The dynamics of finite energy Airy beams modeled by the fractional Schrödinger equation (FSE) with a linear potential are numerically investigated. Different from the propagation properties of Airy beams described by the standard Schrödinger equation, the splitting phenomenon, which is presented under the FSE without potential, is influenced by the quadratic chirp and the Lévy index. As the linear potential is considered, the periodic evolution of Airy beams is shown, and the period is inversely proportional to the linear potential coefficient. The beam width can undergo an ab...