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 $\nabla^2 u$ = $T\nabla^2F$?
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.