Why even solvable problems are often unsolvable
Solvability is a very interesting concept in mathematics by which one can determine, using sophisticated methods of proof and analysis, whether a certain problem can be solved, be it an equation, system of equations, integral, etc. Solvability usually refers to whether a problem can have an “exact or closed-form solution”, but other times it can also refer to whether a problem’s suitability to be solved via numerical or approximate methods.
For instance, certain linear partial differential equations such as the heat equation or wave equation can be solved exactly under certain initial or boundary conditions, and usually in very simple geometries like rectangular or circular domains. However, as soon as you approach the problem on a non-standard domain, or introduce more complicated boundary conditions, the differential equation becomes unsolvable unless you employ some numerical scheme.
This is precisely the issue one faces in the real world. It is easy to fall into the trap of thinking that because a certain mathematical problem can be solved numerically, that there are no computational challenges associated with it. “Oh well, the heat equation is easy to solve, so why do we need to spend all this time doing design and engineering simulation of heat transfer?” is the type of question one might ask while not knowing how a problem scales…
