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Math Help - Comparison principle for inhomogeneous heat equation

  1. #1
    Junior Member beebe's Avatar
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    Aug 2011
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    Comparison principle for inhomogeneous heat equation

    I just finished this problem:
    "Prove the comparison principle for the diffusion equation: If u and v are
    two solutions, and if u ≤ v for t = 0, for x = 0, and for x = l, then u ≤ v
    for 0 ≤ t < ∞, 0 ≤ x ≤ l."

    Like so:
    max(u),min(u),max(v), and min(v) all occur on the boundary of the domain. max(v)>=max(u) and min(v)>=min(u) over the domain.
    Let w=v-u, then max(w)=max(v)-max(u)>=0
    and min(w)=min(v)-min(u)>=0
    so w>=0 in the domain, therefore w=v-u>=0 => v>=u in the domain

    Now the next problem stems off of this:
    More generally, if u_t − ku_{xx} = f, v_t − kv_{xx} = g, f ≤ g, and u ≤ v
    at x = 0, x = l and t = 0, prove that u ≤ v for 0 ≤ x ≤ l, 0 ≤ t <∞.

    And I'm totally stuck. My first thought was to try something similar with w=v-u and h=g-f and going from there, but that doesn't seem to lead anywhere helpful. I need a nudge to get me going on this guy. Thanks.
    Last edited by beebe; October 8th 2013 at 05:05 PM. Reason: fixing tex tags
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