This course provides an introduction to the field of partial
differential equations through a study of the classical equations of
mathematical physics. The methods discussed proceed from classical to
modern. The course begins with the method of characteristics for
first-order equations and then proceeds to look at general
second-order equations in terms of the classification problem and
weak solutions via the theory of distributions. Then the standard
second-order partial differential equations of mathematical physics,
namely, the heat, wave, and Laplace equations, are studied. The focus
is initially on weak solutions, then moves to the qualitative nature
of solutions, what conditions imply uniqueness and why, and what
consitutes well-posedness. In particular, how do these features
change from one class of problems to another. An introduction to
linear functional analysis is covered so as to provide a basis for
the study of distributions and abstract existence methods, and to
serve as a first step into more advanced work.