We shall trace the development of ideas and methods from their origins in the 19th century (and earlier) physics through classical and modern approaches, showing how new methods evolved when older methods exhausted themselves.
Specifically, we treat the following topics:
First order equations - Characteristics and bicharacteristics, variational solutions and generalized solutions.
Higher order equations - Well-posed problems, Cauchy-Kowalewsky's theorem, classification of partial differential equations.
Hyperbolic equations - Fourier transforms of distributions, energy integral methods, Sobolev spaces, non-classical solutions, wave propagation.
Second order elliptic and parabolic equations - review of classical methods.
Application of functional analysis to partial differential equations - from Riemann to Weyl.
Pseudo-differential operators, Fourier integral operators, and wave-front sets.