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In this paper, we consider the application of intelligent reflecting surface (IRS) in unmanned aerial vehicle (UAV)-based orthogonal frequency division multiple access (OFDMA) communication systems, which exploits both the significant beamforming gain brought by the IRS and the high mobility of UAV for improving the system sum-rate. The joint design of UAV's trajectory, IRS scheduling, and communication resource allocation for the proposed system is formulated as a non-convex optimization problem to maximize the system sum-rate while taking into account the heterogeneous quality-of-service (QoS) requirement of each user. The existence of an IRS introduces both frequency-selectivity and spatial-selectivity in the fading of the composite channel from the UAV to ground users. To facilitate the design, we first derive the expression of the composite channels and propose a parametric approximation approach to establish an upper and a lower bound for the formulated problem. An alternating optimization algorithm is devised to handle the lower bound optimization problem and its performance is compared with the benchmark performance achieved by solving the upper bound problem. Simulation results unveil the small gap between the developed bounds and the promising sum-rate gain achieved by the deployment of an IRS in UAV-based communication systems.