# [SOLVED] Fourier transform to solve an ODE

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...

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

and

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

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.