Hello all!

I appreciate it if you can share any thoughts that you may have regarding how to solve the following PDE:

$\displaystyle \frac{\partial U(z,t)}{\partial t}+(1-z)\frac{\partial U(z,t)}{\partial z}=(\frac{1}{z}-1)\left(U(z,t)-U(0,t)\right)$

Initial condition:$\displaystyle U(z,0)=z^{K}$

$\displaystyle U(0,t)$ arises due to a boundary condition, but it is unknown and should be derived by solving the PDE itself (details are irrelevant).

A typical strategy would be using Method of Characteristics to tackle this. But $\displaystyle U(0,t)$ can't be dealt with.

Specifically, I am stuck when integrating both sides of the following subsidiary equation due to $\displaystyle U(0,t)$:

$\displaystyle \frac{dU(z,t)}{U(z,t)-U(0,t)}=(\frac{1}{z}-1)dt$

I greatly appreciate it if you could share any thoughts you might have regarding how I can proceed to solve this through this method or any other method.

Thanks!