Series solution to 2D Poisson's equation on a rectangle with Dirichlet boundary conditions.

Question

I've been following this link in order to try to solve Poisson's equation on a rectangle $[L_x, L_y]$:
begin{equation}
left(frac{partial^2}{partial x^2} + frac{partial^2}{partial y^2}right)varphi(x, y) = f(x, y)
end{equation}
with $varphi = 0$ on the boundaries. They solve the eigenvalue problem $Delta varphi = lambda varphi$ to find that $varphi(x, y) = sin(k_x x)sin(k_y y)$ for $k_x = mpi/L_x$ and $k_y = npi/L_y$. Then they claim that $f(x, y)$ may be expanded as
begin{equation}
f(x, y) = sum_{k_x, k_y} B_{k_x, k_y} sin(k_x x)sin(k_y y)
end{equation}
Then go on to use this to solve Poisson's equation.
I am confused, however, because it seems like we have only pinned $varphi$ at the boundaries, and said nothing about what values $f(x, y)$ should take. However, based on this series expansion it seems impossible that $f(x, y)$ can be anything other than zero at the boundaries. How would we expand an $f(x, y)$ that is nonzero at the boundaries? Is there even necessarily a solution for such an $f(x, y)$? Does it even matter what values it takes at the boundaries? And what if we had a source term which looked like $partial_x f(x, y)$? Could we expand it with:
begin{equation}
f(x, y) = sum_{k_x, k_y} B_{k_x, k_y} cos(k_x x)sin(k_y y)
end{equation}

Kajelad · Accepted Answer

There are some subtleties in this procedure that haven't been addressed. When we say the set
$$
left{sinleft(frac{npi x}{L_x}right)sinleft(frac{npi x}{L_y}right):n,minmathbb{N}right}
$$
forms a basis, what is meant is that it forms a basis for $L^2([0,L_x]times[0,L_y],mathbb{R})$. This space consists of equivalence classes of square integrable functions, that is, two functions $f$ and $g$ are considered equivalent in $L^2$ if $int|f-g|^2=0$, or equivalently if $f=g$ alomst everywhere. In this sense, every square integrable function, even those that do not vanish on the boundary, may be uniquely written as
$$
f(x)equivsum_{n,m}B_{n,m}sinleft(frac{npi x}{L_x}right)sinleft(frac{npi x}{L_y}right)
$$
with the understanding that $equiv$ denotes equivalence in $L^2$. This sum may not converge pointwise, and indeed will not if $f(0)neq 0$. However, with this caveat in mind, there's no loss of generality in writing a square integrable function this way.
When doing calculus on $L^2$ spaces, one should keep in mind that these functions need not be continuous, let alone differentiable (though if a class has a continuous representative, it is unique). Especially in physics, it's common to simply assume that derivatives exist and are will behaved w.r.t. summation; this can ususally be justified after the fact. Making this process more rigorous is the domain of functional analysis.

Series solution to 2D Poisson's equation on a rectangle with Dirichlet boundary conditions.

One Answer

Add your own answers!

Ask a Question