What we call “differential equations” are usually not just differential equations. They also have associated initial conditions or boundary conditions.

With ordinary differential equations (ODEs), the initial conditions are often an afterthought. First you find a full set of solutions, then you plug in initial conditions to get a specific solution.

Partial differential equations (PDEs) have boundary conditions (and maybe initial conditions too). Since people typically learn ODEs first, they come to PDEs expecting boundary values to play a role analogous to ODEs. In a very limited sense they do, but in general boundary values are quite different.

The hard part about PDEs is not the PDEs themselves; the hard part is the boundary conditions. Finding solutions to differential equations in the interior of a domain is easy compared to making the equations have the specified behavior on the boundary.

No model can take everything into account. You have to draw some box around that part of the world that you’re going to model and specify what happens when your imaginary box meets the rest of the universe. That’s the hard part.

**Related posts**:

Three views of differential equations

Nonlinear is not a hypothesis

Approximating a solution that does not exist

This manifests, albeit in a somewhat different way, in many planning models, where the boundary is marked by the beginning and end of the planning horizon. Typically the state is known with some certainty at the start of the horizon, but there is seldom a specific terminal state, and often very little guidance about what the terminal state should look like. This can lead to entertaining solutions that look very effective over the planning horizon but leave you bare naked and unprepared for much at the end of the horizon. “Rolling horizons” and discounting tricks help but are not a panacea.

Why are the boundary conditions hard?

I suppose one explanation for why boundary conditions are hard is geometry. Partial derivatives only depend on function values in a little open set around a point, but boundaries can have complex geometry.

Modern PDE theory works in spaces of functions having generalized (weak) derivatives. This theory is easy on the interior of a domain. But when you want to speak of the values such functions have on a boundary, a lower-dimensional space, you get into technical complications (trace theorems etc.) You get theorems that say the boundary has to have this degree of smoothness in order for the solution to have that degree of smoothness (or for the solution to even exist).