Numerical Schemes for Time-Space Fractional Vibration Equations (Advances in Applied Mathematics and Mechanics, 2021)
In this paper, we present a numerical scheme and an alternating direction implicit (ADI) scheme for the one-dimensional and two-dimensional time-space fractional vibration equations (FVEs), respectively. Firstly, the considered time-space FVEs are equivalently transformed into their partial integro-differential forms with the classical first order integrals and the Riemann-Liouville derivative. This transformation can weaken the smoothness requirement in time when discretizing the partial integro-differential problems. Secondly, we use the Crank-Nicolson technique combined with the midpoint formula, the weighted and shifted Grünwald difference formula and the second order convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and fractional central difference formula are applied to approximate the second order derivative and the Riesz derivative in spatial direction, respectively. Further, an ADI scheme is constructed for the two-dimensional case. Then, the convergence and unconditional stability of the proposed schemes are proved rigorously. Both of the schemes are convergent with the second order accuracy in time and space. Finally, two numerical examples are given to support the theoretical results.