Thread: PDE

**zokomoko** · February 6th 2010, 08:22 AM

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..

**Danny** · February 6th 2010, 08:29 AM

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

$<br /> v_t = a v_{xx}<br />$

$<br /> v(x,0)=f(x),\;\;v(0,t)=0,\;\;v_x(1,t)=0.<br />$

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 $c$ .

**zokomoko** · February 6th 2010, 08:52 AM

thank you very much !

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