Reservoir computing systems utilize dynamic reservoirs having short-term memory to project features from the temporal inputs into a high-dimensional feature space. A readout function layer can then effectively analyze the projected features for tasks, such as classification and time-series analysis. The system can efficiently compute complex and temporal data with low-training cost, since only the readout function needs to be trained. Here we experimentally implement a reservoir computing system using a dynamic memristor array. We show that the internal ionic dynamic processes of memristors allow the memristor-based reservoir to directly process information in the temporal domain, and demonstrate that even a small hardware system with only 88 memristors can already be used for tasks, such as handwritten digit recognition. The system is also used to experimentally solve a second-order nonlinear task, and can successfully predict the expected output without knowing the form of the original dynamic transfer function.