Okay, let me first explain you the heat equation. One-dimensional. So what is the one-dimensional heat equation?
It's like this. You have a rod. You have a thin rod.
And we have to assume that the rod doesn't lose any heat in this direction. Okay, so we will say that it's embedded in some insulation. And the only place that it may acquire heat or lose heat is through these endpoints. And to illustrate my example, I'll just say that in both ends, you put ice.
So let's say initially you have some rod and everything is in 20 degrees Celsius. And then suddenly you make it touch ice on both ends. And ice is at 0 degrees Celsius, right?
So the ice is melting at the endpoints, keeping the endpoints at 0 degrees. And you want to know the progression of the heat distribution or the temperature distribution at each time t. That's what we are trying to figure out.
So I'm going to say u of x. of T is the following. Temperature at location X and at time T.
And we can kind of have some idea of what ux looks like. So ux of 0, for example, in the beginning, so let's say 0 is the left end and L, if L is the length of the rod, so 0 to L, and let's say the temperature is set at 20 degrees. So initially it will be like this.
So because it's everything is at 20 degrees, everything is 20 degrees, and you have this graph, okay? At time t equal to zero, this is the temperature distribution, okay? Now what's going to happen after a short moment? You're going to see some dip in the temperature, right?
Where? Where would you see dip in the temperature? Huh?
At the end points, right here and there, right? So after some time, I don't know, some u of x, let's just say a, I don't know what time. After some time, the graph will be like, like that, right?
And I guess overall it must have decreased. So in the center maybe I became the 20 degree it may came down to 10 degrees but at the end points it became much lower because it's closer to the ice. So that's that's the kind of picture that we're thinking of.
That's that's our UXT. Now In order to figure out uxt exactly, in order to figure out exactly what uxt is, we need to come up with an equation that governs all this system, okay? And the equation for this comes from the Newton's cooling law, which says that change Change in temperature is same as ambient temperature minus the temperature of that thing. So that's basically the Newton's cooling law. And there are nice...
nice work done in the textbooks. You can just search on how this heat equation is derived from the Newton's cooling law. However, we're not going to We're not going to use this. I'll just try to argue by intuition and this argument is probably easier to understand than actually deriving the heat equation from the Newton's polygon.
So here's what I want you to think about. So if you have a function and the second derivative is positive, what does that mean? The function is?
Concave up. And when I teach calculus 1, I usually explain it by this clown face like that. So what does that mean? If the second derivative is positive, then it's concave up like that, okay? And then if the second derivative is negative, it's concave down, and it's a frowning face, just like this.
That's concave down and that's concave up. Now... what does this have anything to do with the the function here?
So look at this. If the heat distribution, okay so this is a good example. Over here, what is a partial derivative of u in the x direction and it's the second partial in the x direction? Here t is fixed, so we're doing a partial differential equation.
t is fixed as the value of a, so this is the you're considering this as a function of x, right? If you take the second derivative here, what do you think? Would it be positive or negative?
Negative. Negative, because it's the frowning one, right? Yeah. So it's a negative.
And do you think the heat will come down or go up? It'll come down, right? Because the nearby ones are lower than the ones.
So what you see is that if this is negative, Then you expect the derivative, the rate of change of the temperature with respect to time is? Negative or positive? Negative because it's decreasing right?
So this is what you expect. Okay. Now think about the other case. What if it was like this? If the temperature distribution is like this, at this point would you expect the heat to go up or down?
It will go up, right? So what you see is that if the second derivative, second partial with respect to time is negative, with respect to X of U is positive, then you would expect round U round T to be also positive. Not only that, doesn't it make sense that the more bend it is, the faster the change of temperature would be?
Yes. Does that make sense? Okay.
So not only their size match, you actually come up with a conclusion just from geometry inspection that The derivative of u, second partial, is proportional, oh, then actually, sorry, it's the other way. The rate of change of the temperature is proportional to what? The second partial in u. That's what you gain. I mean, after drawing a few pictures and thinking about how the temperature would be affected with the concavity, you see that the rate of change of the temperature should be.
proportional to the concavity. You can actually prove this by using Newton's cooling law, but we're skipping that. It just kind of makes sense.
Now can you turn this into an equation? Yeah, how does that work? Round U round T is equal to 2. How do you take proportionality into an equation? If this is proportional to the other one and then this is equal to?
K times that one over. What? K constant.
K, some proportionally constant. K times second partial. Now a lot of times people just write down the partial derivatives using indices. So a lot of times the heat equation is written like UT, U sub T is K times U.
u sub xx. So if you differentiate twice in the x direction, if you differentiate once in t direction, they're equal by some proportionally constant k. And this k has to do with with how fast the heat will be spreading out in the material.
So depending on some material, K will be small. That means the temperature doesn't really change that much. So it's something that doesn't change a lot. I guess if it's a metal, then heat in the metal, they change really fast, right? So in that case, K will be something large.
So K is something that you have to figure out by experiment. But we know that that should be. the equation that would govern it, okay? So this one is called the heat equation.
And we will try to solve that using some boundary conditions, okay?