Calculations using the particle-map perturbative approach predict that fluxons in long Josephson junctions can exhibit chaotic motion when the junction is driven by an external microwave signal, applied to the junction via magnetic field boundary conditions. The chaotic state is reached through a Feigenbaum-like cascade in the fluxon times of flight across the junction, with increasing signal amplitudes. In the present work the existence of such chaotic dynamics is demonstrated via numerical integration of the full perturbed sine-Gordon partial differential equation (PDE) model of the junction. The salient characteristics of the PDE dynamics are compared with the results obtained from the perturbative map approach. The resulting PDE chaos appears to be strictly low-dimensional: fluxons retain their shape without loss of spatial coherence, but their temporal motion is chaotic.