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How do I refresh advanced math I learned at a graduate level?

I once was able to do the full solution of a particle in a parabolic well and other advanced math, however 20 years later I'm struggling to follow again Eugene Butkov: Mathematical Physics.

Now, after 20 years I need for professional reasons to solve PDE's again and all that college knowledge that I had would be handy.

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    $\begingroup$ Welcome to Mathematics Educators! You mention graduate level, but does this question apply to advanced undergraduate level or other levels as well? For instance, I can imagine that after 20 years (or even fewer years!), many people might have forgotten or become rusty on much of calculus and linear algebra, let alone more advanced topics. $\endgroup$ – J W Jun 3 '20 at 7:38
  • $\begingroup$ Also, would you like an answer specific to PDEs or would you like general advice? $\endgroup$ – J W Jun 3 '20 at 9:26
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    $\begingroup$ A general advice. The red flag raised when I was no longer able to solve a PDE the other day. Then when I went to Butkov's book for help I noticed that a lot of concepts were rusty. $\endgroup$ – ppaulojr Jun 4 '20 at 14:43
  • $\begingroup$ Regarding PDEs specifically, you might want to check out this question. $\endgroup$ – J W Jun 4 '20 at 15:03
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Don't underestimate how much your knowledge and skills can fade when unused, even after a couple of years, let alone twenty. I've experienced this personally when trying to study advanced topics for which I had the prerequisites on paper from many years ago, but which were no longer fresh in my memory.

My general advice is first to check the immediate prerequisites for the area you are trying to refresh. If your recall of those is fairly solid, then find an introductory book, lecture notes - maybe your own old lecture notes if you still have them - or video lectures on your target area (in your case, PDEs) and get to work. Be patient with yourself, as it may take some time to get up to speed. As you proceed, you may find yourself recognizing topics and even recalling definitions, examples or methods that you once learned. Take heart and be encouraged!

If you find yourself hazy on the prerequisites, spend some time on those first. How much time is a balancing act. Ideally perhaps, you would study the prerequisites thoroughly, but time may not permit that, or you might simply get bogged down and lose motivation. Monitor how you are going and when you feel somewhat ready, return to your target area. With any luck, you will now be able to make some headway, dipping back into the prerequisites when you need to. (In your case, you might want to revisit some Calculus, Linear Algebra, ODEs and maybe Numerical Analysis.)

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