  We study the collective modes of a one-dimensional harmonically trapped
Bose-Einstein condensate in the presence of the anomalous density using the
time-dependent Hartree-Fock-Bogoliubov theory. Within the hydrodynamic
equations, we derive analytical expressions for the mode frequencies and the
density fluctuations of the anomalous density which constitutes the minority
component at very low temperature and feels an effective external potential
exerted by the majority component, i.e., the condensate. On the other hand, we
numerically examine the temperature dependence of the breathing mode
oscillations of the condensate at finite temperature in the weak-coupling
regime. At zero temperature, we compare our predictions with available
experimental data, theoretical treatments, and Monte carlo simulations in all
interaction regimes and the remaining hindrances are emphasized. We show that
the anomalous correlations have a non-negligible role on the collective modes
at both zero and finite temperatures.
