# Conduction of Heat

• Dec 23rd 2009, 09:31 AM
Apprentice123
Conduction of Heat
Find a general formula for the temperature $\displaystyle u(x,t)$ in the form of a series with general formula for the coefficients of the series.

$\displaystyle \alpha^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}$ with $\displaystyle 0 < x < L$; $\displaystyle t>0$

$\displaystyle u(0,t) = 0$
$\displaystyle \frac{\partial u}{\partial x} (L,t) = 0$ with $\displaystyle t>0$

$\displaystyle u(x,0) = f(x)$

How can I resolve ?
• Dec 23rd 2009, 10:39 AM
galactus
If you look around, you can find this worked out in a DE book or on line.
• Dec 23rd 2009, 05:19 PM
arbolis
Quote:

Originally Posted by Apprentice123
Find a general formula for the temperature $\displaystyle u(x,t)$ in the form of a series with general formula for the coefficients of the series.

$\displaystyle \alpha^2 \frac{\partial^2 u}{\partial x^2} = \frac{\partial u}{\partial t}$ with $\displaystyle 0 < x < L$; $\displaystyle t>0$

$\displaystyle u(0,t) = 0$
$\displaystyle \frac{\partial u}{\partial x} (L,t) = 0$ with $\displaystyle t>0$

$\displaystyle u(x,0) = f(x)$

How can I resolve ?

Here's another guy from Brazil that posted exactly the same question 2 minutes after you did. He received a (great!) reply.
See Conduction of Heat.
• Dec 24th 2009, 04:15 AM
galactus
Zill's Differential Equations with Boundary Problems has this problem outlined. Except, they use k instead of $\displaystyle {\alpha}^{2}$.

That is all I meant. This is a common DE problem and can be found.
• Dec 24th 2009, 07:41 AM
arbolis
Quote:

Originally Posted by galactus
Zill's Differential Equations with Boundary Problems has this problem outlined. Except, they use k instead of $\displaystyle {\alpha}^{2}$.

That is all I meant. This is a common DE problem and can be found.

Maybe TPH did it also... "better than the book". (Happy)
See http://www.mathhelpforum.com/math-he...-equation.html. This thread is worth to be a sticky one I believe.