Consider the initial value problem value problem of finding such that

for and ,

for ,

given data .

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

where H_t is the heat kernel and is defined as

So (skipping to the essentials...),

Taking Fourier transforms...

and

Hence our ODE becomes

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