  In this study, after we have briefly introduced the standard Gross-Pitaevskii
equation, we have suggested fractional Gross-Pitaevskii equations to
investigate the time-dependent ground state dynamics of the Bose-Einstein
condensation of weakly interacting bosonic particle system which can includes
non-Markovian processes or non-Gaussian distributions and long-range
interactions. Only we focused the time-fractional Gross-Pitaevskii equation and
have obtained solutions of the standard Gross-Pitaevskii and time-fractional
Gross-Pitaevskii equations for attractive and repulsive interactions in the
case external trap potentials $V(x)=0$ and optical lattice potential $V(x)
=\pm\sin^{2}x$ by using Homotopy Perturbation Method. We have found that the
Homotopy Perturbation Method solutions of the Gross-Pitaevskii equation for
these potentials and interactions are the same analytical results of it.
Furthermore we have also found that solutions of the time-fractional
Gross-Pitaevskii equation for these potentials and interactions can be given in
terms of Mittag-Leffler function. The solutions of the time-fractional
Gross-Pitaevskii equation provide that the time evolution of the ground state
dynamics of Bose-Einstein condensation of bosonic particles deviates
exponential form, and evolutes with time as stretched exponentially.
