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It should at least include:

Laplace equations, waves and diffusions

reflection, boundary problems, Fourier series

The content of the book I mentioned can also be found on

https://www.wiley.com/en-us/Partial+Differential+Equations:+An+Introduction,+2nd+Edition-p-9780470054567

Indeed, Algebraic structures(Lie groups), Differential geometry, Complex analysis, and PDE are all essential for a theoretical physics student. I think such a student should take all the courses I mentioned above. It's the minimum

But what's better? Are there any alternative paths on course choosing?