Laplace’s Equation

All right. Hello everybody. Now what I want to do is I want to just go over these classical PDEs, as well as a boundary value problem, and then I will go in depth with each of these equations that you're seeing here on the screen. Okay. I'll first start with this row in-depth, same, same, right? But let's get an overall what is going on over here. So the very first one is called k is the thermal diffusivity, and it is larger than zero, and I have over here u is actual temperature in this particular case. Okay. So I'm looking at the temperature distribution in a rod. Okay. That's a typical example of this one, the heat equation, and it is actually, looking at it, the independent variables are x and t. So it's time dependent. Okay. Let's look at the second equation. The second equation is this. a is actually the wave speed, and this is, you can look, well, similar, right? But the right-hand side is second order with respect to time, right? This is like if I have a string vibration, you know, like guitar string. So I can obtain the vibration of that through this equation. Okay. It's still a function of time. Right. Just want to highlight. And let's look at the last one. The last one is 2D Laplace equation. This is a very common equation as well, and the application-wise, you can think of this as, let's say that I have like a rectangular plate over here, a thin plate, and I'm interested in the temperature distribution over here. However, there's no t here, time, so I'm only looking at steady-state temperature distribution for this particular case. Okay. So as I mentioned, this heat and wave equation are time dependent, but there's only one dimension. That's the advantage of it, right? I don't see any y here. And Laplace is time independent, which means it's steady, but I have two dimensions that I need to consider. Okay. Actually one thing, you know this is additional information, but Laplace's equation is extremely common. Actually, we have a mathematical operator for that. If you remember this, you know, we use this in fluid mechanics as well, in undergraduate too. You know, so I have this second del square, whatever the parameter is, and the last one is del square this divided by del z square. I can give an example from fluid mechanics, and this will be velocity potential. I mentioned that up there when I was talking about the subsonic, hypersonic, supersonic. So it's the same thing, the velocity potential. If you look at it, this is the conservation of mass expressed in terms of the velocity potential function. Okay. But also something interesting to note. I want to actually go up here if I can squeeze myself in there. Let's look at these three equations from a different angle. Do you remember that I called the parameter in front of this particular term as A in the previous segment? And in this particular case, A is k, B is zero, C is zero. So I'm trying to calculate this B square minus 4AC. If you don't know what I'm talking about, please visit the previous segment. I discussed this in more detail. Okay. But in this particular case, what happens is, well, B is zero. So that this term is gone. A times C, C is zero. So I get myself zero. Okay. So what I get is here is my heat equation turns out to be a parabolic. Okay. It's a parabolic. Let's look at the wave equation. This time, my A is equal to a square, my B is equal to still zero, and my C is equal to, I'm not going to say one, but minus one. The reason is the way that I write this was this is supposed to be on that side of the equation, and the right-hand side is equal to g of x comma y. Right? So from here you can see my B square minus 4AC happens to be positive, right? Because this a square is positive, the C is negative, negative, negative, positive. So I got myself a positive. So then this is referred to as hyperbolic. And last but not least, let's look at this. A is equal to one now, B is equal to zero, and C is equal to one. And if I do B square minus 4AC, A is positive, C is positive, so I got myself a negative value over here. So then this means that this is elliptical. So you can see that, you know, with these three in-depth studies that I will do, you will be able to approach a parabolic, a hyperbolic, as well as an elliptical equation. Okay. So one thing before I go with. I call this BVP, but I need to go a little bit in depth. So basically, this is the thing. Wherever I have some kind of time dependence, I need to have initial conditions. Okay. So let's write this down. I need to have some type of an initial condition, and this is only applicable for the first two equations, which are the 1-D heat equation, and 1-D wave equation. So 2-D Laplace equation doesn't have any time dependence in it, so I don't really need an initial condition. And typically, you don't have to, but it's much more convenient to give initial condition at t is equal to zero. Okay. But you don't have to. And I'll give you an example. Let's pick up the heat equation. I can say that, hey, u, which is the temperature, x comma t I is my solution. Okay. I can say that u x comma zero, you can see t is equal to zero, is, I don't know, some kind of distribution, temperature distribution, f of x. So I can use this and I can go ahead and find this solution. Once I find my solution, I should ensure that this satisfies this equation. Okay. Or I can give you another example from wave equation. Let's write wave equation. And in the wave equation, I can specify to you that my u, which is the displacement in this particular case, x comma zero, again t is equal to zero, is f of x. Very similar, right? So this time I can prescribe you the displacement, or I can actually go ahead and give you or prescribe you the velocity as well, because I know that is the del u del t, and I have t is equal to zero, and I will have, I don't know, g of x another function like that. Okay. So these are all possible initial conditions, so we will be exposed to that. Okay. And other thing is I also will have boundary conditions. Okay. Boundary conditions. So this prescribes as a function of location. Right, let's also pick up the wave equation while we are there. Actually, let me go ahead and draw this for you. So like this, and it's called plugged string. So I can just, it doesn't have to be right in the middle. Anywhere I can simply just hold this and pull this up, right? This was the equilibrium position and I just pulled it up here. Okay, but you can see what happens with the boundary conditions. Let's say that this is x is equal to zero, let's say this is x is equal to l. So I will have u zero comma t is equal to zero, u now l comma t is equal to zero. So you can see that I'm specifying this where it's not a function of time anymore, it now is a function of the location. It's also important to note that there are three types of boundary conditions that I can specify. These three are, let's write this. The first one is called the, simply I can go ahead and, let's say this is the wave equation, so I can give you the displacement. Okay. So this is called Dirichlet condition. Or I can have del u del n, where n is normal to the, we're talking about boundary, so it's going to be boundary. Okay. So this n is normal to the boundary. And this has a name as well. This is called the Neumann condition. Okay. And the third one, it's kind of like a combination of one plus two. Okay. So there's nothing new that we're introducing in here. So I can have del u del n plus some kind of a constant times u, and this is a constant. And this is called Robin condition. Okay. So why don't I apply this to a heat equation? Okay, let's be specific. Let's start with the first one. Well, I already gave this to you, but it's like this. So basically I have myself a rod, and in this rod what I'm saying is, this is x is equal to zero, this is x is equal to l. What I'm saying is initial condition at this end is equal to t zero. Okay. That is fairly reasonable, you know, in real-life applications we come across this often. The second one is more interesting. Del u del n at x is equal to l can be zero. Right. So I'm not going to go too much in depth because this typically is covered in the heat conduction, and we have a course on that, right? Advanced heat conduction. You know, you go much more deep than what I'm doing now. But basically, this means that we are looking at the heat flux at this end of this rod, and as the heat flux is equal to zero, that means that this is insulated. Okay. So I'm insulating this end of it. Right. And I can combine 1 and 2 and I'll get a something called this. Let me write it. So I can have del u del x at x is equal to l. At the same location I'll have myself minus h, h is the convection heat coefficient, u l t minus u of ambient, ambient temperature. Okay. So what this is, this means that I have a convection. So this is exposed to, let's say air over here, this x. And there's some convection happening at the end of the rod, and this heat is transferred to the air or water, and medium I'm keeping it at a constant ambient temperature that I have over here. You may ask, why do I have a negative? The assumption is that this is higher temperature than the ambient. It's just an assumption by convenience. Okay, so far I did the initial value. Now I did the boundary condition. So now I will look at the boundary value problem. Okay. So it will incorporate both of the initial condition and boundary conditions. Okay, as needed obviously. So let's write this is boundary value problem, BVP. Okay. As I mentioned, it can have initial conditions. It doesn't have to be, right? We discussed that. The Laplace equation doesn't need initial condition, or it will include the boundary condition. All three will have boundary condition. Okay, depending on the case. So I'll give you an example from the wave question. Then I'll also give you an example from the Laplace equation because I didn't really go there yet. If I rewrite the equation, del square u del x square will be equal to del square u del t square. Right. And I'm subjecting this to, let's give you, you know, again, an example. u, and I'll ask you what it means, is equal to zero. u l comma t is equal to zero, where t is larger than zero without saying. Right. So these are the boundary conditions. Right. I'm not specifying the time. Time is variable over here, but I'm prescribing the location. So that's why this is called the boundary conditions. All right. So these are boundary conditions. And I'm saying that just like up here, I don't want to redraw the same thing, but right here, it's the same thing. Right. I already written this. So this is fixed. This is fixed. All right. Okay, that’s the how about the initial conditions? Okay. I can give you initial conditions like this. How many initial conditions do I need to give? Two, right? Because it's a second order. So I'll give you, for instance, u x comma zero. You see what I did now? I fixed the time, and now I'll have f of x. And I need to give one more. Let's say that del u del x at t is equal to zero is equal to g of x. I also talked about this as well. Okay. So let me go up here instead of re-drawing it. But what I'm saying is, at the initial time, you know, it's called the plug string for a reason. I pull this. Basically, I pull this, and this is basically the f of x. Okay. This function I'm giving you. This is f of x, right? Because it's the displacement, basically. But also the g of x is the velocity. So at time is equal to zero, I'm pulling over here. But who knows, maybe I'm just pulling it this way up, or I'm pushing it down. When I'm just letting I’m not just simply letting it go. Okay. So that will be my another initial condition. All right, let's go ahead and analyze the Laplace equation. I didn't really talk about that. Del square u del x square plus del square u del y square is equal to zero. Like 2D Laplace, not 3D. So let's say that this is my boundary. So actually, why don't I go ahead and draw this? So I have this. Let's say that this point is 0, 0. This point will be in the x also, you know, I'm saying that this is x, this is y. Right. a comma 0. This will be a comma b, 0 comma b. So now I will solve this temperature distribution, this particular plate. But I don't see any time, right? Important. So everything that I'm going to give you, these will be the boundary conditions. I'll give you an example. Let's start with the x, and I'll go to the y. In the x I can do this: del u del x at x is equal to zero is zero. We discussed this in the heat. This means that this is insulated. Right. The heat flux is zero. Okay, how about, let's get this. Del u del x and x is equal to a, the other end of the spectrum. I can say this is also insulated. So I insulated these two ends for this particular example. Right. That's all I need for the x direction because you can see it's the second order. I have a second order, so I have to have two boundary conditions for y. I can prescribe them this way. u x comma zero. Did you see what I did? I now fixed the y. Right. I say that, hey, it's at y is equal to zero. So I'm talking about this particular surface. Okay. This is given as zero. So, okay, I know the temperature right over here. T is equal to zero. Temperature is equal to zero right there. And I'll give you another one. u x comma b. Now I'm looking at this top over here. This can be f of x, actually we're going to solve this. All right. So this will be f of x, as an example. So I have insulated here. I have insulated in here. I have T zero here. And I have f of x up there. So this kind of summarizes what I need to talk about. Now you know the classical PDEs. The first thing I'm going to do is I'm going to focus on the heat equation and go and solve it. All right. It'll be fun. You'll see. Thank you for watching this. Take care.