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 $zgtrsim2$, $zgtrsim6$ and $zgtrsim10$, 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 $zgtrsim1$, $zgtrsim2$, and $zgtrsim4$, 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.
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