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I would love to get professional answers on this in general. In the meantime here's my crude attempt using the Simple Harmonic Oscillator as an example..

Consider a function of an integer variable defined by this..

$f(n)=k*f(n-1)-f(n-2)$

where $n=2,3,4,5,..$ and $k$ is a constant.

If $k$, $f(0)$ and $f(1)$ are given then $f(n)$ can be calculated for any $n$.

This equation is an example of a difference equation. An area of mathematics called the Theory of Finite Differences (or Difference Calculus) tells how to solve these equations. The solution is a nice surprise..

$f(n)=sin(na)$

It's the famous sine function, where $a$ is a constant related to $k$, $k=2*cos(a)$. So the solution is a wave and if you plot $f(n)$ for $n=2,3,4,5,...$ you get the beautiful sine wave, and the three numbers $k$, $f(0)$ and $f(1)$ determine the amplitude, wavelength and phase of the wave.

What about $n$? It plays the role of time, because at time=$n$ the function $f(n)$ is the displacement from the origin for a Simple Harmonic Oscillator.

So the difference equation $f(n)=k*f(n-1)-f(n-2)$ replaces the second order differential equation used to describe the Simple Harmonic Oscillator. It's a nice example of Difference Calculus in action.

Of course, things are not exactly the same.. time is no longer a continuous variable!

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    $\begingroup$ Most high school physics is taught with very little calculus. Knowing calculus makes it much much easier. More advanced physics becomes very tedious to say the least without advanced math. $\endgroup$ – Jon Custer Sep 9 '16 at 13:12
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    $\begingroup$ @JonCuster, to be fair, functional equations are very advanced math. Much more complicated and less known than DE, as far as I know $\endgroup$ – Yuriy S Sep 9 '16 at 13:13
  • $\begingroup$ Hi Ken, imo the effort involved in trying to do physics without calculus would be far greater in the long run, than biting the bullet and learning it. I am saying this in general, it's not directed at you personally. And how much more could Faraday have achieved if he did learn calculus... $\endgroup$ – user108787 Sep 9 '16 at 13:15
  • $\begingroup$ You can "do" quite a lot of classical physics (in the sense of explaining common physical phenomena) with very little mathematics. But it is hard to see how quantum mechanics or general relativity could be derived and described without calculus. $\endgroup$ – Floris Sep 9 '16 at 13:17
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    $\begingroup$ I see that most of the comments are missing the point. The OP is interested in the application of functional equations in physics, rather than differential equations. This is quite a complicated topic, and not at all related to the 'simplification' of mathematical methods in physics. It is possible that the OP doesn't understand it himself though. Considering this quote: "The solution is a nice surprise". Difference equations and functional equations in general is not a simple topic. It's very advanced $\endgroup$ – Yuriy S Sep 9 '16 at 13:22
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It is possible to do physics without differential calculus, at least named as such, although I wouldn't recommend it.

Early physics was done almost purely geometrically, as Euclid's geometry was the template of mathematics for quite a while. It is possible to perform physical predictions (at least for classical mechanics) using purely geometric axioms (such as the Hilbert axioms). There have been modern attempts to do this, such as in Science without Numbers. While the work is only done for classical mechanics it is believed that the formalism could be extended generally to other physical theories.

Of course that formalism is very inefficient.

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  • $\begingroup$ Thanks for your answer. "Science Without Numbers" is interesting - but a bit extreme for me. I would be totally happy if someone could write the EFE as functional equations ;-) $\endgroup$ – Ken Abbott Sep 9 '16 at 17:09

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