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 = 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.