System of Partial differential equations

Hi,

I want to solve the attached system of partial differential equations. I have tried to link the variables here. Is it possible to solve this system, or are there any missing links? I am aware that I have not given the boundary conditions. For this system, what boundary conditions do I need? And what kind of solver would I use to solve this system? If I want to solve this system in steady state (d/dt = 0), what would I do differently?

Furthermore, I have looked at the Method of Lines to discretize a system. Is it possible to discretize the system with respect to all the variables? (not just with respect to one dimension). I want to do that because I want to eventually optimize this system on another software. That software does not deal well with differential equations.

I do not have much experience with MATLAB. I have used ode45 in the past, but never solved a system of partial differential equations. Any help would be greatly appreciated.

[Zain Shabeeb]

To clarify, x and z are spatial directions perpendicular to each other. I also realized that it is not possible to use pdepe for all three independent variables (t,z,x). I am okay with solving in steady state (d/dt in first two equations = 0). But please do let me know the procedure to use in the case of a system depending on time and 2 space variables.

[Zain Shabeeb]

One correction: the 'K' in the second equation is not the same as the 'K' in the first equation. The 'K' in the second equation is a variable dependent on c1(dash), which is the time dependent variable in the second equation. Please refer to the snapshot attached in this comment.