Impulse response

You may expect that a burst of input will cause a burst of output. Sometimes that’s the case, but often a burst of input results in a long, smoothly decreasing succession of output. You may not get immediate results, but long-term results. This is true of life in general, but it’s also true in a precise sense of differential equations.

One of the surprises from differential equations is that an infinitely concentrated input usually results in a diffuse output. A fundamental solution to a differential equation is a solution to the equation with a Dirac delta as the forcing function. In a sense, your input is so concentrated that it’s not actually a function. And yet the output may be a nice continuous function, and not one that is not particularly concentrated.

The situation is analogous to striking a bell. The input, the hammer blow to the bell, is extremely short, but the response of the bell is long and smooth. Solving a differential equation with a delta function as input is like learning about a bell by listening to how it rings when you strike it. A better analogy would be striking the bell in many places; a fundamental solution actually solves for a delta function with a position argument, not just a single delta function.

If you’re curious how this informal talk of “infinitely concentrated” input and delta “functions” can be made rigorous, start by reading this post.

Related post: Life lessons from differential equations