I am not a specialist at 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 eq.s 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:
- 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 stackexchange community posts on partial integro-differential equations (some have quite interesting answers and ideas):
- How to solve partial integro-differential equation?
- 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 (numerical, Mathematica).
- Partial integro-differential equations (numerical, code + Mapple).
I do not know of any book containing examples of implementations of Matlab/Mathematica/Mapple 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: