If we are interested in numerical solutions, there is the second part of the series --> https://www.dpmms.cam.ac.uk/~wilf1/publications/ Although the book is more focused on rigorous solutions to first-order PDEs, there are quite a few examples of numerical methods for initial-boundary value problems. I have a hard time deciding which one to use for the course I teach -- perhaps the most useful choice would depend on the particular application.
I would suggest that the Craik text is not the best first text, and certainly not the best as a self-study text. And Dalrymple's text is quite good, but also quite expensive (especially the paperback edition). 827ec27edc