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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.

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    $\begingroup$ An example is the proof that the equilibrium functions of Boltzmann equation are Gaussians. $\endgroup$
    – LL 3.14
    Nov 2, 2020 at 23:47
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    $\begingroup$ "A simple form of functional equation is a recurrence relation." Now primitive recursive functions are big part of computational theory. "The commutative and associative laws are functional equations." These are foundations of whole algebra. Also, differential equations are functional equations too. $\endgroup$
    – Przemek
    Nov 2, 2020 at 23:51
  • $\begingroup$ Thanks for the replies both! I am newer to math, so I am looking for a more "elementary" topic. Also, it would be great if it could be applied to some field other than mathematics. $\endgroup$ Nov 2, 2020 at 23:52

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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.

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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.

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