My choice would be **Boundary Value Problems and Green's Functions** by Ivar Stakgold. It have an introduction to *distribution theory* and them apply it to finding Green's functions.

It includes:

  - ODE
  - PDE with initial conditions
  - PDE with boundary conditions.

I found a preview [here](http://cam.ucsd.edu/~mholst/pubs/dist/StHo2011a-preview.pdf)

Cheers