Climate Policy

[--Pete]

Hi,

. . . in the age of Wellington with the 'Thin Red Line'."

I believe that that was the operator that transformed Napoleon from an emperor to a prisoner. Highly advanced math, indeed.:whistling:

. . . so we're back at the beginning.

OK, let's try an example.

First, the basis of the Fourier analysis is that any continuous periodic function can be expressed as a sum of a specific type of trigonometric terms. The important concept is that it is a sum of terms -- a linear combination.

Second, there are relationships (including some partial differential equations) that have multiple solutions such that the sum of the solutions is also a solution. Again, the important concept is that it is a sum of solutions -- again a linear combination.

So, in this example, if a general term of the Fourier series is a solution of the relationship, then so is the sum of all the terms. In this sense, the whole resulting Fourier series is the 'complete solution' to the relationship, and the problem is reduced to finding the coefficients of the Fourier terms.

However, if the second condition is not met -- if the relationship is such that the sum of two solutions is not a solution -- then the problem is said to be non-linear. A given trigonometric term might be a solution of the problem, but the sum of such terms is not. Thus, the Fourier analysis does not apply and is guaranteed to yield incorrect results.

Note that the fact that the Fourier analysis does not apply does not mean one cannot calculate a Fourier series for the problem. One can. Consider a problem which we can both model and observe.

If the problem is linear, we can calculate the Fourier series of the initial conditions, then we can calculate the time evolution of the Fourier series, and finally we can (in principle) calculate the final condition, a function that is the inverse of the series. If we did that, then the calculated and observed final conditions would be in agreement within experimental error.

If the problem is not linear, we could still go through the process of calculating a Fourier series, of propagating that series through time, and of calculating the inverse. The difference would be that the predicted final condition and the observed final condition would differ, the difference being greater the greater the non-linearity of the system is.

I'm not sure how useful all this is, but I hope it helps.

--Pete