really, of course, solve this problem, you have to be careful about the physics.

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The physics values.

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So here the X equals 0.1.

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Now.

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What we need is we need the vector.

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Vector.

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Equals the x vector.

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So we will have this a road that has to be cut it for a very small portion of 0.1 meter, which is ten

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centimeter.

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So we need to cut it.

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So we say numpy, we will make a basically a list of values or, or more like it's a numpy array of

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these values that will increase little by little.

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So we have a numpy line.

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Space.

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It will start from zero until the length.

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The length of the rod with an interval of.

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A length of the rod.

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Divided by.

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The.

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The x.

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The x is the the step or how this small change of x that this rod will be cut it into it.

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And then we shift enter and we have error.

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What is.

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A line.

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Line.

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Linspace is.

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Oh, sorry.

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This has to go.

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Just this one.

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It's just a syntax error.

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And now we can see this vector, which is you can see it's start from oh point from zero and it will

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change to 0.1, 0.20.3.

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And of course it's just some numerical accuracy.

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It's quite high here, but it's not a problem.

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It's a simple problem.

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So it's maybe not so we don't have to care much about like if it's just too much accuracy regarding

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the X.

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So here we have it Now this is the the road.

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And what we will do now is we will do the same thing, but this time for time.

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So we will have TX equals some change in time, which is make it a bit small just to, to be, to,

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to like be careful about the convergence.

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So we just take the same thing here.

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And here it will be time.

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This will not be time.

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It will be the total simulation time.

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Over the.

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And this is going to be also the total simulation time and.

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We shifted.

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And now we see the vector of time, which is quite long.

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This is just means like zero, like points, points, points and till the end of the total simulation

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time.

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So it will keep changing, like it will keep increasing little by little.

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So.

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Okay, now we have.

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The length of the rod got well made.

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Small parts like kind of become numerical kind of mesh.

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And time is also being now like kind of well described.

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So what we need also what in the equation is of course, in the heat equation, we have to solve the

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we have the time, we have X and now you now you will be dependent on time and space, so T and X,

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So now we have to do it this way.

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So basically u equals.

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And this is also like if we consider this one.

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I will consider boundary condition.

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Just this one.

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And here we can consider.

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Or not in concert is actually the initial condition, which is you.

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We need to initialize the values of you numpy dot zeros.

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And.

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We give it dimension.

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The dimension is the length.

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And well the length of the space and linked with time.

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So we take the length of time, vector and length of.

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Space victim.

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And we put.

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Shift.

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So we see how our solution domain looks.

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And we have this is how many time steps and this is how many space steps.

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So we have 100 space steps and we have quite a lot like 100,000 time steps.

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Of course, this is a toy problem, so we don't really care about the values, but we need to really

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solve what we care about the numerical way of solving this problem.

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So anyway, you have this you if you want to look how you looks and we want to see how you looks, you

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can see that it's all zeros.

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We have a thousand, 100,000 sorry rows and we have 100 columns and it's all filled with zero.

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Now, the problem we have is, well, basically.

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A the rod will have a temperature.

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The road will have a temperature of.

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Just be like this.

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Start from.

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200.

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And then all of this.

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We'll go to 100.

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So the temperature will be a little bit high here and then we'll be 100 here.

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And in between, the temperature will be zero.

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Maybe it's better to use freezing.

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Yeah.

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So this is the initial condition now.

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What we didn't do so far is we didn't really well.

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Uh, sorry.

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This one?

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Yeah.

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We didn't.

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Now everything is zero.

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The whole road is zero.

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All the time.

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So we start from zero and everything.

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Now the values of the whole solution is zero.

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But now let's impose the boundary condition.

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Of course, if you want to change any value, for example, the first value zero zero, you want it

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to be 100.

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You can put it like this.

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And when you read the you now the become, we change this value to be 100.

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So basically this is the row and this is the column and this is how we refer to this array.

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NumPy array.

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Now to apply The boundary condition that we want is we want u.

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And.

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All the time have to be.

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And from this side, from the left side, it has to be the temperature of the left side, which is 100.

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So we will equal it.

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We will equal it twice.

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Not moving.

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Yep.

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Equal it with the left temperature.

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And you.

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Also all the time with.

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The last, not the last one.

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But the way Python works is either we tell them the length of the vector, which is a little bit dangerous.

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We we just put minus one which is related to the final point.

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So we put it minus equals and then we consider.

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The right temperature.

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We go all the way.

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Here and we paste it and then shift enter.

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And now let's see how it is.

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How does it look like?

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So we can see now it starts from the left, which is 100 degrees, and the right, which is will become

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a 200 degrees.

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So the last thing we need to do is we need to see how our solution will or how our actual problem description

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looks like.

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00:14:33,050 --> 00:14:37,520
And here we will do a pie plot by.

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Plot dot plot and we will take.

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Not the T vector.

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The x vector.

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And put it here.

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We need to plot it with.

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0Y0 because we refer to the first time.

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This is, of course, a two dimensional array.

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00:15:07,760 --> 00:15:20,030
And here we have the X and of course we have also need the A, the value of U and by plot.

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Dot y label.

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We can say them richer.

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And.

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Biplot.

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Dot x.

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Label.

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And we can say l x.

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And be a Pi plot.

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Just show the the thing.

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The show and we can have it and shift enter.

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By plot.

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Plot?

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Sorry.

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Yeah.

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This is where.

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00:16:00,960 --> 00:16:10,110
So what do we have here is at the rod end of ten meter, we will have the temperature of 200 and the

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at zero, which is the rod end of the the left side of the rod, we will have the temperature of.

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A 100.

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The rest will be 100.

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So this way we define our solution space, which is the U.

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We define the boundary condition.

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We define the initial condition, and we are ready to actually solve it.

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And of course, later we just see how the solution will will be.