TITLE: How to use Fourier's transform to solve differential equation
QUESTION [0 upvotes]: I have to solve following problem:
$$ u_t(t,x) = \Delta u(t,x) $$
$$ u(0,x) = f(x) $$
I've started:
$$ \frac{\delta}{\delta t} F(u(t,\xi))=F(u_t(t,\xi))=F(\Delta u(t,x))$$
and here I've stoped, because what I also know is:
$$ \frac{\delta^2u}{\delta x_k^2}=(i\xi_k)^2F(u) $$
Unfortunatelly here I've got $$ \Delta u(t,x) $$ not $$ \Delta_x u(t,x) $$
Could someone help me what should I do in the next step?

REPLY [1 votes]: Hint 2:
use a test function $e^{i(\omega t-k x)}$ and see what the differential operators do in this function.
Then the (whole) Fourier solution is the (function represented by the) sum over all these (fourier basis) functions.
In other words if the required solution is expanded in a 2-D fourier space of the above basis functions:
$$f(x,t) = \sum_k \sum_{\omega} c_{\omega k} \times e^{i(\omega t - k x)}$$
One needs only solve the diff. equation for each $e^{i(\omega t - k x)}$ and combine. (note the sums above are formal, they may represent integrals if needed)
In most cases of linear partial differential equations this also leads to a separation of parameters solution, where the equation is split into loosely-coupled parts each related to one parameter (i.e $x$ or $t$)
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