[SOLVED] Fourier transform to solve an ODE

Oct 2007
Consider the initial value problem value problem of finding \(\displaystyle u : \mathbb{R} \times [0, \mathbb{C}) \to \mathbb{C}\) such that

\(\displaystyle \partial_t u(x,t) - \partial^2_x u(x,t) = 0\) for \(\displaystyle x \in \mathbb{R}\) and \(\displaystyle t>0\),

\(\displaystyle u(x,0) = f(x)\) for \(\displaystyle x \in \mathbb{R}\),

given data \(\displaystyle f \in S(\mathbb{R})\).

Use the Fourier transform to derive a formula for a solution to be,

\(\displaystyle u(x,t) = (H_t * f)(x)\)

where H_t is the heat kernel and is defined as

\(\displaystyle H_t(x) = \frac{1}{\sqrt{4 \pi t}}e^{-x^2/4t}\)

So (skipping to the essentials...),

Taking Fourier transforms...

\(\displaystyle \hat{u}''(\xi) = -4\pi^2\xi^2 \hat{u} (\xi)\)


\(\displaystyle \hat{u}(\xi,t)\)

Hence our ODE becomes \(\displaystyle \partial_t \hat{u}(\xi,t) + 4\pi^2\xi^2 \hat{u} (\xi,t) = 0\)

So it's a bit easier now but I fail to see where the whole convolution part comes in.