Suppose we have Laplace's equation for some $u(x,y)$ as:
$$\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$$
for $x \geq 0$, and $0 \leq y \leq a$, and that $u(x,y) \to 0$ as $x \to \infty$. We have the boundary conditions:
$$u(x,0) = u(x, a) = 0$$ $$u(0, y) = \sin (\pi y/a) + 2 \sin (2\pi y /a )$$
I want to solve this equation using separation of variables.
What I've done:
- I let $u(x,y) = X(x)Y(y)$ to transform the equation into
$$ Y\frac{d^2 X}{dx^2} + X\frac{d^2 Y}{dy^2} = 0$$
- Then I solve for the following ODE:
$$ \frac{d^2 X}{dx^2} - \lambda X = 0$$ $$ \frac{d^2 Y}{dy^2} + \lambda Y = 0$$
where we have
$$ \frac{1}{X} \frac{d^2 X}{dx^2} = - \frac{1}{Y} \frac{d^2 Y}{dy^2} = \lambda$$
However note that in this step I have no clue where the $\lambda$ eigenvalue comes from as it is something I adopted from a similar question.
- Till now I have no idea how to apply the boundary conditions or to check if $\lambda$ should be positive or negative in order to express $X$ or $Y$ as exponential and sinusodial functions respectively.
How should I proceed?
If you have an equation $f(x)=g(y,z,\cdots)$, and there is no $x$ dependence on the right, then fixing $y,z,\cdots$ results in the right side being constant in $x$, which means the left side must be constant in $x$. So there is a constant $\lambda$ such that $f(x)=\lambda$, $\lambda=g(y,z,\cdots)$. The permissible constants $\lambda$ are then determined by the equation $f(x)=\lambda$ and endpoint conditions in $x$. Once the possible values of $\lambda$ are known, then must solve $\lambda=g(y,z,\cdots)$ for all such possible $\lambda$. Now you separate again, and find a new parameter $\mu$. Etc.