This is probably the very best upper undergrad/graduate level textbook on applied partial differential equations.

The book is written by leading academics with extensive experience in applied mathematics and industrial engineering problems, who worked and taught the subject for decades. What I particularly like about it is that gives not only the technical details, but also a very broad overview of the subject.

The first 6 chapters, which constitute a self contained upper level course, start (very naturally) with quasi linear first order PDE's, then systems of first order PDE's, and from that move to the classification of linear 2nd order PDE's, then to the canonical topics of hyperbolic, elliptic and parabolic equations (in that order). Following that, 3 more chapters cover a number of more advanced topics: free boundary conditions, non quasi linear equations, and miscellaneous topics.

The book is written by applied mathematicians for applied mathematics students. It hardly contains any functional analysis or distribution theory, but emphasizes effective methods for approaching practical problems. However, the book is NOT light reading, but is meant for serious students at serious universities.