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wxstall
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Homework Statement
Solve:
∂u/∂t = k ∂2u/∂x2 - ζu
with the initial condition
u(x,0) = f(x)
where k and ζ are constants.
x is on an infinite domain.
Homework Equations
Define Fourier transforms:
f(x) = ∫[-∞,∞]F(w)e-iwxdw
F(w) = 1/2∏ ∫[-∞,∞]f(x)eiwxdxFrom tables of Fourier Transforms:
∂2f/∂x2 = (-iw)2F(w)
The Attempt at a Solution
I have little experience with transforms so please don't berate me if this is completely wrong.
Began by taking transform of entire eqn:
F(∂u/∂t) = k F(∂2u/∂x2) - ζF(u)
I will call F(u) = U*
∂U*/∂t = w2k U* - ζU*
∂U*/∂t = (w2k - ζ) U*
So the general solution is:
U* = C(w) e(w2k-ζ)t
Where C(w) = 1/2∏ ∫[-∞,∞]f(x)eiwxdx
Is this even remotely correct? It seems too easy, but then again, I suppose that is the point of transforms, to put something into a simpler form.
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