What are some real-world applications of functional equations?

Question

My question is simple: What are some real-world applications of functional equations?

I recently started working with more and more functional equations, and was pondering this. Most fields in math have some practical application to the real world, however I just can't see how this applies to functional equations as well.

Note: For functional equations, I mean this.

Answer 1

My Ph.D is on econo-physics, so I will try to answer your question based on my own (which is biased) understanding. Basically, econo-physics is a subbranch of statistical physics which apply concepts of statistical physics to finance and economics, see for instance the pioneering paper "Statistical mechanics of money" (written by Adrian Dragulescu & Victor M. Yakovenkoon) regarding this area of applied math. The line of research can be very "practical-driven" in the sense that researchers often use their results to test against real-world data (on wealth/money distribution across different countries). The main mathematical object of study are nonlinear evolution equations describing the evolution of the pdf/pmf of the wealth.

Answer 2

One nice example of a functional equation is the 'Euler-Lagrange Equation' in physics. A specific case of this tries to find the 'shortest path' between two points on a manifold. If the two points are on the Cartesian plane, the equation is a line. If the two points are on a sphere, the equation is a 'great circle'. This is the reason why airline routes tend not to be on a 'straight path' because it is actually makes the journey further.

In both cases, the shortest distance/path between the two points is called a 'geodesic'. This is a nice example of a functional and its usefulness in the world.