Detonation-based engines have a higher thermal efficiency than deflagration-based engines and are therefore widely studied. In this study, the effect of initiation conditions on the detonation wave propagation and the performance of a pulse detonation engine, one type of detonation-based engine, was numerically investigated. Hydrogen-oxygen mixtures with equivalence ratios of φ=0.8, 1.0, and 1.2 were defined as initiation conditions with constant pressure and temperature. The numerical simulations were conducted using the transient explicit density-based solver in the ANSYS Fluent commercial software. The adaptability of the adaptive mesh refinement method in detonation-based engines was explored in the numerical studies, reducing the average total cell count by a factor of 2.617, and obtaining consistent results according to validation studies. The adaptive mesh refinement method was also used in numerical simulations where different equivalence ratios were defined. It was determined that an increase in the equivalence ratio resulted in an increase in the detonation wave velocity. Also, an increase in thrust distribution at the nozzle exit was observed before the blowdown stage, and the calculated thrust values for φ=0.8, φ=1.0, and φ=1.2 were 248.28 N, 264.5 N, and 270.83 N, respectively.