# Solving a PDE in four variables analytically without using separation of variables

Within a cylinder with length $\tau \in [0,2\pi]$, radius $\rho \in [0,1]$ and angular range $\phi \in [0,2\pi]$, we have the following equation for the dynamics of a variable $K$:

$$\left( - \frac{1}{\cosh^{2} \rho}\frac{\partial^{2}}{\partial\tau^{2}} + (\tanh\rho + \coth\rho)\frac{\partial}{\partial\rho} + \frac{\partial^{2}}{\partial\rho^{2}} + \frac{1}{\sinh^{2} \rho}\frac{\partial^{2}}{\partial\phi^{2}} - m^{2} \right) K = \frac{\partial K}{\partial u}.$$

Here, $u$ is the time variable. I need to solve this differential equation subject to the boundary conditions

$$K(\tau = 0) = K(\tau = 2\pi) = 0$$ $$K(r = 1) = 0$$ $$K(\phi=0) = K(\phi=2\pi)$$ $$K(u = \infty) = 0$$

The first two boundary conditions simply state that the variable $K$ vanishes at the boundary of the cylinder, the third boundary condition is simply a periodic boundary condition on the angular coordinate and the final condition simply states that the variable $K$ must vanish at late times.

How do I solve this differential equation analytically without using separation of variables?

• Your time constraint might lead to a good use of a Laplace transform – DaveNine Dec 11 '17 at 18:32
• I’m not sure why you’d want an explicit solution to this with such a bad looking left hand side, actually. Can you provide more context? – DaveNine Dec 11 '17 at 18:34