Question

How would you describe the transition from studying ODEs to studying PDEs to an undergraduate student?

Now for the real question, is anyone at school today? I'm on the 5th floor Student Center Reading Room, I'm starving and I don't want to eat alone. ^^

Answer 1

It's a bit like going from (elementary) algebra to calculus in high school.

- You need to understand ODEs in order to understand PDEs, as the solutions of many PDE problems reduce to ODE problems.

- But like moving from algebra to calculus, the range of problems and things we can tackle greatly increase and so you should expect to see a lot of new things. (And in particular, once you move beyond simple equations like the wave equation or heat equation, the theoretical complexity greatly increases and we've obsessed with figuring out existence and uniqueness of the solutions of PDEs, and the machinery to tackle those issues in step change.)

- As a practical matter, PDEs come up all the time in the sciences and understanding them will greatly expand the range of the situations you can analyze.

Answer 2

That is superb. Thank you very much, James. :)

Answer 3

thanks. Wish I could type however:

"And in particular, once you move beyond simple equations like the wave equation or heat equation, the theoretical complexity greatly increases and we're obsessed with figuring out existence and uniqueness of the solutions of PDEs, and the machinery to tackle those issues is a step change."