Abstract: Based on the theory of fractional derivative and elastic mechanics theory, the model of oil and gas wells tube is simplified into a viscoelastic cylindrical tube. The motion control equation of the incompressible viscoelastic cylindrical tube under axial symmetry is established. The displacement solution is directly obtained from the incompressibility assumption. The Laplace transformation is used to solve the circumferential displacement and stress, and analytical solution of the oil and gas wells tube is obtained. The numerical results show that the order of fractional derivative, the ratio of model constant and the ratio of inner radius to outer radius of oil and gas wells tube have great influence on the radial displacement of oil and gas wells tube, and the curve of radial displacement varying with frequency has a peak value. The larger the order of fractional derivative and the ratio of model constant, the smaller the peak value, the larger the frequency corresponding to the peak value, while the influence of the ratio of inner radius and outer radius of oil and gas wells tube is opposite. The order of fractional derivative, the ratio of model constant and the ratio of inner radius to outer radius of oil and gas wells tube have great influence on circumferential stress and vertical stress, but little influence on radial stress.