A linearized ADI scheme for two-dimensional time-space fractional nonlinear vibration equations

A nonlinear initial boundary value problem with a time Caputo derivative of fractional-order beta is an element of (1, 2) and a space Riesz derivative of fractional-order alpha is an element of (1, 2) is considered. A linearized alternating direction implicit (ADI) scheme for this problem is constructed and analysed. First, we use the classical central difference formula and a new 3 - beta-order formula to approximate the second-order derivative and the Caputo derivative in temporal direction, respectively. Meanwhile, the central difference quotient and fractional central difference formula are applied to deal with the spatial discretizations. Then, the proposed ADI scheme is proved unconditionally stable and convergent with the 3 - beta-order accuracy in time and the second-order accuracy in space. Furthermore, the effectiveness of the ADI scheme and theoretical findings are illustrated by numerical experiments.