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Partial Differential Equations Intro · Acedra
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Partial Differential Equations Intro
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0:09
welcome back so i am incredibly excited today to tell you about partial differential equations and so in this video what i'm going to do is essentially motivate what are partial differential equations why do we need to solve them and i'm going to give you a lot of examples of types of pdes that come up throughout physics and engineering and science okay so this is absolutely one of my favorite topics um and i gotta say i am just so
0:37
grateful to have the opportunity to teach this to you it's something that has you know partial differential equations have changed my life you know it allows you to solve problems that uh you can't solve without them and so i really really hope that this is kind of opens a door to you like it did for me okay calculus is kind of the same way it's one of those things that once you see the world through that lens you can't unsee it okay so a partial differential equation
1:04
there's a definition here it's essentially a set of equations in terms of a multivariate function
1:11
and its partial derivatives and so the multivariate function um the multivariate function u that i'm going to talk about u
1:21
is often a function of space and time okay so this is my function um x is space and t is time and this might be a this might be a scalar function this might be the temperature distribution on this piece of glass here so x would be a two dimensional x and y vector so in general this could be a vector u could be a vector and t could be time or you could be a scalar like this could be the temperature distribution like i said this could be the fluid flow vector field of the error
1:56
in this room in which case you would have three components the x velocity u the y velocity v and the z velocity w and x would be a three-dimensional space vector x y and z so this is a very flexible function it's a function of space and time typically and the partial differential equation is going to relate this function and its partial derivatives so a good example would be the wave equation so i could say the second partial derivative of u with respect to time is equal to some constant
2:35
uh some some positive constant times the second partial derivative of u with respect to x
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so in this case u is a scalar function this is just a scalar function and x is a scalar you know position along the x-axis and this would be the equation for a wave uh you know kind of a wave
3:00
u of x and t propagating at some constant velocity c okay and i'm going to you know we can derive this we can analyze this this is just one example of what a partial differential equation looks like so again it is a relationship between some function u and its partial derivatives here i have a partial derivative and t and a partial derivative and x now that's different than an ordinary differential equation so um you know if i looked at my ode this is just a sidebar if i look
3:34
at an ordinary differential equation it is usually a single derivative of a function you know x of t equals some uh you know function of x like lambda x of t and in this case there is only one variable that's being varied time so i have a an independent variable time a dependent variable x and i'm just taking derivatives only with respect to that independent variable time whereas for partial differential equations it has multivariate dependencies it's a function of multiple variables
4:10
x and time in this case or maybe even multiple spatial dimensions and so i have to think about partial derivatives with respect to each of those those independent variables okay good
4:23
uh so that is just kind of an example or a definition of what a pde is and this is an example what i want to do is i want to start building up some of the most useful pdes that we use like all the time in physics and engineering and science and then we're going to start thinking about what are some of those properties of those pdes that are particularly useful so maybe one of the things i'll start with is i'll just kind of describe some of the canonical canonical pdes
4:53
and all of these canonical pdes that i'm going to describe have some common properties so they are going to be linear and i'm going to define what i mean in a minute so they're linear they are second order
5:08
and they are what is known as homogeneous or homogeneous depending on homogeneous
5:16
okay so i'm going to define what each of these means so linear just means that um the partial differential equation can be written in terms of linear operators where there's no nonlinear terms like u squareds or u times u sub x and things like that let me make a quick quick quick sidebar that oftentimes we use subscripts like u sub t to denote partial u partial t and things like u sub x or u sub x x would equal the second partial derivative of u with respect to x squared okay so
5:53
often times partial derivatives are denoted by subscripts and i'm going to do the same thing here so linear basically means that there are no nonlinear terms there's no things like no
6:04
no products or cross products no products between my various u u sub x u sub x x terms so i don't have any u squareds or u times u x's or u x times u x x's those would be non-linear and i'm going to talk actually about linear more later so just to remember that second order means that there's no more than a second no derivatives higher than second order no derivatives higher
6:39
than second order okay so you know i could technically have a fourth derivative and x a fourth order diffusion term there are physical systems that have fourth order diffusion uh but at least in these canonical pdes i'm going to write down here we're only going to think about second order pdes and uh homogeneous means that there is no time forcing or spatial forcing so the the governing rules of the system are the same throughout time and they're the same throughout
7:12
space now for example in this wave equation this is essentially the equation for a guitar string okay and you could imagine building a guitar string where the first half of it is nylon and the second half of it is steel and in that case that will be a non-homogeneous differential equation because the the rules of the equation will change on those different halves of the guitar string because the material properties will mean that the c is different in one part
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of space than in another part of space so what homogeneous means is that there is no um time dependent forcing or spatial dependent forcing no time dependent forcing
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or space dependent forcing
8:04
and you know now that i'm thinking about it that example i gave you is actually not a good example of an inhomogeneous equation so i can have an equation where the constants themselves are functions of space um you know that that's okay but i guess uh here what i mean is if i actually actively force my system in time or in space if i have some plus forcing function that's a forcing of time and space and i'll give you lots of equations examples of this for example in the
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heat equation if i have a blow torch and i'm you know actively heating up a metal plate that would be an inhomogeneous term in that equation so we'll we'll talk through all of this in a minute but these are the three kind of basic properties for these canonical pdes i'm going to write down now
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and so i want to write down three basic partial differential equations we have we have the wave equation
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which i've actually already written down before we have
9:06
the heat equation this is one of the most important equations in all of physics the heat
9:11
equation and we've already seen before when we looked at potential flow we have laplace's equation laplace's equation and laplace's equation and heat equation are very very closely related so here what i'm going to do is i'm actually going to write these down and then we're going to talk about what their properties are we're going to confirm that they're linear second order homogeneous equations so the first thing i'm going to do is i'm going to write it down in
9:41
kind of a specific form for two independent variables so the wave equation is u sub t t equals a positive constant c squared u sub x x that's actually exactly what i wrote down here just using subscript notation and this is for a one dimensional wave equation i have two independent variables i have time and one spatial coordinate x okay so this is a 2d version the heat equation is another super important equation so this one again for for a simple case is u sub t a
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single time derivative a single time derivative of u equals another positive constant we're going to call this alpha squared times u sub x x now these look extremely close to each other all that not
10:44
i pluck it you know i available these these standing waves or if i take a a rope and i whip the rope it will send this traveling wave traveling along that rope and the shape will kind of remain constant whereas for the heat equation if i have some temperature distribution in a one-dimensional piece of metal immediately it's going to start to diffuse and average out until it all becomes a constant temperature which is very very very different behavior
11:09
just because of the single additional partial derivative in this wave equation okay and the the third canonical pde that we always study uh in physics is laplace's equation which is just the heat equation if i set u t equal to zero it's the steady state heat distribution uh like let's say i have a metal plate and i heat one side you know in a hot bath and one side in a cold bath and i insulate the other two sides there will be a steady state temperature
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distribution that is given by laplace's equation so again i'm going to write this in two dimensions
11:49
as u sub x x plus u sub y y equals zero now notice i'm actually writing this in two dimensions this has two so so there's no time in laplace's equation because it's for steady state heat distribution and so this would be how the the steady state heat distribution has to satisfy this for a two-dimensional uh heated object for example with boundary conditions
12:15
and again we already saw that you know we already saw this from the potential flow equations also solutions of laplace's equation are potentials that we can use for potential flows
12:28
good so these are three canonical pd's i also want to show you what this looks like in vector calculus notation for arbitrary dimensions because i think you know we've been essentially building up partial differential equations like laplace's equation using vector calculus div grad and curl and so i want to show you what this looks like for those cases as well okay good um so the wave equation in nd is going to look like u uh u sub t t equals again some positive constant now times
13:01
the laplacian of u okay and uh this is going to be so if u is a function of x y and z this is now you know second partial of u with respect to x squared plus second partial of u with respect to y squared plus second partial of u with respect to z squared and you could in fact write down an n dimensional a 10 dimensional laplacian operator if you like it's easy same thing for the heat equation it's just u sub t equals a positive constant alpha squared again times the laplacian operator of u
13:33
and laplace's equation aptly named is just the laplacian operator of u del squared u set equal to zero and so these are kind of the n dimensional this this would give you the steady state heat distribution not just in a two-dimensional plate but in a three-dimensional object then you take the three-dimensional laplacian or an n-dimensional object sometimes for example in machine learning you might add some kind of artificial diffusion on an
14:02
n-dimensional vector and you would need to be looking at an n-dimensional laplace's equation
14:07
now there are more complicated pdes that we use in our daily lives like the navy or stokes equations are non-linear equations okay so they are they don't satisfy this kind of nice property that the solutions uh satisfy superposition there's other equations that are you know not second order like i mentioned that if you had u quadruple x there are systems that have that kind of fourth order diffusion i can have systems
14:32
that are inhomogeneous i'm forcing them in space or in time okay actively forcing those systems and so we can essentially relax all of these but These are the canonical building block partial differential equations that we use to build intuition to understand solution techniques and so we'll actually spend a lot of time thinking about how do you solve Laplace's equation in fact we can derive separation of variables one of the most powerful techniques to solve PDEs
14:58
we can demonstrate it on Laplace's equation and then use it to solve the heat equation you can use the Fourier transform techniques to solve the wave equation and so these all become very very useful as building blocks to understand how partial differential equations work good... are there other things I want to tell you? I'm sure there are... in fact I think I'm going to tell you lots more in other videos maybe one of the things I'll mention is what does it mean
15:26
to be a linear PDE. So we've talked a lot about this for ordinary differential equations. We know that we like linear ordinary differential equations specifically because it means that linear superposition holds. If I have two solutions of a linear differential equation, and I add up that solution, it is still a solution. And that also is what linear means for PDEs. So let me just write this out... so linear is just a way of saying that superposition holds
16:06
and so what that means... let's take the wave equation, for example, if I have two solutions of the wave equation I have two waves that are traveling at some constant speed, I've got two solutions u1 and u2 and I add that solution up... I superimpose that solution, it will also be the solution of the wave equation, because this is a linear partial differential equation. And in fact, like I mentioned, all three of these equations are linear. That is super super useful.
16:34
So Laplace's equation tells me what the gravitational potential or the electric potential is going to be in a medium away from charge densities or point masses. And again because linear superposition holds I can have 10 charges and just add up the fields from all of those charges to figure out what the field is going to be at some distant location x Linear superposition is one of the most powerful tools we have in all of mathematics
17:03
to handle differential equations. I'm going to say it again: When we say superposition holds, this means if u1 and u2 are solutions to the PDE,
17:21
then in fact any constant alpha*u1 + beta*u2 is also a solution. And so that allows us to essentially cook up basis solutions or "eigen" solutions and then take linear combinations of those for more complicated problems, for more complicated boundary conditions, geometries, etc., etc. And, in fact, that's how Fourier analysis was invented... Fourier himself invented this Fourier transform specifically to provide a basis of eigen solutions for the heat equation.
17:59
So the heat equation, you can essentially transform this into some diagonalized system using these sines and cosines... this Fourier transform that Fourier invented, and it simplifies the heat equation and it provides a family of unique orthogonal basis solutions, so that any combination of them also satisfies the heat equation. Really, really powerful stuff. So we'll talk about all of this more, but I really, really just love this idea that linearity
18:29
allows you to add up independent solutions of these equations and there's still a solution.
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And linearity also relates to the operators themselves. So remember we're talking about
18:43
this Laplacian operator, del squared is equal to the divergence of the gradient of whatever function... so if I have the laplacian of "f" it is equal to the divergence of the gradient of "f" for some scalar value to "f" and
19:04
essentially these the gradient and the divergence operator and therefore the laplacian operator are linear operators the partial derivative of a function u is a linear operator on u and so that's something i really want to to convey to you this is a kind of an abstract concept but very important is that i'm just going to give you some examples of linear operators so
19:30
a linear operator on u one example would be if i just multiply it by a constant if i take u times 5u that's a linear operator and you can verify this because you can plug in u1 plus u2 and that's still equal to 5u1 plus 5u2 it's the linear operator of u1 plus the linear operator of u2 so it's linear another linear operator and this is really important and i want you to verify this this is extremely important is the partial derivative of u with respect to x
20:02
in fact the partial derivative of u with respect to any of its uh the variables it depends on partial u partial t partial u partial x second derivative of u with respect to x the laplacian of u the gradient the divergence you know all of these are linear operators and so i want you to actually verify that uh this l of you know alpha u1 plus beta u2 right so if if u1 and u2 are solutions then i want you to plug
20:34
in this and confirm that this is alpha times the operator of u1 plus beta times the operator
20:46
okay this thing is kind of a pain to erase uh plus beta times the operator of u2 and i'm just going to say this again so what linearity means is that if u1 and u2 are solutions
21:02
to the pde then alpha u1 plus beta u2 is also a solution and what that means for a linear operator we build up this pde using these linear operators this is a linear operator on u this is a linear operator on you multiplying by c squared is a linear operator on u this is all kind of l of u equals zero a linear operator of u equals another linear operator of u and so to confirm that an operator like a partial derivative is in fact linear we verify that if we take two functions
21:33
u1 and u2 and we add them up in some combination a plus b then we hit that with the operator you can essentially you know pop the operator out you can pop these constants out and you can split the sum so that it is equal to that constant a times the linear operator of u1 plus that constant beta times the linear operator of u2 and i'm gonna have you like get a piece of paper actually verify this is true for partial u partial x verify it's true for second derivative
22:02
of u with respect to x or for the laplacian of u with respect to a multivariate function of let's say x and y really really really important okay and these are all linear pdes
22:15
maybe the last thing i want to show you before you know we'll talk more about pdes a lot in the next few lectures so i don't want to overload you here but maybe what i will do for the last thing is just give you an example of a non-linear so this is not going to be a canonical pde it's a non-linear pde and this non-linear pde is going to be berger's equation so burgers burgers equation it sounds delicious i'm quite hungry so this is kind of a non-linear
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burgers equation and i'm just going to write out what this pde is and then we'll talk about why it's non-linear okay so the partial differential equation for burgers equation in one component is the time derivative u sub t plus u u x equals some viscosity uh u x x so again this is for the one-dimensional burgers equation this u is just a function of time and a single spatial variable x and it also is going to be describing how waves propagate uh in space and time but because we have
23:25
this non-linear term here this is a product this is a product of u and u x this is not linear you can try to do this for u u x and this is not true so that's another thing i'd really like you to verify that this is non-linear and it's called a nonlinear uh convection
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okay that's a really important term here and we'll talk about burger's equation and the physics more later but i also want to point out in vector form you can write this as a vector u sub t this is for a higher dimensional burgers equation plus u dot
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grad of u plus my sorry equals i i have to erase twice in one video okay good
24:17
this equals my laplacian operator my new times laplacian of u and i'm going to say that this is for a vector valued u so now u could be the velocity in the x and the y component so this u could be
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u equals you know u in the x direction and v in the y direction and it's a function of x and y and t so now i have to use these partial these kind of gradient and laplacian operators to account for those those partial derivatives okay good so let's just take a step back and do a quick summary uh to wrap up so partial differential equations like ordinary differential equations like calculus is one of the most powerful tools we have to describe systems that change in space and time
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the world around us changes in space and time so your brain has you know it's changing as a function of x and t constantly the atmosphere the climate is changing as a function of space and time the ocean is moving and flowing as a space function of space and time and this partial differential equation framework allows us to relate that function we care about and how it changes in space and time through these relationships of partial derivatives generally
25:36
speaking we want to solve for this unknown function u we want to solve this equation for u that's usually the goal and that is typically easier to do for these linear pdes because if we can solve for kind of a basis of solutions then linear combinations of those solutions also are solutions uh but there's a lot of interesting equations like this burgers equation is basically
26:00
uh kind of a 1d analog of the navy or stokes equations and so you know for really nonlinear fluid flows it you're not going to have that linear superposition and we're going to have to have other other solution techniques okay a lot more on this coming soon stay tuned thank you
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