I know you said you don't want an undergraduate text, but as a physicist (BS+PhD), trust me, you do want an undergraduate text. The textbooks used in introductory classes (100-series, or equivalent) won't go beyond basic Calculus, because the students haven't had time to learn anything more, and even the intermediate classes (200-series) won't go past ordinary differential equations. Also, looking at the topics you mentioned (heat equation, wave equation, etc), I think what you want is not just a physics textbook, but specifically a mechanics textbook, since that's where these topics are normally covered.
So why do I say you want an intro text? Because you already know the mathematics; what you're trying to learn are the basic concepts of classical mechanics. These are what you'll find in an introductory textbook. More advanced book, on the other hand, will assume you already the difference between a force, an energy, and a momentum, and therefore they don't bother explaining it; they'll instead move on to more sophisticated treatments of the subject building on the student's previous exposure (variational methods, field theory, and the like).
My advice, then, is to go to the university library and check out a random introductory textbook. "Halliday and Resnick" [those are authors, not the title] was the standard intro text when I was a student, and it's a good choice. "Marion" and "Webster" are even older (and a bit more advanced) but they would also be good choices. Anything with the title "Intro to Mechanics" is probably a second-year book, but you might luck out and find something you like there too. "Landau and Lifschitz" is good too, but the series is probably too advanced to just pick up and read. Also, I would avoid the Feynman lectures at all costs -- they're great after you know the subject but piss-poor if you're trying to actually learn physics from them.
If you're bound and determined to use a graduate level text, then Goldstein's Classical Mechanics and Jackson's Classical Electrodynamics would be the standard graduate-level textbooks and will find plenty of partial differential equations in either one.
-JS
