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A Fourier Transform for 3rd kind of boundary conditions?

  1. Nov 18, 2017 #1
    I am studying online course notes from University of Waterloo on 'Analytical mathematics in geology' in which the author describes a 'modified fourier transform' which can be used to incorporate 3rd kind of boundary conditions. The formula is
    ## \Gamma \small[ f(x) \small] = \bar{f}(a) = \int_{0}^{\infty } f(x) [a \cos(ax) + h\sin(ax)] dx ##
    where $$ \Gamma $$ be the fourier operator.
    with operational property

    ##\Gamma [\frac{d^2f}{dx^2}] = -a^2 \bar{f}(a) - a [ \frac{df}{dx}|_{x=0} - hf|_{x=0} ]##

    I am trying to look for the detailed derivation of this equation in literature but have not found so far! Can anybody tell me in which book I can find detailed derivation of this formula and possibly with application?
  2. jcsd
  3. Nov 20, 2017 #2
    The formula is straightforward to derive it by adopting your definition of ##\Gamma## for ##d^2/dx^2## and using integration by parts twice. How you tried to derive the formula yourself?
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