# Thread: Sep of variables in PDE

1. ## Sep of variables in PDE

Hi,

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

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

not going to list the BC's here.

So, I start by letting

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

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

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

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

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

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

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

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

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

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

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

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

2. Your next equation after (1) works just fine. Just continue in that vein:

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

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

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

Can you finish?

3. Can you finish?
Yes I can. Thanks for your help.

4. Great. You're welcome, and have a good one!