 Hi, I'm Zor. Welcome to a new Zor education. I would like to address partial differential equations in this lecture. We have already spoken about ordinary differential equations. That's the differential equations where function of one argument with its derivatives participates. Now, obviously the natural extension is to allow functions of more than one attribute argument with partial derivatives participate in this equation. So, that's basically what it is all about. Now, this lecture is part of the course of advanced mathematics for teenagers and high school students presented in Unisor.com. It's better to view this lecture from this website because every lecture has detailed notes and in some cases exams which you can just use to check yourself. And the site is completely free, there are no advertisements, it's just pure knowledge. So, partial differential equations. Now, what's interesting is, and it's basically about both ordinary differential equations and partial differential equations, but probably it's even more about partial. The whole physics is actually filled with different laws which are expressed in terms of differential equations. One of the examples we were talking about acceleration for ordinary differential equations, that's basically the second Newton's law. And also we were talking about the spring, how it behaves and that's also differential equation of the second order. Now, as far as partial differential equations, again as I was saying physics is just filled with all these examples. So, today instead of talking about purely mathematical approach to partial differential equations, I will exemplify it with one particular problem. Very physical in nature, very kind of a common place. You will definitely recognize the process because you were dealing with this many times. And it leads to a differential, partial differential equation. So, in every aspect of physics, in thermodynamics, in electricity, magnetism, gravitation, wherever you go, you will find differential equations. And I would like actually to spend some time to explain it to you, although regular high school curriculum very rarely includes this topic. Anyway, it's not much more difficult than, let's say, ordinary differential equations. And I think it's very useful to know that this is basically the foundation of the whole physics. Okay, so what I am going to talk about is dissipation of the heat in one particular case. Okay, so let's consider you have a very thin rod, which at some point has some temperature. And then we started heating it at this particular end. Now, as soon as we applied the heat to this particular end, the heat will dissipate towards the other end. Now, my purpose is to find temperature as a function of two arguments. One is time and another is the distance from the beginning. This is zero. So, at the end of this lecture, I will basically come up with a differential equation, partial differential equation, because this is the function of two arguments now, time and x coordinate. And this partial differential equation basically defines how the heat dissipates. And it's called heat equation. So that's my goal for this particular lecture. Okay, so how can I approach this? I will try to use only logic. I will not present you with any fact without the proof, except there are certain experimental facts which I will explain right now. So, we are dealing with certain physical entities and I would like to make sure that everybody understands what we are dealing with. So, the first entity is the heat. Well, heat is a form of energy. It's inner energy within a particular object. It basically reflects how the molecules inside this rod, let's say it's a metal rod, how the molecules inside of this metal rod are vibrating. The more intensely they vibrate, the more heat actually was infused into this object and its temperature is higher. So, heat is a form of energy and again, in a simple kind of view, you can always consider it as a vibration of the molecules inside of this body. So, that's what heat is all about. Now, what is the temperature? Well, temperature is a measure of how intensely these molecules are vibrating. So, if you touch this with a thermometer, for instance, then the vibration will actually be transferred into the thermometer itself and the thermometer will show the intensity of this vibration. So, the heat is energy, temperature is a measure of the intensity of the vibration of molecules inside of this. Alright, now, what's interesting is that obviously, the more energy you pour into an object, well, the more intensely molecules will start vibrating, right? And therefore, the temperature will rise. So, the amount of heat which you infuse into the object and its temperature are related somehow. So, experimentally, experimentally was established that if you take one particular material, let's say iron or anything else, and you will take a unit of this material in mass, so one kilogram, let's say. And you apply certain amount of heat, certain amount of energy to this particular object to raise its temperature. Then the amount of heat which you pour into this object and the temperature are proportional to each other. Basically, the formula, and again, this is an experimental formula. So, if you have some kind of an increment of the heat which you supply into this object, and you have certain temperature by which, well, you have certain increase in temperature, then what's important is that it's proportional to the mass of this object. Where C is certain coefficient, it's called a specific heat capacity for this particular material, whether it's iron or plastic or anything like this. So, every material has its own specific heat capacity. How much energy is necessary, basically, to supply to increase its temperature by, let's say, one degree. So, if you have, obviously, if you have twice as much of this material to raise by the same temperature, you need twice as much energy to supply. So, that's why it's proportional to the mass. And again, the coefficient, this coefficient which is called specific heat, specific heat capacity, is specific for each particular material. So, let's say if you want to raise the temperature of the water by one degree, you need certain amount of heat to supply. But if you want to increase the same mass, one gram, let's say, of iron to increase the temperature by one degree, you need a different amount of heat. And this is what specific heat capacity actually means. I mean, it's amount of heat you need to supply to a unit of mass to raise the temperature by unit of temperature, right? So, this is, again, experimental fact. I mean, maybe there is certain theory which is based on some molecular structure, etc. And it can explain this formula, but first it was derived basically experimentally. People realize that if you supply twice as much heat to the same body, the raise of the temperature would be twice as much. So, that's what it is. And different objects have this different number. But anyway, it's specific for each one of them. So, that's a very important experimental thing. Now, another, so this is specific heat. Another, again, experimental fact is conductivity, the thermal conductivity. Now, you obviously had this experience. If you have a teaspoon made of steel and you put it in hot tea, it will start heating up. You need certain time to feel that this is really very hot. But if you have a silver spoon, you will feel it much sooner. So, the silver conducts the heat much faster than the steel, let's say. So, different materials, not only they have different specific heat capacity, but they also have specificity of conductivity. So, how fast they transmit the heat from one point to another. Now, this is a much more, I would say, delicate issue, this thermal conductivity. Now, obviously, if you have certain amount of heat applied to one particular place of this object made of certain material, then you obviously understand that if the difference between the temperature of this source of the heat is significantly higher than the temperature of the body which you are attaching to, the heat will be transferred faster. So, the bigger difference in temperature, the faster heat will be transferred. And, again, there is a proportionality. So, what's very important is the following. If your temperature between this point and the next point is very different, the difference is very big. So, it's very much analogous to you have the high water and the lower water. Whenever you are having this water coming down, the engine speed would be faster. So, the water will be faster going from this reservoir to this reservoir than if they are much closer to each other. Same thing with the heat. So, what's very important, it depends on how temperature is changing. So, if the temperature is changing with certain rate, then the amount of heat which is actually conduced, transferred from one point to another, would be greater. So, the greater the difference, the greater the speed of transfer. So, this particular thermal conductivity is very much dependent on the rate of change of the temperature from point to point. Now, if we are talking about the thin rod, we have only basically one dimension. So, our movement from point to point has only one coordinate, which is x coordinate. Temperature is changing somehow. This is the higher temperature, this is lower temperature. But if you take two different points, this one and this one, let's say this is x and this is x plus delta x, then the transfer of energy from here to here would be actually more, would be faster if the difference in temperature is greater. So, on any particular slice of this, this conductivity depends on the rate of change of the temperature. This is the rate of change of temperature. Well, temperature is function of two arguments. So, right now we are talking about argument x. So, if I will take a partial derivative of this function by x coordinate, I will have the rate of change, right? I don't know this function, obviously, yet. But I do know that this is a very important factor in the speed of transfer of the heat from left to right. Now, how can I represent it in the formula? Here's how. The amount of heat which goes through this particular point from left to right, from slice at point x is proportional to this, obviously. Then it also depends on the time we are applying this heat. I mean, obviously, the greater the time, the greater the heat is needed or will be transferred. And another thing is, obviously, we need the area of the cross cut. So, obviously, again, the wider connection between two slices, the faster the heat transfer will occur, right? Because it's actually transferred from one slice where molecules are vibrating with one speed to another slice. So, the more points of touching these two slices, the faster the heat will be transferred, right? So, obviously, the amount of heat which will be transferred is proportional to, as I was saying, rate of change of the temperature and time this heat is applied and the cross cut. And there is a coefficient of proportionality which is called thermal conductivity. So, this is another formula which is purely experimental and this thermal conductivity is obviously dependent on the material. Steel has one conductivity, silver has another conductivity. But if it's established, then we can always tell exactly that if my temperature rate is changing as this particular derivative, partial derivative shows, then, obviously, the amount of heat will be proportional to the time, the longer we apply the longer the more energy will be transferred. And the more points of touching when we are transferring the heat from one to another place, obviously, the more heat will be transferred. So, if you wish, this K is amount of heat which will be transferred per unit of time, per unit of area of touching, per unit of rate of speed changing of the temperature. Well, basically, I have set up completely whatever the problem we are talking about right now. So now, what we will do, we will just use these concepts which I was just explaining, the thermal capacity and conductivity which are relating our amount of heat and the temperature and the amount of heat and speed of the transfer of the energy. I will use them to derive what exactly is differential equation of this thing using some simple logic. So, here is the logic. Consider this particular piece from X to X plus delta X and we are assuming that this piece is small enough. Well, actually, we will definitely have this delta X converging to zero as well as delta T converging to zero. So, we will consider this to be small enough. Here is the logic. The amount of heat which this particular piece of rod consumes is equal to the amount of heat which is going from left to right, which is incoming, and we have to subtract the amount of heat which is outgoing. Right? Now, let me make a small adjustment to this formula. I mean, as it is written right now, consider it this way. If my temperature is decreasing, like in this particular case, this is the source of the heat, which means that this thing is negative. We would like it to be positive. So, if we want this coefficient to be positive, I really have to put minus in front of it. So, that's a small adjustment to make our constant K positive. So, now I can say that K is positive, as well as C is positive. That makes my life a little bit easier. Alright, so let's talk about amount of heat which is consumed by this piece of rod and which contributed to the rise of the temperature. Well, this amount is equal to amount of heat which comes in minus amount of heat which comes out. Right? Okay, so, how much heat goes through this particular border, left border of this interval? Well, let's assume that our rod has area of crosscut S. So, we have an area of crosscut. We have coefficient which represents our thermal conductivity. So, we can basically say that it's equal to delta Q in equals 2. Basically, it's this. During the time delta T, that what will be going through this point, through this border, I would say, inside. Now, what goes outside? Well, the outside would be minus and this minus would be plus. Also K, T of temperature of, it's a rate of temperature change in the next point, right? At this point. During the same amount of time and through the same area of the crosscut. So, that difference is what remains within this particular piece. It contributes to rise of this, of temperature of this thing. Now, if I have increased the amount of energy by this particular thing. I don't really need this. In is this one and out is that one, right? So, this is just delta Q. So, if I have increased the amount of heat which this particular piece consumed, it's inside of this particular piece of rod. Its temperature has risen by how much? Well, this is the law. So, what I have to do is I have to equate this with C times mass. What's the mass of this? Well, again, I don't know the mass because I don't know what it's made of. But I know that there is a unit of material. This material has certain number which basically represents its mass per unit of volume, right? And the volume I don't know. The volume of this piece is a cylinder, right? So, it's area times its length times increase of the temperature. Well, what is increase of the temperature? Let me just leave this as delta T. That's increase of the temperature during the time delta T. Okay? I can actually put T of T plus delta T X minus T of T X. This is increase of the temperature. Well, basically I have finished or almost finished. Because this particular equality allows me to construct this differential equation. And here is how. What I will do is I will, instead of this, let me just make it a little bit shorter. It's K times delta T times S times difference between T X plus delta X dx minus D T T X by dx. So, what I will do now, I will do the following. I will put delta X down here and delta T down there. So, what I will have is C times rho which is mass per unit of volume times, well, S is not actually participating anymore. And here I will have T of T plus delta T comma X minus T of T X divided by delta T, right? This delta T goes to the denominator. And this delta X goes to this denominator. And here I will have D T of T X plus delta X dx minus D T of T X by dx times K, right? And what do I do now? Well, obviously I will use delta T goes to zero and delta X goes to zero independently. What happens? What is this? If delta T goes to zero, that's obviously C times rho times partial derivative of this function by DT. And what is on the right? This is partial derivative by X if delta X goes to zero. And then I again divided by delta X, so difference between two values of partial derivative, first partial derivative by X. If I divide by the X, this is the definition of the second derivative, right? So, what is second derivative? It's the derivative of the first derivative, right? So, that's what it is. And this constitutes the heat equation. That's what it is. This is the heat equation. It connects together second derivative by the second argument by linear argument X and the first derivative of the temperature by time. And we have these coefficients of proportionality which usually are combined together somehow. And the final equation usually is expressed as this, where alpha square is equal to K divided by C rho. And alpha is equal to square root of this. Alpha square is important because if you will start solving this equation, you will have something which depends on alpha, so that's why it's important. Alpha or A sometimes, different letters are used. So, what was important here is, well, first of all, I spent some time explaining certain concepts of thermodynamics, if you wish. What is the heat? What is the specific thermal capacity? What is thermal conductivity? And there are two experimental laws which combine together the amount of heat and rise of the temperature. And in the case of conductivity, amount of heat which goes through certain point is related to a rate of change of the temperature at this particular point. So, we had these experimental laws, if you wish. And after that, using pure logic, we have combined these conditions together and derived certain differential equation which function temperature as dependent on the time and distance from the source of the heat. Now, this function should obey this differential equation. That's what's important. Now, if you start solving this differential equation, and we might actually speak about this in another lecture, you will find that there are certain constants which are always kind of involved when you are solving differential equations, which must be somehow defined. And for this reason, you need, obviously, initial conditions. Now, one of the initial conditions, for instance, in this particular case, is what is the material our syn rod is made of. That defines actually all these constants. It's mass per unit of volume, it's specific thermal capacity, and it's thermal conductivity. So, these are specific for material it's made of. Let's say it's a steel, for instance, or something. But then, there are some other constants our solution will depend upon. And obviously, it includes, okay, what's the temperature of the rod initially, and what's the temperature of the source of the heat which we are applying to the end, to the left end of this rod on my picture. So, these are initial conditions, and if we apply these conditions, then basically everything should be defined. Then this particular equation, after we solve it, will no longer depend on certain unknown constants, but these constants will definitely have certain concrete value. All right. So, I do suggest you to read notes for this lecture. It's on Unizor.com. It's a little bit involved, I would say, because I was trying to do something, and I had to slow down in certain cases because I myself had to really think about it twice before every new formula. But if you will read it in the notes, it's really explained relatively well. And again, you're always welcome to go back to this lecture to make sure that you understand all the details of this. So, it's important to understand that this is one of the many, many differential equations which basically conduct the way how all our world around us is working, living, moving, doing whatever it's doing. Everything in this world is somehow related to certain differential equations. In most cases, it's partial differential equations. And it's very important for you to understand the philosophy behind it. In the same way as simple, second Newton's law tells you how exactly one particular point mass will move if there is a force which we apply to this mass. So, these are simple things, but the real laws of the physics and nature are much more complicated. And partial differential equations is basically the most important kind of laws which all these movements in our world actually supposed to obey. Alright, that's it. Thank you very much and good luck.