Finite parallel impedance network general solution
You can use the Modified Nodal Analysis method for solving the nodal voltages, and then back-compute the branch currents.

The beauty of this technique is that it generalizes very well to any well defined netlist and is quite easy to program. This technique can give either an analytical or a numerical solution, depending on your solver, and can be used to solve DC, steady state AC, or full transient problems.

The first link gives good resources on how to solve linear DC and steady-state AC circuits, and if you want to solve linear transient problems I have a blog post about how to approximate the inductors and capacitors to solve a transient problem.

edit:

In response to the OP's comments,

Matrix inverses can be solved analytically, though the degree of difficulty increases drastically for larger matrices (note: difficulty = time, just follow Gaussian elimination).

Some methods for finding matrix inverses analytically:


Gaussian Elimination
LU Decomposition
QR Factorization


... take your pick. The reason I suggested using a computer (even if the computer is solving your system analytically for you) is because for any appreciable size matrix it will take you a very long time to solve by hand.

This is the exact solution if you have a discrete/finite number of nodes, and there's no way around this linear algebra problem.

Maxwell's equations can have a "simple" looking solution because they don't apply to a discrete/finite domain: they rely on continuous functions and thus all the tools of calculus are available.

If you can make the approximation that the number of nodes tends towards infinity and don't care for the discrete nature of your network, you can simulate your transmission line using the Telegrapher's Equations. These will have solutions of a similar form to Maxwell's equations. For example, the solution to a lossless transmission line is:
$$
V(x, t) = f_1(x-u t) + f_2(x+u t)
$$
$$
I(x, t) = \frac{f_1(x-u t)}{Z_0} + \frac{f_2(x+u t)}{Z_0}
$$
A.k.a. the homogeneous wave equation solution.

A lossy transmission line can be described as a non-homogeneous wave equation, who's solution can be found by applying Green's Function to find the particular solution and adding on the homogeneous solution, or by only using Green's Function (details omitted).

Edit 2:

One of the limitations with using Nodal Analysis is you need to fully specify the problem, i.e. specify the entire network and boundary conditions. You have not specified what's connected to the input/output of your "transmission" line, which is why I don't know what your full network will look like and consequently can't offer any advice other than the "general" linear algebra techniques.

In my blog post, I noticed that even though my entire network results in a non-tri-diagonal matrix, I could easily reduce it into one by not solving for IS at the same time as all the nodal voltages, and re-arranging the matrix form. Thomas's Algorithm provides a very efficient method for expressing the solution to a tri-diagonal system.