I'm very new to the field of numerical analysis of PDE and integro partial differential equations.

My advisor (who is not a specialist in this area) highly recommended to read

Randall J. LeVeque's Numerical Methods for Conservation Laws,

which is immediately relevant to my interests.

However, I'd like to gather further reading recommendations that range from basic topics to more sophisticated and recent advancements in numerical PDE and integro partial differential equations. I'm also interested in books that have extensive examples of implementations with software like Matlab and Mathematica.