It is not an error. It is a fact of life.
It is trivial to formulate equations or systems of equations that have no analytical solution, or at least no solution that could be found. In fact, you did just that.
It always helps to apply common sense. Exactly why was no solution found? What happens if you log each of those equations? Nothing changes in theory. If a solution existed before, it will still exist.
eqn1 = log(c) + m*log(Re_1) + n*log(Pr) == log(Nu_1);
eqn2 = log(c) + m*log(Re_2) + n*log(Pr) == log(Nu_2);
eqn3 = log(c) + m*log(Re_3) + n*log(Pr) == log(Nu_3);
Look carefully at the problem. We have a constant term, in the form of log(c). We have n times a constant, the same constant in each equation, thus log(Pr).
Effectively you have a rank 2 system of 3 linear equations, in 3 unknowns. This system will almost always NEVER have a solution. This is not the case where infinitely many solutions exist. It is the case where no solution exists at all.
What did MATLAB tell you? It returned an empty array for the solutions. So, as I predicted, NO solution exists. Period. Wanting magic to happen is not sufficient.
