The diffusion equation $u_t=\alpha u_{xx}$ has a similarity solution with $\eta=x/\sqrt{\alpha t}$. If $u(x,0)=e^{-x^2}$, find $u(0,1)/u(0,4)$ to 1 decimal place.
The intended similarity scaling for a diffusing localized profile has centre amplitude proportional to $t^{-1/2}$, so the ratio is $\sqrt{4/1}=2.0$. The printed initial profile and omission of α require a qualification: the exact infinite-line Gaussian solution is $u(x,t)=(1+4\alpha t)^{-1/2}\exp[-x^2/(1+4\alpha t)]$, giving ratio $\sqrt{(1+16\alpha)/(1+4\alpha)}$, not identically 2 for finite α. The official interval 1.9–2.1 reflects the intended similarity limit; no unique exact finite-time ratio follows from the supplied data.
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