Consider the one dimensional heat equation on a semi-infinite bar such that
$u_t=u_{xx}$
with initial condition of
$u(x,0)=0$.
What is the solution for all x and t if the boundaries are
$u(0,t)=e^{-t}$ and $u(x,t)=0$ as x approaches infinity?
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You must take the 'Laplace Transform of your equation' but, in that way, it's not clear hot to handle "$\displaystyle\mathrm{u}\left(x,t\right) = 0$ as x approaches infinity". – Felix Marin Nov 14 '18 at 16:58