It will help if the PDE is written completely - define variables, boundary conditions. Also, same symbols can be used in the FD scheme of your Fig 1 and your PDE. Finally, knowing how other variables (e.g. T) varies will also help.

Importantly, if the systems matrix is under-determined, then no method gives a unique solution.

Thank you Sanjay Kharche for your response.

It is a non-linear model of a cable whose vertical planar motion only is under consideration. Velocities [u, v], cable inclination angle ϕ and the elongation (strain component) ε are the dependent variables defined along the cable length. In the model, s is the length along the cable. The model involves both time and space derivatives.

Following is the PDE,

m[(∂u/∂t)-(v ∂ϕ/∂t)]=-0.5 ρa dπ CDt u|u|(1+ε)+(∂T/∂s)-wc sin⁡ϕ

m(∂v/∂t+u ∂ϕ/∂t)=-0.5 ρa dπ CDn v|v|(1+ε)+T (∂ϕ/∂s)-wc cos⁡ϕ

with their compatibility equations,

(∂u/∂s)-(∂ϕ/∂s)*v=(∂ε/∂t)

(∂v/∂s)-(∂ϕ/∂s)*u=(∂ϕ/∂t)*(1+ε)

Considering Y=[u v ϕ ε]^T as a vector, the governing equation can be written as in figure 2.

By applying the finite difference method on this equation yields the discrete form as shown in figure 1 (in the question).

I want to solve for the (k+1)th value of Y. There are boundary conditions for the cable. The lower end is having zero velocity and the upper end is having the same velocity as of a vehicle (aerostat) as shown in figure 3.

I don't know this specific physical problem but you have to be aware of the mathematical character of this system of partial differential equations. Try to see from the matrix form if the eigenvalues problem admits real solutions and the system is hyperbolic. That will define the type of BCs and the numerical discretization.

Hint: The second compatibility equation can be substituted directly in the first two equations so that you have only time derivatives of u and v.