Solving system of nonlinear equations using MATLAB (or NumPy/SciPy, or Mathematica)

I seek help replicating a result in the attached paper, using MATLAB to solve a system of nonlinear equations. Specifically, I seek to replicate Table II. This comes from solving the system of equations given by (11) and (12) on p. 266, along with the data in Table I. If the work is good on this initial replication, I will have some additional work available on an extension of the results. I prefer MATLAB, but will also accept proposals to do the work in NumPy/SciPy or Mathematica. The attached code reflects the work of another freelancer who started on the project, and mostly wrote the setup correctly, but could not finish the actual solution of the nonlinear equations. You can use it as a starting point. With your proposal, it is helpful if you can give me an example of a project you completed which was similar (ideally with sample code).