Hi,

Solve $\displaystyle \displaystyle u_t = k \Big( u_{xx} + u_{yy} + u_{zz} \Big) $ with $\displaystyle \displaystyle 0 < x < L, 0 < y < H, 0 < z < W$

IC's: $\displaystyle \displaystyle u(x,y,z,0) = f(x,y,z)$

not going to list the BC's here.

So, I start by letting

$\displaystyle \displaystyle u = \Phi (x,y,z) T(t)$

so $\displaystyle \displaystyle \frac{T'}{k T} = \frac{\Phi_{xx} + \Phi_{yy} + \Phi_{zz}}{\Phi} = - \lambda$

so we have $\displaystyle \displaystyle T' + \lambda k T = 0$ and $\displaystyle \Phi_{xx} + \Phi_{yy} + \Phi_{zz} + \lambda \Phi = 0$

Now I let $\displaystyle \Phi = X(x) Y(y) Z(z)$

so $\displaystyle X''YZ + XY''Z + XYZ'' + \lambda XYZ = 0$ (divide by $\displaystyle XYZ$)

but I'm not too sure how correct this is because if I carry on I get

$\displaystyle \displaystyle \frac{X''}{X} + \frac{Y''}{Y} + \frac{Z''}{Z} + \lambda = 0$......(1)

Now Im not sure how to proceed from here....I think I need to somehow add another two separation constants....but I'm not sure how to exactly...

when I try.... I might get something like

$\displaystyle \displaystyle \frac{X''}{X} = \frac{-Y''}{Y} - \frac{Z''}{Z} - \lambda = - \mu$

so I end up with $\displaystyle \displaystyle X'' + \mu X = 0$.

Now the answer in the text book says that I should end up with the three equations

$\displaystyle \displaystyle X'' + \mu X = 0$, $\displaystyle \displaystyle Y'' + \nu X = 0$, and $\displaystyle \displaystyle Z'' + (\lambda - \mu - \nu) Z = 0$

But I can't see how to get from ....(1)...to the three equations above. Can anyone please help me out?