## [SOLVED] Fourier transform to solve an ODE

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

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

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

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

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

$u(x,t) = (H_t * f)(x)$

where H_t is the heat kernel and is defined as

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

So (skipping to the essentials...),

Taking Fourier transforms...

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

and

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

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

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