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Define f(x) where it is odd and even at two points

  1. Sep 24, 2011 #1
    1. The problem statement, all variables and given/known data:

    A function f : R → R is called “even across x∗ ” if f (x∗ − x) = f (x∗ + x) for every x and is called “odd across x∗ ” if f (x∗ − x) = −f (x∗ + x) for every x. Define f (x) for 0 ≤ x ≤ ℓ by setting f (x) = (x^2) . Extend f to all of R (i.e., define f (x) for all real x) in such a way that it is odd across x∗ = 0 and even across x∗ = ℓ.


    2. Relevant equations:

    1. conservation equations (transport): concentration, flux
    (a) flow (flux = vu ) — fluid, traffic, etc.
    (b) mixing: diffusion/dispersion (probability; flux = D ∇u )
    reaction/diffusion systems
    2. mechanics (Newton’s 3rd Law): force, potential energy, momentum
    (a) wave equation; ICs and BCs
    (b) beam, plate equations
    3. steady state (equilibrium: balance equations)
    4. some other examples . . . (e.g., Cauchy-Riemann equations)

    **Also studying the heat equation/etc**


    3. The attempt at a solution:

    I understand the difference between "even" and "odd". I have created the following:
    f(ℓ-x)=f(ℓ+x) even at "ℓ"
    f(0-x)=-f(0-x) OR f(-x)=-f(x) odd at "0"

    I think I need to use the above IC's to setup the BC's. Once I have the BC's determined I am not sure how to combine them to find the full equation for f (x) or what to do with f (x) = (x^2).

    Please help!!!
     
  2. jcsd
  3. Sep 24, 2011 #2

    LCKurtz

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    Can you draw a picture of the graph that is y = x2 on [0,ℓ] and that is odd across 0 and even across ℓ? That's your first step.
     
  4. Sep 25, 2011 #3
    Thank you for your help. I was able to plot the graph for x^2 and the odd/even BC's not sure where to go from there. Any help is appreciated.
     
  5. Sep 25, 2011 #4

    LCKurtz

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    If your function comes out to be periodic (did it?) all you have to do is define it over one period and extend it periodically.
     
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