Such equations are solved by searching an approximate solution which is a function that belongs to a restricted class of functions, where each function can be described by a finite number of parameters, for instance functions that are piecewise polynomials (splines), truncated polynomial series, truncated Fourier series etc. All the various methods for computing approximate solutions, e.g. finite differences, finite elements, boundary elements, spectral methods and so on, are equivalent with this.

When the equations are well-behaved, you can be certain that it is possible in principle to obtain an approximate solution that can be as close as you want to the true solution. Otherwise, it may happen that no function belonging to the restricted set of functions where you search solutions can approximate well enough the true solution, e.g. because the true solution can grow faster than any function in that set.

This research establishes conditions that can be verified for PDEs to ensure that the methods that you intend to use for solving them will work correctly, instead of providing misleading results.