In the series of lecture notes provided above, all of the examples are PDEs with 2 independent variables
(time and one spatial dimension for the parabolic and hyperbolic PDEs and two spatial dimensions for the elliptic PDEs).
The methodology used can be straightforwardly extended to PDEs with more than 2 independent variables
(PDEs with 2 or 3 spatial dimensions).
Given below is one example of how the numerical method employed for solving linear parabolic
PDEs with one spatial dimension is extended to solve the problem of a linear parabolic
PDE with variation in two spatial dimensions.
It will be clear after viewing this how the extension is made from a 1-D linear parabolic PDE to a 2-D linear parabolic
PDE that any such extension can be made in a precisely analogous matter.
Thus, any of the subjects above--i.e. linear or nonlinear; single equation or systems;
parabolic, hyperbolic, or elliptic--can be solved in higher spatial dimensions using this form of the extension.
I don't work them all out by hand because there is no new intellectual content in doing so.
