# Heat equation

1. Feb 10, 2009

### dirk_mec1

1. The problem statement, all variables and given/known data
http://img3.imageshack.us/img3/5020/84513876dm0.png [Broken]

3. The attempt at a solution
I found that $$f(t) =exp \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right)$$

Is this correct?

Last edited by a moderator: May 4, 2017
2. Feb 11, 2009

### dirk_mec1

I thus only solved the ODE -problem.

3. Feb 11, 2009

### cristo

Staff Emeritus
Show your working, and it will be easier to check whether you are correct or not.

4. Feb 11, 2009

### dirk_mec1

OK, you got it:

$$\frac{ \partial{u} }{ \partial{t} } = \frac{df}{dt} \sin(\frac{m \pi x}{L})$$

$$\frac{ \partial{^2 u} }{ \partial{x^2} } = f(t) \left( \frac{-m^2 \pi ^2 }{L^2} \right) \sin(\frac{m \pi x}{L})$$

Therefore

$$f(t) =\mbox{exp} \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right)$$

Conclusion:
$$u(x,t)= u_0 + \mbox{exp} \left( - \frac{m^2 \pi ^2 \kappa t}{L^2} \right) \cdot \sin(\frac{m \pi x}{L})$$

5. Feb 11, 2009

### Redbelly98

Staff Emeritus
Looks good, I'm just wondering one thing.

Should ρ, c, and k be in there somewhere, or do we assume units such that those are all =1?