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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.

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I am not a specialist on the subject, but having faced similar problems in the past, I know the situation can be quite tricky, especially since you put the question in general, including thus non-linear cases as well. (I will only refer to partial integro-differential equations since the topic of the numerical solution of PDEs has a quite extensive literature on its own).

Here are some articles, in which you may find some interest:

and an indicative list of Stack Exchange community posts on partial integro-differential equations (some have quite interesting answers and ideas):

I do not know of any book containing examples of implementations of Matlab/Mathematica/Maple code for solving (partial) integro-differential equations. However, you could take a look at this post from Mathworks and this one from PhysicsForums as well.
Finally, some classic texts on implementing PDEs through Maple/Mathematica:

See also: Books and resources on PDEs that use Mathematica and Matlab

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