# How to write the ODEs of a PDE in Sturm-Liouville form, when there are no boundary conditions

Consider the PDE,

$$\frac{1}{\sin{\theta}}\frac{\partial}{\partial\theta}\bigg(\sin{\theta}\frac{\partial u}{\partial\theta}\bigg)+\frac{1}{\sin^2{\theta}}\frac{\partial^2 u}{\partial\phi^2}=-\lambda u.$$ We can do separation of variables to obtain the following ODEs: $$\frac{1}{f}\bigg(\sin{\theta}\frac{d}{d\theta}\bigg(\sin{\theta}\frac{df}{d\theta}\bigg)\bigg)+\lambda\sin^2{\theta}=m$$ $$\frac{1}{g}\frac{d^2g}{d\phi^2}=-m$$

In this case, the latter equation determines the possible values of $m$, and the former the possible values of $\lambda$, which we can find after knowing the constraints on $m$ by solving the latter.

Now consider the following instead: $$\frac{1}{\sin{\theta}}\frac{\partial}{\partial\theta}\bigg(\sin{\theta}\frac{\partial u}{\partial\theta}\bigg)+\frac{1}{\sin^2{\theta}}\frac{\partial^2 u}{\partial\phi^2}=\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}.$$

We can again do separation of variables to obtain the following:

$$\frac{d^2 T}{dt^2}=-c^2\lambda T$$

$$\frac{1}{\sin{\theta}}\frac{\partial}{\partial\theta}\bigg(\sin{\theta}\frac{\partial Y}{\partial\theta}\bigg)+\frac{1}{\sin^2{\theta}}\frac{\partial^2 Y}{\partial\phi^2}=-\lambda Y$$

My question is this:

There are no boundary conditions on $t$, which represents time (as far as I can tell), so we cannot determine $\lambda$ using the $T$ ODE. Instead, we must solve the first PDE I considered and determine $m$ and $\lambda$ using those solutions instead as there are well-defined boundary conditions (i.e. $f(\theta)=f(\theta+\pi)$ and $g(\phi)=g(\phi+2\pi)$). So, how do we write the $T$ ODE in Sturm-Liouville form, if there is no parameter it determines? Is it just an extraneous equation with parameter $\lambda$?

• Whether the constant for separation is quantized (discrete) or not has nothing to do with the method of separation of variables... it's not entirely clear what you're asking...? Solve the second equation exactly the same as you did the first... – ziggurism Dec 5 '17 at 2:48
• @ziggurism I suppose my question is, how would we write the the ODE for $T$ in Sturm-Liouville form, as there are no boundary conditions we can impose (as far as I can tell)? – quanticbolt Dec 5 '17 at 2:50