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Can someone please help me with the following problem?

The heat flow equation is $\nabla^2 u = \frac{1}{\alpha^2}\frac{\partial u}{\partial t}$, where $u(x,y,z,t)$ is the temperature, $\alpha^2$ is constant and $t$ is time.

Separate variables in the heat equation by assuming a solution in the form $(x,y,z,t)$ = $F(x,y,z)T(t)$, and obtain two distinct equations, one for F and another for T by using a separation constant. Solve for T (t), and explain physically how the sign of separation constant should be chosen.

I assume the right hand side can be written as $\frac{1}{\alpha^2}\frac{\partial F(x,y,z)T(t)}{\partial t}$ = $\frac{F}{\alpha^2}\frac{dT}{dt}$, however Im stuck on what to do with the Laplacian on the left hand side. Can I write $\nabla^2 u$ = $T\nabla^2F$?

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Of course, since:

$$\begin{align}\nabla^2 u = & \nabla \cdot \nabla u = \\ = &\nabla \cdot \nabla (F(x,y,z) \, T(t) ) = \\ = & \nabla \cdot (T(t) \, \nabla F(x,y,z)) = \\ = & T(t) \, \nabla \cdot \nabla F(x,y,z) = \\ = & T \, \nabla \cdot \nabla F = \\ = &T \, \nabla^2 F, \end{align}$$ which can be also proved using index notation (and assuming cartesian coordinates):

$$ \nabla \cdot \nabla u = \partial_{x_j} \partial_{x_i} u = \partial_{x_j} \partial_{x_i} (T(t) \, F(x_1,x_2,x_3)) = T \, \partial_{x_j} \partial_{x_i} F = T \, \nabla^2 F, \quad i,j = 1,2,3;$$

since $t$ and $\mathbf{x} = (x_1,x_2,x_3)$ are independent variables.

Hope this helps!

Cheers.

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  • $\begingroup$ That's great, thanks! $\endgroup$ – Mr Man May 4 '14 at 19:27
  • $\begingroup$ You're welcome @MrMan. Now the next step would be to set $F(x,y,z) = P(x) Q(y) R(z)$ and solve for one of them. Then expand in terms of the eigenfunctions of the problem, but... this is not being asked in the question. Let us know your progress. Cheers! $\endgroup$ – Dmoreno May 4 '14 at 19:59

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