Reference request: numerical analysis of PDEs and integro partial differential equations 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.
 A: 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: 


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*Partial Integro-Differential Equations (PIDES),Ekaterina Voltchkova 

*Partial Integro-Differential Equations: Classification & Solutions

*Solving Partial Integro-Differential Equations Using Laplace Transform Method, J.Thorwe, S. Bhalekar

*Solution of Partial Integro-Differential Equations by Elzaki Transform Method, Mohand M. Abdelrahim Mahgob, Tarig M. Elzaki 

*A New Numerical Method for Fast Solution of Partial Integro-Differential Equations, P. Dourbal, M. Pekker

*Two Numerical Algorithms for Solving a Partial IntegroDifferential
Equation with a Weakly Singular Kernel , J.-Mi Yoon, S. Xie and V. Hrynkiv 


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


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*How to solve partial integro-differential equation? (Matlab)

*Partial integro-differential equation

*finding solution to a partial integro differential equation

*Solution of a partial integro-differential equation

*Solving a nasty partial differential equation (Mathematica). 

*Partial integro-differential equations (code + Maple). 


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: 


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*Numerical Solutions for Partial Differential Equations: Problem Solving Using Mathematica

*PDEs, an introduction with Mathematica and Maple
See also: Books and resources on PDEs that use Mathematica and Matlab 
