The Airy pulse dispersion effect by the split-step Fourier transform method (SSFM) is sumulated and analyzed. The wave transmission characteristic of Airy pulse under second and third-order dispersion effects in temporal domain is discussed. Furthermore, the influence of the initial chirp and weakly nonlinear effects is studied. The results show that the positive third-order dispersion effect leads to pulse compression and amplitude increase. On the other hand, the negative third-order dispersion effects leads to pulse broadening and amplitude decrease. When taking the effect of the initial chirp and weakly nonlinear effects into account. It is found that as the third-order dispersion effects growing, the quality of Airy pulse reduces.