Solution
= Solution
Setting $\epsilon=0$ makes the inhibitor equation quasistatic:
$$
0=v_{xx}+u-v.
$$
Taking the spatial <Fourier transform> and using the <Fourier transform of a derivative> gives
$$
0=-k^2\widehat v+\widehat u-\widehat v.
$$
Therefore the <fast-inhibitor elimination in a reaction-diffusion system> is
$$
\boxed{\widehat v(k,t)=\frac{\widehat u(k,t)}{1+k^2}.}
$$