  A calculation of the non-singlet part of spin dependent structure function,
$xg_1^{NS}(x,Q^2)$ and associated sum rule, the Bjorken Sum rule up to
next-next-to-leading order(NNLO) is presented. We use a unified approach
incorporating Regge theory and the theoretical framework of perturbative
Quantum Chromodynamics. Using a Regge behaved model with $Q^2$ dependent
intercept as the initial input, we have solved the
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation up to
NNLO at small-$x$ for $xg_1^{NS}(x,Q^2)$ and the solutions are utilised to
calculate the polarised Bjorken sum rule(BSR). We have also extracted the
higher twist contribution to BSR based on a simple parametrisation. These
results for both of $xg_1^{NS}(x,Q^2)$ and BSR, along with higher twist
corrections are observed to be consistent with the available data taken from
SMC, E143, HERMES, COMPASS and JLab experiments. In addition, our results are
also compared with that of other theoretical and phenomenological analysis
based on different models and a very good agreement is also observed in this
regard. Further a very good consistency between our calculated results and
theoretical QCD predictions of BSR is also achieved.
