  We present a novel semi-analytical method for parton evolution. It is based
on constructing a family of analytic functions spanning $x$-space which is
closed under the considered evolution equation. Using these functions as a
basis, the original integro-differential evolution equation transforms into a
system of coupled ordinary differential equations, which can be solved
numerically by restriction to a suitably chosen finite subsystem. The evolved
distributions are obtained as analytic functions in $x$ with numerically
obtained coefficients, providing insight into the analytic behavior of the
evolved parton distributions. As a proof-of-principle, we apply our method to
the leading order non-singlet and singlet DGLAP equation. Comparing our results
to traditional Mellin-space methods, we find good agreement. The method is
implemented in the code $\texttt{POMPOM}$ in $\texttt{Mathematica}$ as well as
in $\texttt{Python}$.
