  It is quite common that several different phases exist simultaneously in a
system of trapped quantum gases of ultra-cold atoms. One such example is the
strongly-interacting Fermi gas with two imbalanced spin species, which has
received a great amount of attention due to the possible presence of exotic
superfluid phases. By employing novel numerical techniques and algorithms, we
self-consistently solve the Bogoliubov de-Gennes equations, which describe
Fermi superfluids in the mean-field framework. From this study, we investigate
the novel phases of spin-imbalanced Fermi gases and examine the validity of the
local density approximation (LDA), which is often invoked in the extraction of
bulk properties from experimental measurements within trapped systems. We show
how the validity of the LDA is affected by the trapping geometry, number of
atoms and spin imbalance.
