Solving a PDE in four variables without separation of variables

[highflyyer]

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?

 

[Gigaz]

Can you give a bit more context, please? Is that a physics problem? And why would you not want to separate variables?

 

[highflyyer]

Can you give a bit more context, please? Is that a physics problem? And why would you not want to separate variables?

This is, in fact, a physics problem. The physics is that the cylinder is a radially cut-off (at radius $\rho=1$) anti-de sitter cylinder in global coordinates.

Please ignore the following (in italics) if you are not interested. This is just motivation and is just some dense physics that's of little use to solving the differential equation above :

I am solving for the heat kernel $K$ taking inspiration from section 1.1 of the paper https://arxiv.org/abs/0804.1773.

As stated in that paper, the computation of the one-loop correction in equation (1.9) requires a computation of the spectrum $\lambda_{n}$. The computation of the spectrum can be circumvented if we solve equation (1.11) for the heat kernel and plug it into the equation just above equation (1.11). Using separation of variables is just going to lead us back to equation (1.9) and defeat the purpose of using the heat kernel approach.

What I am looking for is some means of solving the differential equation analytically without separation of variables. That's all. Please do not let the abstruse physics that I wrote in this post deter you from adding any useful hints to solving the very simple differential equation above.

Thanks in advance for any useful comments. :)

 

[Gigaz]

What I am looking is some means of solving the differential equation analytically without separation of variables. That's all.

Then all I can tell you is this: Partial differential equations are really complicated. As a general rule, many of the most common PDE's (Schrödinger, Boltzmann, Laplace, Einstein, Navier-Stokes...) have entire books written on the solution of just this particular PDE and still in most cases you have to use numerical methods. If you have a good intuition about the physics of the system you can make a smart ansatz that maybe solves the PDE. Sometimes you can do a Greens function that just leaves you with a complicated integral. But you need to find the Greens function for this particular PDE.

 

[highflyyer]

Then all I can tell you is this: Partial differential equations are really complicated. As a general rule, many of the most common PDE's (Schrödinger, Boltzmann, Laplace, Einstein, Navier-Stokes...) have entire books written on the solution of just this particular PDE and still in most cases you have to use numerical methods. If you have a good intuition about the physics of the system you can make a smart ansatz that maybe solves the PDE. Sometimes you can do a Greens function that just leaves you with a complicated integral. But you need to find the Greens function for this particular PDE.

If Green's function analysis just leaves a complicated integral, I'd rather skip that approach.

Is there any way this equation looks similar to one of the many PDEs that's already been solved?