  Q^2 evolution equations are important not only for describing hadron
reactions in accelerator experiments but also for investigating
ultrahigh-energy cosmic rays. The standard ones are called DGLAP evolution
equations, which are integrodifferential equations. There are methods for
solving the Q^2 evolution equations for parton-distribution and fragmentation
functions. Because the equations cannot be solved analytically, various methods
have been developed for the numerical solution. We compare brute-force,
Laguerre-polynomial, and Mellin-transformation methods particularly by focusing
on the numerical accuracy and computational efficiency. An efficient solution
could be used, for example, in the studies of a top-down scenario for the
ultrahigh-energy cosmic rays.
