For something that has a little bit of everything, check out Partial Differential Equations by Walter A Strauss
It is a great intro to all of these topics.
For more in depth references, I reccommend these to anyone studying this field:
Partial Differential Equations- Lawrence C Evans
Numerical Solution of Partial Differential Equations: An Introduction- Morton, K. W.
Numerical Methods to Conservation Laws- Randall J. Leveque
Green's Functions and Boundary Value Problems - Ivar Stakgold