An equation for an unknown function f involving partial derivatives of f is called
a partial differential equation. Essentially all fundamental laws of nature are
partial differential equations as they combine various rate of changes. We are affected
by partial differential equations on a daily basis: light and sound propagates according to the wave equation
which is a consequence of the Maxwell equations, the fabric of space and time are described by the Einstein
equations which tells how mass affects distance. The heat equation describes diffusion,
the propagation of energy or is used in smoothing procedures of computer vision.
Laws like the Navier-Stokes equations govern the motion of fluids or gases, the currents
in the ocean or the winds in the atmosphere. On a fundamental level, the laws of particle motion is
not given by ordinary differential equations like the Newton equations which describe the motion of planets
but by partial differential equations, the Schrödinger equation in particular.
In quantum mechanics, a physical configuration is modeled by a complex-valued function like the wave function
of a particle or the wave function of the universe. Its time evolution is the Schröodinger
or Dirac equation. Unexpectedly, partial differential equations also appear in finance.
The infamous Black-Scholes equation for example relates the
prices of options with stock prices. In the course-wide introduction lecture of this Math 21a
course on September 3, 2019, one slide illustrated the front page of a book of Ian Stewart
"In Pursuit of the Unknown, 17 equations that changed the world".
You see the book cover again on the left. At the end of the present lecture
we want to see in a worksheet whether we can identify a few laws. The goal of this lecture is to get
you exposed to partial differential equations. You should also know a few partial
differential equations personally. They should be your friends in the sense that you
know what they do and for what adventure you can join them. By the way, you already know
one partial differential equation: it is the Clairaut equation
The transport equation
P.S. Here is something for those of you who have seen Taylor series in Math 1b: given any nice function g of one variable, the function f(t,x) = g(x+t) satisfies the transport equation. Check this by taking the derivative with respect to t and the derivative with respect to x! We can now write the transport equation as ft = D f, where D means taking the derivative. You might also remember what the solution of the differential equation f' = a f was. It is f(t) = eat f(0) = (1 + a t + a2 t2/2 + .... ). If we replace a with D, then this becomes f(t) = e D t f = f(0) + t D f(0) + t2 D2/2 f(0) + ... = f(0) + t f'(0) + t^2 f''(0)/2 + ..., which is the Taylor series of the function. We say, that the derivative generates translation. Applying that to the function g gives g(x+t)= f(x) + t D f(x) t + t^2 f''(0)/2 + ..., which is Taylor's theorem. The transport equation can also be seen as the Schrödinger equation i h d/dt ψ(t) = P ψ for the momentum operator P = iD in quantum mechanics where the constant h is the Planck constant. Isn't that cool? The motion of a free particle without being exposed to an external force is just uniform translation. This is one of Newton's laws who formulated it as ``Every object in a state of uniform motion will remain in that state of motion unless an external force acts on it." The language of quantum mechanics describes this in terms of functions and this is the Taylor theorem. Using a Taylor series with a few terms already gives a good approximation of the real situation. In quantum mechanics, this idea is pushed much further and leads to Feynman path integrals describing the interaction of particles. Some Schrödinger equation animations from 2001, the C program has computed also the animation to the right. I wrote that program shortly after writing this paper on quantum dynamics.
The heat equation
The wave equation
The Burgers equation is the partial differential equation
|The Burgers equation can be generalized. First of all, one can add a viscosity term μ fxx which adds a diffusion. Then one can add more forces due to pressure and possible external forces. These are complicated equations. One has to determine the pressure from an additional partial differential equation. To the right, we see a scene from the movie ``Gifted" (see a few clips from gifted here in larger format), in which the Millenium problem about Navier Stokes appears.|
The Laplace equation
P.S. Harmonic functions play an important role everywhere in mathematics, for example when calculus is done in the complex. Let us look for example at the function g(z) = z3 in the complex, where z=x+i y is a complex number. You might not have seen complex numbers but i stands for the square root of -1. One can calculate with complex numbers like with other numbers but just has to keep in mind that i2 can be replaced by -1. For example, with g(z) = (x+i y)3 then this is x3 + 3 i x2 y - 3 x y2 - i y3. This is the complex number with real part u(x,y) = x3 - 3 x y2 and imaginary part v(x,y) = 3 x2 y - y3. As you can check, both of these functions u(x,y) and v(x,y) satisfy the Laplace equation.
The Laplace equation can also be studied on graphs. See this
A ``dangerous idea seminar" framework has been a MIT tradition for decades, where the speaker
has to structure the talk around five questions based on Fear, Joy, Mom, Cool, New.
The questions can be adapted also to each lecture. Lets do that: |
1. Why should I fear the topic? Partial differential equation ideas are used in any technology, this includes face recognition, building weapons etc. But this can be said about essentially any science. But you know what is probably the most scary thing about PDE's: the topic is not easy! It is a quite technical area of mathematics. Also when studying the topic with a computer, one has to deal with complicated numerical frame works. One has to work hard in order to make numerical approximations which are robust and for which the numerical solution is close to the actual solution one sees when one makes the experiment. 2. Why should I rejoice it to be done? Partial differential equations are the fabric we are made of. Understanding the Schrödinger equation allows us to understand elementary particles. Just an example. If one looks at the Energy operator L of a Hydrogen atom, then the structure of the eigenvalues describes the periodic system of elements. Partial differential equations are used to predict the weather, the paths of hurricanes, the impact of a tsunami, the flight of an aeroplane. They are used to understand complex stochastic processes. Partial differential equations appear everywhere in engineering, also in machine learning or statistics. A generalization of the transport is the gradient flow which is used to get good solutions to problems. As mentioned at the beginning, all fundamental laws in nature (classical mechanics can be described by variational problems leading to partial differential equations, the Euler equations, general relativity is described by the Einstein equations, gravitational waves emerging from a black hole merger recently measured out allows us to see what happened billions of years ago in an other part of the universe, diffusion equations can help to track the spreading of a virus or environmental disaster like an oil spill). 3. What should I tell my mom about it? Partial differential equations allows us to look into the future and allows us to take action in order to avoid difficult situations. Similarly as ordinary differential equations allow us to predict how far an asteroid zooms by the earth, we can build and use models to predict how the climate changes, we can take measures to soften the impact of a storm, or use it even for rather mundane things like how to make money (or lose some ...) on the stock market. 4. What is a cool discovery in that field? One of the fundamental results is the theorem of Cauchy-Kovalevski which assures a system of partial differential equations with analytic functions as coefficients has a unique solution. This is quite subtle, as analyticity is stronger than just smoothness. Analytic functions are functions which have a Taylor series which converges. 5. What is a recent discovery in the subject? A rather recent discovery is a result of Gregory Perelmann which tells that a simply connected bounded three dimensional space must be a three dimensional sphere. The theorem was proven using some sort of heat equation acting on a curvature functions. Given the space, the space is deformed by applying the heat flow. The flow smooths out the space, making it round. The limiting shape is then a sphere, like the bubbles seen in the lava lamp placed on the table during the lecture.
|Sofia Kowalevskaja (general PDE)||Augustine Cauchy (general PDE)||Johannes Martinus Burgers (Burgers eqn)|
|Pierre-Simon Laplace (Laplace eqn)||Jean d'Alembert (Wave eqn)||Joseph Fourier (Heat eqn)|
|This was a lecture given on Monday, October 7th 2019 at Harvard university in the multi-variable calculus course Math 21a. The page is filed under pedagogy, which is defined as ``the method and practice of teaching, especially as an academic subject or theoretical concept". I believe that pedagogy is an extremely applied and practical subject and that examples and real demonstrations are more important than theory. Video is a great tool to learn from a lecture (one immediate take away is that without microphone, it is necessary to speak slower and louder). This PDE lecture illustrates, that lecture, experiments, worksheet work and slides can be used in the same lecture. It might be a bit too much for one lecture, but the PDE topic leads to an unusual lecture which is not covered in most multi-variable courses. Thanks to Daniel David and Daniel Rosenberg from the Science demonstrations group for building and borrowing the experiments (wave machine and Chladni figures). Thanks also to the Bok center and math department for two independent recordings. I have placed myself a go-pro camera into the class to see an other angle. The quality of the go-pro is astounding, the side view of the old Panasonic cameras worked better however in this case. Ideally, (and with more cash available), I would probably place 3-4 go pro cameras at different places. It would generate a Terabyte of video data but would produce great and uniform 4K quality. One critique one can make about the lecture: ``Too many notes!"|