  The fractional Schr\"{o}dinger equation (FSE) -- a natural extension of the
standard Schr\"{o}dinger equation -- is the basis of fractional quantum
mechanics. It can be obtained by replacing the kinetic-energy operator with a
fractional derivative. Here, we report the experimental realisation of an
optical FSE for femtosecond laser pulses in the temporal domain. Programmable
holograms and the single-shot measurement technique are respectively used to
emulate a \textit{L\'evy waveguide} and to reconstruct the amplitude and phase
of the pulses. Varying the L\'evy index of the FSE and the initial pulse, the
temporal dynamics is observed in diverse forms, including solitary, splitting
and merging pulses, double Airy modes, and ``rain-like'' multi-pulse patterns.
Furthermore, the transmission of input pulses carrying a fractional phase
exhibits a ``fractional-phase protection'' effect through a regular
(non-fractional) material. The experimentally generated fractional time-domain
pulses offer the potential for designing optical signal-processing schemes.
