Advice for Learning Partial Differential Equations

In this video I discuss learning partial differential equations. I talk about all of the prerequisites you need to know in order to learn this wonderful subject. Do you have any advice or recommendations for people? If so, please leave a comment below.

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[Kasual_]

hey math sorcerer, i’m a 14 almost 15 year old from the U.S and during covid in 6th grade i took a huge interest in math and it’s been the biggest interest in my life for the past 3 years. i’ve taught myself and am now trying PDE’s. i’m a freshman now taking honors geometry because that’s the highest math class i can take. wish me luck!

[synocx]

Hey Math Sorcerer, like the approach to get a book in the first way. Sometime I made the experience, that using a book which fits not to you, could be disapponting. Therefore I like your approach, using several books.

My advice to learn PDEs:

1. Start with ODEs in the first place -> it will help to solve and understand PDEs
2. Start with the Heat Equation
3. Look for applied science applications like thermodynamic and electrical engineering problem sets on YouTube or books -> solving a real problem will help
4. Solve PDEs with an analytic solution
5. Learn numerics and solve the same PDEs by numerical methods and compare the results
6. Start solving more complex Equations by numerical methods
7. Go on to other PDE classes like the Wave Equation
8. Look for classes and people you can share your knowledge

Have a great time learning PDEs!

[alexandroskyriakis]

Great video brother, the topic addressed is very critical. In order to model various phenomena that occur in nature we use PDEs (Partial differential equations). PDEs are equations that involve the function and the partial derivatives of the function. In some cases, we can reduce the PDE to a canonical form and obtain the solution doing some tedious calculations. As a general rule, in most cases obtaining a closed form of the solution is impossible. So the next step is to obtain estimates that reveal the behaviour of the solution. In most cases we have a PDE and we want to see if the problem is well posed (existence and uniqueness of solutions). Here is a list on how you can enjoy the journey step by step:
1.Learn the definition of a PDE
2.Learn how to classify it. Is it linear, non linear, semi linear? Is it parabolic, hyperbolic or elliptic PDE?
3.Get familiar with three PDEs ubiquitous in science and engineering (Laplace equation, Wave equation, Heat equation)
4. Get familiar with the type of boundary conditions (Dirichlet, Neumann, Robin)
5.Learn to reduce the PDEs to canonical form.
6. Try to obtain solutions in closed form for some elementary problems just to warm up
7.Get familiar with some basic theorems (Maximum principle, Energy methods)
8. Review some tools from Analysis such as Cauchy Schwarz inequality, Minkowski inequality, Holder inequality, Young inequality, Gronwall inequality, Poincare inequality.
9.Study some basic theorems such as Lax Milgram and Reilich Konrakov.
10. Get familiar with function spaces like L2 and Sobolev spaces.
11. Refresh your memory of Fourier Analysis
12. Read some basics from Functional analysis like Hilbert spaces, Measure spaces, Lebesgue integrals. Also some basic theorems :Hahn Banach theorem, Fubini Theorem, Tonelli theorem, Caratheodory theorem.
13. Get familiar with the variational formulation of a PDE
I think it an exhaustive list, but you can break the material into smaller bits.
There are some excellent books in PDE analysis. One classic is the book by Heim Brezis. Another one is by Arne Broman in PDEs and Fourier Analysis. There are also two great books, one by Evans and one by Krylov.
Also there is an excellent book about Functional Analysis by B. Rynne and M. Youngson(I had the honor to interact with them during my Msc in Mathematics ). PDE courses might be hard but they are fascinating😉.

[anthonyymm]

As a grad student working in PDE, I feel like Strauss’s book is pretty good for getting intuition and is a pretty solid introduction. If you have analysis background (say baby Rudin at least, but preferably measure theory) then Evans is the gold standard.

[guidichris]

I remember dealing with PDEs in senior level EE courses. Very intense, and I've forgotten so much after 40 years.

[agni]

For a rather deep knowledge:
(1) Introduction to Partial Differential Equations: Zachmanoglou and Thoe
(2)A first course in partial differential equations: H F Weinberger
(3)Methods of Mathematical physics vol 2: Courant and Hilbert (yes that Hilbert)
All are great and very clear but Courant and Hilbert is quite demanding.
Taylor and Mann, Apostol calc 2 and Mathematical Analysis should provide all prerequisites.

[douglasstrother]

Boyce and DiPrima's "Elementary Differential Equations and Boundary Value Problems" (3rd ed.) discusses PDEs in Ch. 10, and Sturm-Liouville Theory in Ch. 11. This is a math text.

Boaz's "Mathematical Methods in the Physical Sciences" (3rd ed.) discusses ODEs in Ch. 8, Series Solutions in Ch. 12 and PDEs in Ch. 13. This is a more applied text and typically used in a "Mathematical Methods in Physics" course.

[thorliebhammer]

Introduction to Partial Differential Equations by Olver is terrific. It is in Springer's UTM series. I've read several of the Dover books which were helpful, also.
Strauss's book, btw, has a second edition. There is a pdf out there that serves as a solutions manual.

[dannywelds]

My friendly advice would be to remember to pde's is a technique class... Learn to identify the three beginning types, and what techniques solves it.
Work as many problems as you can with the solution supplied, good luck!

[corysuzuki]

The Adomian Decomposition method is a pretty powerful method for solving both linear and nonlinear odes, pdes, and integral/integro-differential equations. It decomposes the pde into iterations of integrals you can solve to form a series solution of all the integral components. The "Partial Differential Equations and Solitary Waves Theory" book by Abdul Wazwaz is the text that introduced this technique to me after my pdes class.

[cunningba]

I never took a course in PDEs. However, while in grad school, I taught ODE courses a number of times and did a lot of tutoring. One of my tutoring students eventually enrolled in a PDE course for which I ended up tutoring him. It turned out it was pretty easy. He'd come over with an exercise he was having trouble with. I'd look to see what the main theorem was in the section of the book where the exercise appeared and use that to set up the problem. Usually that was straightforward and he could get that easily. Then one had all these expressions involving a bunch of Fourier series. That was usually where he was stuck. I showed him all the usual tricks for manipulating expressions involving series to simplify them. That usually took care of the problem.

[aikidograndmaster]

My math PHD was in partial differential equations .. Start with Farlow it is the best book(Dover publications) Recently there has also been a complete solution manual published to this book also.

[jserink]

I used David Powers "Boundary Value Problems and Partial Differential Equations" when in university and to this day is was the best math course I ever took. I was in EE and we were supposed to take an EE based PDE course but I attended one lecture and it was:
1. Given by an EE prof,
2. Was a cookie cutter course.
I dropped it and scoured the course calendar to find a match course that I could get credit for and found were a few chem grad students, lots from geophysics and a few civil engineering grad students....Taught by a math prof.... ALL math courses should be taught by math profs. Engineering profs should teach digital logic, cct analysis, heat transfer, materials, that sort of thing. Pure physics should be taught by physics profs, pure chem by chem profs etc....
Anyhow, my two bits worth.

Cheers,
Jim

[paulkarch]

This is, I guess, way off topic but I'd like to recommend a book on probability and statistics. It's titled "The Probability Lifesaver" by Steven J. Miller. I am not a teacher but I would say it's intermediate to advanced. It is written in a entertaining style with examples, exercises and Mathematica code. In a some cases he walks you through a solution to a problem in detail he even shows you a few plausible solutions to problems that are wrong and he tells you why they are wrong. The first chapter is about: The Birthday Problem, Basketball, and Gambling.

As far as Partial Differential Equations the book I took a course from is fairly good: A First Course in Partial Differential Equations by Hans Weinberger. The first chapter analyses the one dimensional wave equation and it's pretty interesting. One topic that might be a little entertaining is to look up the topic "Can you hear the shape of a drum?" A normal drum is a membrane with a boundary - so it's a 2 dimensional wave equation. Hearing the shape of the drum means by looking at a solution you can determine the boundary. If memory serves, this is possible for 2 and 3 dimensional drums but not possible in higher dimensional drums.

[byronwilliams]

I really could have used this advice when I took it. Most of us got murdered the first time. I don't feel as though we were well enough prepared to solve PDEs. I ended up finding a copy of the book by Strauss and was absolutely amazed at how much clearer the subject was presented. Whatever you do, please keep these book recommendations coming. They do make a difference.

[dannywelds]

Great video
These two books are good:
Larry Andrews partial differential eq.
Donald trim partial differential eq.

[sicko_the_ew]

No advice from me, just a comment. PDEs was where I first properly hit "an out of depth error", doing a course in them. I passed it, but the other guy doing it at the exam hall we used also said it felt like he survived, but that he didn't know what he was doing. Quite frustrating. Nice not to be alone in the feeling, though.

I suppose part of surviving the course was at least knowing some of it, so the perception was a bit skewed by the tough parts.

One thing I remember struggling to grasp the Why of was Bessel functions. This is a perceptual thing, mainly, because I think I have something of the same problem with Fourier series. I know/ knew "how to do the trick" (good dog sit, get pat on head), but how the trick worked under the hood was something my reading around to satisfy my curiosity about this never unearthed that elusive Why.

Hmm ... I do have a kind of "half-tip" for PDEs (something I taught myself just for them, and which I've since forgotten how to do): You have to "automate your integration by parts", so that you can do most of it mentally. (But you're going to have to ask someone who knows how to show you How to do that. Although if I could figure it out, there's no question that you can, since I'm not the sharpest pencil in the box.) Had a young cousin who did a Masters in engineering not long after who told me he'd had to learn the same trick to get through that. I don't know why, but I guess it must be quite generally useful for PDEs.

It's a bit weird that the integration technique that is probably the most challenging part of Calc I can one day become an "inline operation" you need to be able to deal with quickly, so that you can get on with solving the real problem.

[dylanparker]

Sounds crazy but, when it comes to the Heat equation, learn how to solve it in a discrete context. E.g. Think about heat spreading across a chess board-like grid. Because the heat equation is so much easier to understand when you employ the `discrete approach' (that's probably a horrible misuse of terminology) of Finite Differences. Gives you an intuition for the continuous case.

[ryancreedon]

Strongly recommend "Advanced Mathematics for Applications" by Prosperetti. It's not a well known book, but it should be. It lays out everything you need to know about PDEs for engineers, scientists, and mathematicians alike. I'd also recommend to take a fluid dynamics class. A lot of the concepts and solution methods encountered in a PDE class are inspired by fluids.

[SimsHacks]

In 3rd year of undergraduate, we have an Introduction to PDE and we use Evans (only first two chapiters though, which is already enough for undergraduates 🤣) In Europe we have Real analysis in 1st year and Measure theory in 2nd year.