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The dynamics of strongly interacting trapped dilute Fermi gases (dilute inthe sense that the range of interatomic potential is small compared withinter-particle spacing) is investigated in a single-equation approach to thetime-dependent density-functional theory. Our results are in good agreementwith recent experimental data in the BCS-BEC crossover regime. It is also shownthat the calculated corrections to the hydrodynamic approximation may beimportant even for systems with a rather large number of atoms.Comment: Resubmitted to PRA in response to referee's comments. Abstract is  changed. Added new figure

Time-dependent density-functional theory for trapped strongly interacting fermionic atoms