  Using repeated Laplace transform techniques, along with newly-developed
accurate numerical inverse Laplace transform algorithms, we transform the
coupled, integral-differential NLO singlet DGLAP equations first into coupled
differential equations, then into coupled algebraic equations, which we can
solve iteratively. After Laplace inverting the algebraic solution analytically,
we numerically invert the solutions of the decoupled differential equations.
Finally, we arrive at the decoupled NLO evolved solutions
F_s(x,Q^2)=calF_s(F_{s0}(x),G_0(x)) and G(x,Q^2)=calG(F_{s0}(x),G_0(x)), where
calF_s and calG are known functions - determined using the DGLAP splitting
functions up to NLO in the strong coupling constant alpha_s(Q^2). The functions
F_{s0}(x)=F_s(x,Q_0^2) and G_0(x)=G(x,Q_0^2) are the starting functions for the
evolution at Q_0^2. This approach furnishes us with a new tool for readily
obtaining, independently, the effects of the starting functions on either the
evolved gluon or singlet structure functions, as a function of both Q^2 and
Q_0^2. It is not necessary to evolve coupled integral-differential equations
numerically on a two-dimensional grid, as is currently done. The same approach
can be used for NLO non-singlet distributions where it is simpler, only
requiring one Laplace transform. We make successful NLO numerical comparisons
to two non-singlet distributions, using NLO quark distributions published by
the MSTW collaboration, over a large range of x and Q^2. Our method is readily
generalized to higher orders in the strong coupling constant alpha_s(Q^2).
