How to Self-Study Mathematical Methods?

The courses and book you mentions above is for basic engineering, physics problems. I think in your case, it's more about the applications than theorem. But, here are some books that you should check out and it combines a lot of pure math:

Single and Multi Variable Calculus: Calculus, 7ed by James Stewared

Fourier Transform and Analysis: A First Course in Wavelets with Fourier Analysis by Albert Boggess and Francis J. Narcowich (You should have basic knowledge about linear algebra and some Differential Equation in order to understand Fourier Transform)

Linear Algebra: Linear Algebra and its Application by Gilber Strang. More Advance we have, Linear Algebra Done Right by Axler

Differential Equation: Ordinary Differential Equation by Morris Tenenbau and Harry Pollard. More advance: Introduction to Ordinary Differential Equation by Agarwal, or even deeper: Ordinary Differential Equations by Edward L. Ince

Partial Differential Equation: Partial Differential Equation for Scientists and Engineer by Stanley. More advance: Partial Differential Equation by Strauss (he talks about Fourier Transform in this book too), or Partial Differential Equation by Lawrence C. Evans (I love this book because it's very detail)

About reading papers,let says a paper in Mathematics about PDE; it requires that your knowledge must be very deep in PDE in order to understand that paper, not just reading some basic books.