  We present a detailed QCD analysis of nucleon structure functions $xF_3 (x,
Q^2)$, based on Laplace transforms and Jacobi polynomials approach. The
analysis corresponds to the next-to-leading order and next-to-next-to-leading
order approximation of perturbative QCD. The Laplace transform technique, as an
exact analytical solution, is used for the solution of nonsinglet
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations at low- and
large-$x$ values. The extracted results are used as input to obtain the $x$ and
Q$^2$ evolution of $xF_3(x, Q^2)$ structure functions using the Jacobi
polynomials approach. In our work, the values of the typical QCD scale
$\Lambda_{\overline{\rm MS}}^{(n_f)}$ and the strong coupling constant
$\alpha_s(M_Z^2)$ are determined for four quark flavors ($n_f=4$) as well. A
careful estimation of the uncertainties shall be performed using the Hessian
method for the valence-quark distributions, originating from the experimental
errors. We compare our valence-quark parton distribution functions sets with
those of other collaborations; in particular with the {\tt BBG}, {\tt CT14},
{\tt MMHT14} and {\tt NNPDF} sets, which are contemporary with the present
analysis. The obtained results from the analysis are in good agreement with
those from the literature.
