In Response To 'Re: Re: Re: Re: PDE with two variable'
NDSolve wants to return a single unique numeric solution.
A first order DE needs one initial condition and a second order DE needs two initial conditions to do this. Thus you need one more initial condition for z.
With your earlier code I experimented with your two initial conditions for z and got different errors, depending on how I tried to specify the conditions.
Perhaps you can find an initial condition that satisfies your problem and satisfies NDSolve.
Getting Mathematica to accept initial conditions for DSolve and NDSolve has been a problem for many many users over the years.