We present the influence of second order dispersion(GVD), third-order dispersion(TOD), and initial chirp on femtosecond Airy pulse propagation in Kerr media by solving the Nonlinear Schrodinger Equation with the split-step Fourier Method. In the time duration of femtosecond pulse, the effect of TOD should not be neglected. TOD can lead to waveform distortion and lower the quality of optical pulses. We also study the propagation of femtoscond Airy pulse in anomalous dispersion Kerr media. According to the numerical results, we show that when the parameter of the TOD and the propagation distance are selected as some typical values, the pulses will broadening first and then appear a process of compression. Finally, we discussed the influence of the initial pulse chirp on the propagation of the pulse profile and broadening factor.