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: - [Partial Integro-Differential Equations (PIDES)][1],Ekaterina Voltchkova - [Partial Integro-Differential Equations: Classification & Solutions][2] - [Solving Partial Integro-Differential Equations Using Laplace Transform Method][3], J.Thorwe, S. Bhalekar - [Solution of Partial Integro-Differential Equations by Elzaki Transform Method][4], Mohand M. Abdelrahim Mahgob, Tarig M. Elzaki - [A New Numerical Method for Fast Solution of Partial Integro-Differential Equations][5], P. Dourbal, M. Pekker - [Two Numerical Algorithms for Solving a Partial IntegroDifferential Equation with a Weakly Singular Kernel ][6], 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): - [How to solve partial integro-differential equation?][7] (Matlab) - [Partial integro-differential equation][8] - [finding solution to a partial integro differential equation][9] - [Solution of a partial integro-differential equation][10] - [Solving a nasty partial differential equation][11] (Mathematica). - [Partial integro-differential equations][12] (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][13] from Mathworks and [this one][14] from PhysicsForums as well. Finally, some classic texts on implementing PDEs through Maple/Mathematica: - [Numerical Solutions for Partial Differential Equations: Problem Solving Using Mathematica][15] - [PDEs, an introduction with Mathematica and Maple][16] See also: https://mathoverflow.net/q/277002/85967 [1]: http://onlinelibrary.wiley.com/doi/10.1002/9780470061602.eqf12020/abstract?systemMessage=Wiley%20Online%20Library%20will%20be%20unavailable%20on%20Saturday%207th%20Oct%20from%2003.00%20EDT%20%2F%2008%3A00%20BST%20%2F%2012%3A30%20IST%20%2F%2015.00%20SGT%20to%2008.00%20EDT%20%2F%2013.00%20BST%20%2F%2017%3A30%20IST%20%2F%2020.00%20SGT%20and%20Sunday%208th%20Oct%20from%2003.00%20EDT%20%2F%2008%3A00%20BST%20%2F%2012%3A30%20IST%20%2F%2015.00%20SGT%20to%2006.00%20EDT%20%2F%2011.00%20BST%20%2F%2015%3A30%20IST%20%2F%2018.00%20SGT%20for%20essential%20maintenance.%20Apologies%20for%20the%20inconvenience%20caused%20. [2]: https://iasj.net/iasj?func=fulltext&aId=44143 [3]: http://article.sapub.org/10.5923.j.ajcam.20120203.06.html [4]: http://www.m-hikari.com/ams/ams-2015/ams-5-8-2015/mahgobAMS5-8-2015.pdf [5]: https://arxiv.org/ftp/arxiv/papers/1602/1602.00001.pdf [6]: https://www.pvamu.edu/mathematics/wp-content/uploads/sites/49/08_xie-aam-r460-sx-111411_vol_7_issue_1_june_12.pdf [7]: https://math.stackexchange.com/questions/381171/how-to-solve-partial-integro-differential-equation [8]: https://math.stackexchange.com/questions/137214/partial-integro-differential-equation?rq=1 [9]: https://math.stackexchange.com/questions/938861/finding-solution-to-a-partial-integro-differential-equation?rq=1 [10]: https://math.stackexchange.com/questions/1169291/solution-of-a-partial-integro-differential-equation?rq=1 [11]: https://mathematica.stackexchange.com/questions/73150/solving-a-nasty-partial-differential-equation [12]: https://stackoverflow.com/questions/33204323/partial-integro-differential-equations [13]: https://www.mathworks.com/matlabcentral/answers/95224-can-matlab-solve-integro-differential-equations?requestedDomain=www.mathworks.com [14]: https://www.physicsforums.com/threads/help-solving-a-set-of-integro-differential-equations-with-matlab-maple-mathematica.372504/ [15]: http://www.itam.nsc.ru/users/libr/eLib/1/1996/Ganzha96.pdf [16]: http://ee.sharif.edu/~em/Partial%20Differential%20Equations%20-%20An%20Introduction%20With%20Mathematica%20and%20Maple.pdf