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?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

Have something to add?

Draft saved
Draft deleted

Loading...

Similar Threads for Solving four variables | Date |
---|---|

A Solving an ODE Eigenvalue Problem via the Ritz method | Mar 14, 2018 |

A Solve a non-linear ODE of third order | Feb 20, 2018 |

A Solving linear 2nd order IVP non-constant coefficient | Jan 10, 2018 |

I Method for solving gradient of a vector | Jan 3, 2018 |

Numerically solving system of four PDEs | Jan 21, 2015 |

**Physics Forums - The Fusion of Science and Community**