(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Solve:

∂u/∂t = k ∂^{2}u/∂x^{2}- ζu

with the initial condition

u(x,0) = f(x)

where k and ζ are constants.

x is on an infinite domain.

2. Relevant equations

Define Fourier transforms:

f(x) = ∫[-∞,∞]F(w)e^{-iwx}dw

F(w) = 1/2∏ ∫[-∞,∞]f(x)e^{iwx}dx

From tables of Fourier Transforms:

∂^{2}f/∂x^{2}= (-iw)^{2}F(w)

3. 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(∂^{2}u/∂x^{2}) - ζF(u)

I will call F(u) = U*

∂U*/∂t = w^{2}k U* - ζU*

∂U*/∂t = (w^{2}k - ζ) U*

So the general solution is:

U* = C(w) e^{(w2k-ζ)t}

Where C(w) = 1/2∏ ∫[-∞,∞]f(x)e^{iwx}dx

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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# Homework Help: Solve the PDE by Fourier Transforms

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