The existing physical implementations of reservoir computing are constrained mainly by time-delay architectures that lack capabilities for spatial data processing. This study presents a multifunctional memristor-based reservoir computing system, the memristive echo state network (MESN), which enables spatiotemporal computation within a single device crossbar array. Utilizing a reconfigurable Ta/HfO₂/RuO₂ memristor, three distinct switching modes are realized: stochastic for input masking, bistable for sigmoidal activation, and analog for precise readout. A full in-memory implementation is experimentally demonstrated using a one-transistor-one-resistor crossbar array integrated with indium oxide thin-film transistors. Spatial inference is validated through cellular automata, confirming reliable hardware operation. High-level simulations based on the hardware results demonstrate the performance of the proposed MESN, achieving high accuracy in predicting the Lorenz attractor and classifying attention-deficit/hyperactivity disorder. The system also predicted the Kuramoto-Sivashinsky equation, representing the first memristor-based reservoir to model complex spatiotemporal partial differential equations. These results highlight the potential of multifunctional memristor arrays for scalable in-memory spatiotemporal computing.