1. ## PDE

Ut=aUxx+cU
U(x,0)=f(x)
U(0,t)=0
Ux(1,t)=0

(Ut - derivative of U according to t
Uxx - second derivative of U according to x)

Question:
Find the smallest C possible, for which the solution U(x,t) does not increase at t increases, for every f(x).

any help would be appreciated ! I have no idea how to begin..

2. Originally Posted by zokomoko
Ut=aUxx+cU
U(x,0)=f(x)
U(0,t)=0
Ux(1,t)=0

(Ut - derivative of U according to t
Uxx - second derivative of U according to x)

Question:
Find the smallest C possible for which the solution U(x,t) does not increase at t increases.

any help would be appreciated ! I have no idea how to begin..
If you use the transformation $u = e^{ct}v$ your problem becomes

$
v_t = a v_{xx}
$

$
v(x,0)=f(x),\;\;v(0,t)=0,\;\;v_x(1,t)=0.
$

Solve with the usual separation of variables and consider the first term of the series solution as the other will die out faster. This should you an idea on the choice of $c$.

3. thank you very much !