  Fast radio bursts (FRBs) are mysterious extragalactic radio signals.
Revealing their origin is one of the central foci in modern astronomy. Previous
studies suggest that occurrence rates of non-repeating and repeating FRBs could
be controlled by the cosmic stellar-mass density (CSMD) and star formation-rate
density (CSFRD), respectively. The Square Kilometre Array (SKA) is one of the
best future instruments to address this subject due to its high sensitivity and
high-angular resolution. Here, we predict the number of FRBs to be detected
with the SKA. In contrast to previous predictions, we estimate the detections
of non-repeating and repeating FRBs separately, based on latest observational
constraints on their physical properties including the spectral indices, FRB
luminosity functions, and their redshift evolutions. We consider two cases of
redshift evolution of FRB luminosity functions following either the CSMD or
CSFRD. At $z\gtrsim2$, $z\gtrsim6$ and $z\gtrsim10$, non-repeating FRBs will be
detected with the SKA at a rate of $\sim10^{4}$, $\sim10^{2}$, and $\sim10$
(sky$^{-1}$ day$^{-1}$), respectively, if their luminosity function follows the
CSMD evolution. At $z\gtrsim1$, $z\gtrsim2$, and $z\gtrsim4$, sources of
repeating FRBs will be detected at a rate of $\sim10^{3}$, $\sim10^{2}$, and
$\lesssim10$ (sky$^{-1}$ day$^{-1}$), respectively, assuming that the redshift
evolution of their luminosity function is scaled with the CSFRD. These numbers
could change by about one order of magnitude depending on the assumptions on
the CSMD and CSFRD. In all cases, abundant FRBs will be detected by the SKA,
which will further constrain the luminosity functions and number density
evolutions.
