When something is truly impossible, you can either give up or lower your standards.

Consider the 1D transport equation \(u_{t} + u_{x} = 0\) on \(\mathbb{R}\) with initial conditions \(u(x,0) = g(x)\). Solutions simply translate the initial conditions right with speed 1. We find- using the method of characteristics- that solutions exist1 and look like \(u(x,t) = g(x - t)\). Let \(g(x)\) be the famous bump function, smooth with compact support:

A bump function travels right

If we plug the solution into the PDE, everything works out:

Conceding contrasting commencing conditions can cause curious conundrums. In other words, let’s try some different initial conditions. Let \(g\) be a square wave. We expect the solution to be analogous to the bump function’s: the initial condition traveling right with speed 1.

A square wave travels right
What we expect

Indeed, our formula returns the following,

but validating this solution by plugging into the PDE will be problematic. \(u\) isn’t everywhere continuous let alone differentiable in \(x\) nor \(t\)! This problem arose out of a regularity assumption on \(g\) when using the explicit formula. One might think there exist other solutions that may be plugged in without issue, but we can convince ourselves that this solution is unique since the PDE’s characteristic curves never intersect.

Thus ends our attempt to find a “classical solution” to the PDE. A classical solution is a sufficiently smooth function that satisfies the PDE when plugged in. Our solution is sufficiently smooth on most, but not all of \(\mathbb{R}\).

When something is truly impossible, you can either give up or lower your standards. Finding a classical solution is truly impossible, but the fact that our solution holds almost-everywhere gives us hope. It may be that our solution is “good enough” in that it satisfies many nice properties except that of being classical. A good wish list of nice properties would include the following:

  1. Existence
  2. Uniqueness
  3. Varies continuously with the initial data2

A PDE problem satisfying all of these is called well-posed, and it’s easy to see that our problem is well-posed for sufficiently smooth \(g\). For arbitrary \(g\) however, we lose existence. Obtaining existence requires us to redefine what it means to be a solution. Since our solution is sufficiently smooth almost-everywhere, perhaps we can define a “good enough” solution to be a function that satisfies the PDE almost-everywhere. While this guarantees existence, we give up the rest of our wish list: uniqueness and continuous variation3. Most would argue that this is a fair trade-off.

Existence precedes essence

-Jean-Paul Sartre

Since the notion of measure-zero doesn’t make sense in a discrete space, the loss of uniqueness and continuous variation doesn’t manifest in the same way on a computer. By obtaining solution existence for arbitrary initial conditions, we afford a notion of approximate solutions and hence, a notion of numerical solution of PDEs.

Up next:

  • Theory of distributions. Differentiating the non-differentiable.
  • Weak forms. The Galerkin method for numerical solutions.

  1. Since our initial condition manifold intersects transversally with all characteristic curves. 

  2. In our case, this means that we can bound changes in \(u\) by bounding perturbations of \(g\). 

  3. For instance, you are free to change a countable number of points to whatever value you like.