  Using the formalism of generalized fractional derivatives, a two-dimensional
non-relativistic meson system is studied. The mesons are interacting by a
Cornell potential. The system is formulated in the domain of the symplectic
quantum mechanics by means of the generalized fractional Nikiforov-Uvarov
method. The corresponding Wigner function and the energy eigenvalues are then
derived. The effect of fractional parameters $\alpha$ and $\beta$ with the
ground state solution is analyzed through the Wigner function for the
charm-anticharm, bottom-antibottom and $b\overline{c}$ mesons. One of the
fundamental achievements of such Cornell model is the determination of heavy
quarkonia mass spectra. We have computed these masses and the
