In finding analytical and numerical solutions for numerous forms of anharmonic oscillator potential, the asymptotic iteration method (AIM) is shown to be a powerful and favored technique. In this paper, we consider certain conditions in this method for selecting the main parameters and find that convergence speed and process stability in the AIM are significantly improved. Based on these conditions, we use the AIM to deduce accurate solutions to some examples of anharmonic potential and problems with Morse potential. Focusing on the anharmonic potential problem, we also represent a comparison of different methods such as the airy function approach to the solutions provided by the AIM discussed here and the results for the same problem.