Exponential sum approximation for Mittag-Leffler function and its application to fractional Zener wave equation

The Mittag-Leffler function (MLF) is fundamental to solving fractional calculus problems. An exponential sum approximation for the single-parameter MLF with negative input is proposed. Analysis shows that the approximation, which is based on the Gauss-Legendre quadrature, converges uniformly for all non-positive input. The application to modelling of wave propagation in viscoelastic material with the finite element method is also presented. The propagation speed of the solution to the approximated wave equation is proved to have the same upper bound as the original. The discretized scheme, based on the generalised alpha method, involves only a single matrix inverse whose size is the degree of freedom of the geometric model per time step. It is proved that the scheme is unconditionally stable. Furthermore, the solution converges to the true solution at O (T² ), where T is the interval each time step, provided that the approximation error of the MLF is sufficiently small.