Finding Solution to Laplace Equation

[Nick R]

Usually, we use the technique of "separation of variables" as follows:

In a "separable coordinate system", we assume a separable solution

$$\Phi=A(a)B(b)C(c)$$

Then we obtain 3 ODEs for A(a), B(b), C(c)

We note that there are actually entire families of solutions to each ODE, that happen to be complete/orthonormal. So we can write A(a), B(b), C(c) as linear combinations of these guys:

$$A(a)=\sum E_n A_n(a)$$
$$B(b)=\sum F_n B_n(b)$$
$$C(c)=\sum G_n C_n(c)$$

Then we use the boundary conditions and exploit the orthogonality of $$A_n, B_n, C_n$$ to find the coefficents $$E_n, F_n, G_n$$. This involves integrating at the boundaries.

So we get

$$\Phi=\sum_i\sum_j\sum_k D_{ijk} A_i (a)B_j (b)C_k (c)$$

Where

$$D_{ijk} = E_i F_j G_k$$

OK. Easy.

NOW HERE COMES THE QUESTION

WHY can't I expand $$\Phi$$ in some other set of functions?

In general, I can expand any 3 variable function like so

$$\sum_n \sum_m \sum_l A_{nml} \zeta_n (x) \nu_m (y) \mu_l (z)$$

Where the coefficients are

$$\int \int \int \Phi \zeta_n^* (x) \nu_m^* (y) \mu_l^* (z) dx dy dz$$

Well, I suppose I answered my own question... You have to KNOW $$\Phi$$ to find the coefficients in this general case: but the technique of separation of variables allows us to find the coefficients without knowing $$\Phi$$ in the first place...

And the other way to do it would be to find a set of orthogonal functions that are eigenfunctions of the Laplace operator... Then we don't need to know $$\Phi$$ apriori to find the coefficients either - and this works for finding solutions to Poisson's equation as well (which is good because we can't use separation of variables there).

Because I spent so long thinking about this, I am going to press "submit" for the heck of it and see if I get any interesting comments. This was motivated by wondering WHY separation of variables actually works (without a doubt) and the answer seems to be, because it is really equivalent to expanding $$\Phi$$ in a complete set of functions (a very specific set of complete functions - so that we can actually find the coefficients in the linear combination).

 

[Nick R]

OK so yesterday I somehow lost sight of how you actually do separation of variables when I typed out the above.

Yesterday I said we assume

$$\Phi = \Phi=A(a)B(b)C(c)$$

Where

$$A(a)=\sum E_n A_n(a)$$
$$B(b)=\sum F_m B_m(b)$$
$$C(c)=\sum G_l C_l(c)$$

And then find coefficients using boundary conditions/orthogonality then write the answer as a triple sum.

No!

Actually, we find

$$\phi_{nm} = A_n(a)B_m(b)C_{nm}(c)$$

and represent $$\Phi$$ as a double sum of these two index "separable solutions"

$$\Phi = \sum_n\sum_m D_{nm}\phi_{nm}$$

And use orthogonality/boundary conditions to find the coefficients $$D_{nm}$$

This is distinctly different from the general expansion for a function of three variables. I.e. doing separation of variables and doing an expansion of a function in orthogonal functions are totally different things.

So apparently, the only way to know that separation of variables works is, if you find a solution that satisfies the boundaries, and that obeys $$\nabla^2\Phi = 0$$ in the region under consideration, then that's the only possible solution...

Gosh what a waste of time.