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Deep XD is a library that is designed for pinns computation.

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Of course, using a library is not as flexible as using PyTorch in order to achieve our like solving

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the pens.

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However.

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It is sometimes quite useful for, let's say, sample testing or just somebody who's trying to learn

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pens will just use it as a as a reference for our computation, or just to compare between our belt

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code and the library.

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So this library is built on.

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Top of TensorFlow and actually PyTorch.

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And this is its main backend and of course with, with other like other libraries that is back into

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this deep.

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So what we will learn today or this class is how to solve heat equation using deep.

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So the first thing is well just a quick thing is here of course again like it's a temporal difference

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temporal component and diffusion component.

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The spatial component of of you changing with x square and k is diffusion coefficient or kinematic diffusion

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coefficient.

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So this is a constant.

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And this is we're trying to solve you with an input of t and x.

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And we need to solve this equation.

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So.

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How to do it the same as we always do.

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So we do.

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We start by, we say deep XD.

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Um.

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Heat or even 1d.

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1D.

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Heat and main, let's say.

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Of course you can just name it anything you want.

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And we start, as always, by importing the needed libraries.

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So import of course deep x XD as d and from deep x d.

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But.

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Back end.

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Back end import.

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TensorFlow.

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And import numpy as NP as always and import matplotlib dot pi.

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Plot.

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For plotting.

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Plotting if needed.

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And of course, you have to enter.

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So this is basically the needed libraries for us.

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We take them.

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And now what we will do is we will have a first defining the constant.

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And this diffusion coefficient this k.

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So k equals 0.4.

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Length of the domain is one, length of time is one, and this is what we are going to use like soon.

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So geometry geom equals e dot.

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Of course this is we have to follow the rules and how we write the code for deep.

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So the geo.

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Mitre.

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Dot interval from zero to L.

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And time domain equals dee dee dee.

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Of course.

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This dot.

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Geometry.

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Dot.

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Time.

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Capital letter domain also capital letter domain.

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And it goes from zero to this in.

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And sorry.

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Not in.

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Jim.

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Time, which is geometry and time dot.

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In geome geometry.

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Basically dot.

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Another method.

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Geome Mitre.

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X.

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Time.

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And Jim and the time domain.

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So here we defined our geometry and we put a push shift enter and we are done.

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That's it.

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It's very quick very kind of shortcuts using this library however.

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We need to define the boundary condition and the initial condition.

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So.

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And defining the boundary condition or the initial condition will require us to have a lambda function.

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So well, for example, I would say I will define the initial condition.

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But and then I will explain how to deal with lambda function.

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And then I will define the boundary condition based on a function lambda function but also based on

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external function.

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So we can play with it whatever we want or however we want to define.

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So.

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I c equals d dot I c.

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Initial condition.

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And.

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And initial condition and boundary condition.

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Dot c means also initial condition and then we pass on the geometry.

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And we define the lambda function.

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And lambda.

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Lambda Lambda.

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And X.

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NP dot sign.

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In multiplied.

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In p dot.

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Pie.

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And of course, is the length of the time domain.

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Multiplied x.

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Everything to.

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021.

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And divide everything by L.

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Which is this one?

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And we have to put another function lambda as do nothing function.

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Uh, this way.

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And the last one on.

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A nickel.

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On initial.

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So this way what what do we have is a sine function in which we are going to apply this function.

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Basically it's x its only initial condition.

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And it will take x and multiply it well based on the end value and the value with a pi.

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And we will take the sine of the whole thing.

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And this is going to be our function.

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So basically what we have so far is we have the domain.

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It looks like it's a little bit small.

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We have the domain that maybe if there are bigger.

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Okay, this we have the domain that looks like this.

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This is x and this is T.

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And it goes like this.

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X is from.

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X is from zero to here one and t is from also zero.

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So here to one here this is t.

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Now the.

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The the initial condition it will go it's going to be a sine wave.

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So it will look like.

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It.

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This.

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And then we will define also a boundary condition that is going to be only one at the beginning.

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And at the end we will set it as.

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To.

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So basically this is our boundary condition.

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And what does it mean.

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It means that a.

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Well, it's basically the.

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We start the boundary condition from here, it will be zero and.

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But say this.

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This is the initial condition.

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And the boundary condition is going to be two here.

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So also at the end of this thing.

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The initial condition.

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The boundary condition will be zero and the here the boundary condition is going to be two.

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So this is going to all the boundary condition.

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And basically also here.

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So we have a line.

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The line in between we have to calculate.

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But the boundary condition between the the beginning and the end is basically two.

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So it's in the left side.

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This one is always going to be 000.

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And on the right side the boundary condition is going to be two, two, two.

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So this is what we will do.

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For the initial condition.

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We already defined this as a function.

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A sine function that will deal that will be related to n and l.

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Of course we can remove n.

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It doesn't really matter.

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We can do whatever we want and as we define here is initial condition.

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He will apply this equation only on the initial condition.

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So he will take the whole domain and only take the boundary condition or sorry, the initial condition,

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and then apply it to the equation.

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And then he will get his initial condition.

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Now what we need to do of course, to consider the lambda.

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It's not very common for us to use lambda function.

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So what we will do is or now just define using lambda function only the initial condition.

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But for the boundary condition I will define it using a lambda function, but also a normal function

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to basically do whatever we want, we change whatever we want.

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So here we shift enter and it's already entered.

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Now before that let's understand a little bit about lambda functions.

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So let's imagine we have an input.

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Just a little light.

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Take some spaces.

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So input.

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Array.

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Of NP dot array.

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And let's consider it as let's say it's like a grid so or a mesh.

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Here we have like kind of or it is 2D and.

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This one.

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Will be.

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This four, five six.

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And that we have this.

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The last one will be, let's say seven, eight, nine.

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Yeah.

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So we have now a.

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An array.

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And we will define a lambda function.

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So first lambda.

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Function equals.

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Lambda.

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And then we define what we will change.

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So x we multiply it by.

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X only taking.

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The first column.

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So we take everything.

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We take.

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Everything.

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This column and it will change from 0 to 1 means 0 to 1 is basically zero because.

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While we cannot take one.

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So in this case.

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Results of this lambda function.

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Result.

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Uh.

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Well.

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Array.

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Equals the lambda function.

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And when we pass the input array.

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And let's say, print.

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Results.

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So what's happening here?

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The input array.

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This part of the code we already know.

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This is what db db is going to provide our to our Lambda function.

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And it will pass it to basically our lambda function.

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So what will happen is it will take this input array and we'll consider it.

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This is going to be the x.

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So we said lambda x.

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So it's going to be x.

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And then what will happen is we will take this x and then do this operation which is multiply.

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First we we we extract the component we want which is the first column from 0 to 1 because it will change

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from 0 to 1.

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Let's say like the just the the first, basically the first component.

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And then what we will do is we multiply it by two and that's it.

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So the result will be two.

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And this four multiplied by two, it will become eight and seven multiplied by two it becomes 14.

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If we can do another things with lambda functions.

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So if we have a conditional lambda function.

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So conditional a conditional.

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Well.

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Lambda.

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Lambda.

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Equals.

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The same thing, same structure.

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So lambda x.

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Lambda x, and then we change the what we want if let's say.

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Uh, even, even.

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If.

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X.

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The remaining of dividing by two is zero.

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Else it's, let's say, an odd number.

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Odd number and print.

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Conditional lambda we pass for.

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And do the same with passing.

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Uh, three, let's say.

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So.

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He takes the four and then apply it.

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So this is his X, this is his.

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The value he cares about is x.

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And then he will pass it through the through the system through lambda function.

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And then if it's of course, if it's you know, the remainder is zero, it will just do whatever the

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if condition is there and then it will just give us what, what it whatever we want.

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So this is what we can say like a conditional lambda function.

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Now what we want to do is we want to use the lambda function to actually pass on.

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Well, basically.

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Boundary condition.

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It through a function.

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So in order to do that.

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We first define the function define function.

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Let's say it is double.

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We said about the from 0 to 2 the first.

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Column.

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Input.

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Ray.

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Of course, this lambda function is quite not usual for engineers and maybe scientists to use, but

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this is how this is what the people who wrote the well, basically the code the deep want us to do.

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So we have to to do it.

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So here, for example, is.

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The input array.

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Input equals to input array from 0 to 1.

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So we we write the same function here and.

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Well, basically boundary condition equals.

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The will use the same like just will change some what what is needed.

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So DDE means dxt.

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And here we have Di.

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Ric lit.

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Boundary condition.

271
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It's also syntax geometry.

272
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Lambda function has to be.

273
00:18:21,560 --> 00:18:24,170
Well, I'll try to do it this way.

274
00:18:25,090 --> 00:18:26,560
It's a little bit cleaner.

275
00:18:30,450 --> 00:18:38,310
So the lambda function we need to have input array rather than x input array.

276
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And.

277
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Basically we pass on the function of input array, and here on we have to change it also syntax bound

278
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dairy.

279
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And here also on boundary.

280
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Yeah.

281
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This way.

282
00:19:01,710 --> 00:19:10,470
So now we define the initial condition and the boundary condition using of course lambda function.

283
00:19:12,090 --> 00:19:22,950
One thing to note regarding how I treated this regarding 0 to 1 is we need to to understand it fully.

284
00:19:22,950 --> 00:19:27,540
We need to print what the things we are writing.

285
00:19:27,540 --> 00:19:33,800
So the first thing is this is the input array that is going to be from the boundary.

286
00:19:33,810 --> 00:19:35,760
The first thing is input array.

287
00:19:35,760 --> 00:19:42,480
Let's see what kind of input array they are providing to this function.

288
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So the input array we put it in here which is going to be of course our boundary condition.

289
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And next thing is the input array.

290
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We need to see the shape of it.

291
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In case if it's big.

292
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Of course it's not so big.

293
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It's it's 80.

294
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Multiply two.

295
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We will say that shape and.

296
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What we will do is also.

297
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We put the input array in this way.

298
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The where we are passing it to our system.

299
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And here is the same.

300
00:20:23,280 --> 00:20:30,990
And the last thing, what if we didn't do that and we just simply providing this kind of input array?

301
00:20:34,740 --> 00:20:35,940
Um, yeah.

302
00:20:35,940 --> 00:20:40,680
We put here zero and here zero.

303
00:20:41,140 --> 00:20:48,490
So this is going to be printed as we run the our computation and.

304
00:20:50,220 --> 00:21:02,880
Yeah, let's let's just keep it this way until we have our code ready and we can see how these are changing.

305
00:21:02,880 --> 00:21:12,300
So basically what we need to understand or I can provide let's say the Rand software.

306
00:21:13,830 --> 00:21:17,400
Is this is going to be the result.

307
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Of course.

308
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And this is input array.

309
00:21:33,110 --> 00:21:35,420
So this is how it will look.

310
00:21:35,450 --> 00:21:38,400
The input array of the the input array.

311
00:21:38,420 --> 00:21:43,110
The our input array is going to look like a like this.

312
00:21:43,130 --> 00:21:47,720
Basically you have a the value and you have the time.

313
00:21:47,720 --> 00:21:51,200
So this is this is all a boundary values.

314
00:21:51,230 --> 00:21:54,440
Of course this is we will see it when the when the program runs.

315
00:21:54,440 --> 00:21:55,670
We just will not it.

316
00:21:55,670 --> 00:22:01,760
But it's very important to explain it here in the in the section we are talking about the boundary condition

317
00:22:01,760 --> 00:22:02,990
and of course the initial condition.

318
00:22:03,170 --> 00:22:07,220
So we have the input array which is going to be all like this.

319
00:22:07,940 --> 00:22:13,790
And then the input shape is 80 and to why it's 80.

320
00:22:14,150 --> 00:22:22,820
Well, because again this is later we will decide how many number of boundary points we will put.

321
00:22:22,850 --> 00:22:25,250
Of course these are also random points.

322
00:22:25,250 --> 00:22:28,580
We are also going to see that now input.

323
00:22:29,000 --> 00:22:37,940
When we when we put this thing like this we will have the input array to be looks like this.

324
00:22:37,940 --> 00:22:41,450
Basically you have a list.

325
00:22:41,450 --> 00:22:46,040
And inside the list we have another list and all the values that we need to change.

326
00:22:46,040 --> 00:22:50,090
Basically apply the to multiply them by two.

327
00:22:51,360 --> 00:23:00,960
And this is going to be the dimension is 8180 columns and 180 rows and one column, as you can see here.

328
00:23:00,960 --> 00:23:07,980
However, if we put this way, if we put input array this one and zero, what we will get is we will

329
00:23:07,980 --> 00:23:11,280
get a list that looks like this, which is not what we want.

330
00:23:11,910 --> 00:23:15,510
This is not physics, this is simply programming.

331
00:23:15,510 --> 00:23:24,540
This is how we have to write it in order to get the correct shape of the, of the of the input value.

332
00:23:24,540 --> 00:23:32,400
So this is a well in detail why we put here from 0 to 1.

333
00:23:33,410 --> 00:23:34,790
One thing to note.

334
00:23:35,610 --> 00:23:36,750
Is there.

335
00:23:37,230 --> 00:23:43,770
When we use this definition of boundary condition, we will need three needed information.

336
00:23:43,950 --> 00:23:45,800
The first one is x.

337
00:23:46,610 --> 00:23:48,080
The location in space.

338
00:23:48,260 --> 00:23:50,520
The second one is time.

339
00:23:50,840 --> 00:23:58,550
The boundary condition will end at a specific time, and the third one is the value at this boundary

340
00:23:58,550 --> 00:23:59,240
condition.

341
00:23:59,240 --> 00:24:03,410
So how did we well define them?

342
00:24:03,950 --> 00:24:10,520
Well, this lambda function that we talked maybe a lot about how it is structured.

343
00:24:10,580 --> 00:24:15,860
We can see the input array which is will give us two informations.

344
00:24:15,860 --> 00:24:19,130
That is x and time the x.

345
00:24:19,130 --> 00:24:22,070
We can see it here 0 or 1.

346
00:24:22,070 --> 00:24:31,190
And then here 000111 at different time location that we will basically randomly generate it or going

347
00:24:31,190 --> 00:24:31,820
to generate it.

348
00:24:32,910 --> 00:24:35,340
So these are the two informations.

349
00:24:35,340 --> 00:24:40,740
And they are going to input in this lambda function in this input array.

350
00:24:41,550 --> 00:24:43,860
Now we have the two information.

351
00:24:43,860 --> 00:24:45,330
How about the third information.

352
00:24:45,330 --> 00:24:46,770
And the third information.

353
00:24:46,770 --> 00:24:49,230
We get it by the double first column.

354
00:24:49,230 --> 00:24:51,660
This basically the.

355
00:24:52,400 --> 00:24:58,430
Function and how this information is, is going to be used.

356
00:24:58,430 --> 00:25:05,900
Well we will this function will give us the third information, but well how we put the input array

357
00:25:05,900 --> 00:25:07,250
in that function.

358
00:25:07,250 --> 00:25:10,880
And we use only the space location.

359
00:25:11,570 --> 00:25:16,520
As a to describe the output function which is.

360
00:25:16,520 --> 00:25:22,610
In this function, the space multiplied by two will equal the boundary condition.

361
00:25:22,730 --> 00:25:25,850
Let's see in the this graph.

362
00:25:26,690 --> 00:25:27,010
That.

363
00:25:27,050 --> 00:25:28,610
So our solution will be like this.

364
00:25:28,610 --> 00:25:36,740
Basically, we will have a boundary condition and in the one direction is going to be, well zero and

365
00:25:36,740 --> 00:25:38,810
the other direction which is here.

366
00:25:39,720 --> 00:25:46,800
And at x equals one we will have the boundary condition of well two x basically.

367
00:25:46,800 --> 00:25:53,340
And with time this initial condition will start to diffuse into its final solution.

368
00:25:53,430 --> 00:25:55,200
So x and time.

369
00:25:56,010 --> 00:26:04,080
Is related with X and time, we have a boundary condition value that is related with this equation.

370
00:26:04,110 --> 00:26:07,620
Boundary condition value is two multiplied x.

371
00:26:07,620 --> 00:26:13,740
If the x equals zero, the boundary information or the boundary condition will be zero.

372
00:26:13,740 --> 00:26:21,810
Two multiplied zero is zero and at x equals one it will be two multiplied x, which is one and equals

373
00:26:21,810 --> 00:26:22,410
to two.

374
00:26:22,440 --> 00:26:25,500
This is why we can see this value is two.

375
00:26:25,680 --> 00:26:26,610
So.

376
00:26:27,440 --> 00:26:28,550
At the initial condition.

377
00:26:28,550 --> 00:26:37,160
This one we will start this initial condition diffuse through this well basically the final solution

378
00:26:37,160 --> 00:26:38,420
of the heat equation.

379
00:26:38,570 --> 00:26:46,490
So the well just a minute here we can see that output is going to.

380
00:26:46,520 --> 00:26:48,740
We will take x location.

381
00:26:48,740 --> 00:26:52,850
This one only zero one which is only this value.

382
00:26:52,850 --> 00:26:59,780
And we will multiply it by two which is two x will equal the boundary condition.

383
00:26:59,780 --> 00:27:09,740
And we will add this as one one information which is we can see the values will be this way.

384
00:27:09,920 --> 00:27:13,280
So these are the will the input array.

385
00:27:13,280 --> 00:27:18,620
But if we multiply it by this is the input between 0 and 1.

386
00:27:18,620 --> 00:27:21,740
But we set to multiply this input array which is.

387
00:27:21,740 --> 00:27:22,850
This will be two.

388
00:27:22,880 --> 00:27:23,840
This will be zero.

389
00:27:23,870 --> 00:27:24,740
This will be two.

390
00:27:24,740 --> 00:27:29,480
Because the location every location will be multiplied by two.

391
00:27:29,480 --> 00:27:40,610
So this is the other input x and t output is an array that will take this input function and multiply

392
00:27:40,610 --> 00:27:41,930
it by two.