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##:(adsbygoogle = window.adsbygoogle || []).push({});

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

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 equationanalytically without using separation of variables?

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# A Solving a PDE in four variables without separation of variables

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