For a function $f(x,t)$ with given initial condition $f(x,0)$ and which satisfies $$\frac{\partial f}{\partial t}=\alpha\frac{\partial^2 f}{\partial x^2}$$
and $\left\vert f(x,t)\right\vert \to 0, \ \left\vert \partial f/\partial x\right\vert \to 0$ as $\vert x\vert \to \infty$ for all $t$, use Fourier transforms to derive the solution for $f(x,t)$ in terms of $f(x,0).$ Evaluate this explicitly in the case $f(x,0)=\exp(-x^2)$.
I assume you're supposed to take the Fourier transform of each side.
I've got that the Fourier transform of the left hand side is equal to $-\alpha k^2\tilde f(k)$ but I'm not sure how to evaluate the left hand side $(\int_{-\infty}^\infty \frac{\partial f}{\partial t} dx)$ or how to put it in terms of $f(x,0)$. Sorry if I'm missing something obvious.