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Infinite dimensional PDE

  1. Jan 3, 2008 #1
    Is there any established theory concerning infinite dimensional PDE?
  2. jcsd
  3. Jan 3, 2008 #2
    Do you mean that the function has infinitely many variables, or that it is an infinite dimensional function of a finite number of variables?
  4. Jan 3, 2008 #3
    Infinitely many variables.

    For example a quantum mechanical real Klein-Gordon field, if I have understood correctly, can be pretty much described by the infinite dimensional non-homogenous heat equation (the Shrodinger's equation, with certain constants and with the harmonic potential). Something like this

    i\partial_t \Psi(t,\phi) = \sum_{k\in\mathbb{R}^3} \Big(-\alpha \partial^2_{k} + \beta |k|^2\Big)\Psi(t, \phi)



    It can be solved by a separation attempt

    \Psi(t,\phi) = \prod_{k\in\mathbb{R}^3} \Phi_k(t) \Psi_k (\phi(k)),



    This is total honest pseudo mathematics, motivated by physics, don't complain about it! :biggrin:

    In fact his is a very vague example with uncountable set of variables. There could be more rigor examples with only countably many variables.
    Last edited: Jan 3, 2008
  5. Feb 29, 2008 #4
    It could be these are supposed to be called functional differential equations, but I'm not sure. Some quick google hits were slightly confusing.
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