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Method of successive approximation

  1. May 16, 2005 #1
    Last semester in my course QM 2 we discussed the Rayleigh-Schrödinger perturbation theory. A very elegant theory, based on a general principle: when terms are depending on a certain factor to the nth order, where the factor is very small if not infinitesimally small, you can collect the terms that have the same nth order dependence, and that state that:

    the terms of the 0th order are the most prominent
    the terms of the 1th order are second most prominent

    (I'm a bit clumsy at wording this in English, my apologies.)

    Now, this semester I'm working on a individual project about thermoacoustics, and I'm using a similar method of perturbation theory - the principle of order division is the same, only the context is different. I've been using this trick without ever questioning where it came from - however, it's time to write a report about it, so I feel I really should know where it came from. However, I can't seem to find a good site about this, and considering I don't even know the name of such a theory, it's not easy to look for it either.

    So, my question is: what is the name of the abstract theory that describes this method of succesive approximation? It's always nice if you can provide a lucid link as well, but providing me a name would be a great help already. Thanks.
  2. jcsd
  3. May 16, 2005 #2


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    I can't really help you with the name, sorry. It is really the core of perturbation theory, but I can't really put a name on it. You could look up Fredholm or Volterra integral equations, but I don't know if that's what you're looking for.

    However, I'd like to point out something else. The fact that you can equate the different orders has nothing to do with your parameter being "small". The smallness of the parameter only comes in to cut off the series.
    But the idea behind the perturbation series is in fact different. Instead of solving ONE problem, you solve a WHOLE family of problems, parametrized in your small parameter (say, lambda). If you can now write out equations for that whole family of problems, then things should be ok, because the solution you're looking for is then found by putting in your "correct" value of lambda. But if you leave lambda "free" because you look for the whole family of solutions, and you arrive at an equation:

    blabla + lambda . blah1 + lambda^2 . blah2 +... = blublu + lambda. bluh1 +...

    then you are in fact solving a POLYNOMIAL equation in lambda, which looks trivial:

    (blabla - blublu) + lambda . (blah1 - bluh1) + lambda^2 (blah2 - bluh2) + ... = 0.

    Now, for a polynomial to be 0 FOR ALL VALUES OF LAMBDA, this can only happen when all of its coefficients, individually, are 0.

    So from this follows that:
    blahblah = blublu
    blah1 = bluh1

    which are nothing else but your perturbative equations of different order.

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