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 $O(T^2)$, where T is the interval each time step, provided that the approximation error of the MLF is sufficiently small.

Mittag-Leffler函数（MLF）是解决分数演算问题的基础。提出了带有负输入的单参数MLF的指数和近似。分析表明，基于高斯-勒根德勒（Gauss-Legendre）正交的近似值对于所有非正输入均一收敛。还介绍了有限元方法在粘弹性材料中波传播建模中的应用。证明了近似波动方程解的传播速度具有与原始波动上限相同的上限。基于广义alpha方法的离散方案仅涉及单个矩阵逆，其大小是每个时间步长的几何模型的自由度。证明了该方案是无条件稳定的。此外，$Ø（{Ť}^2）$，其中T是每个时间步长的间隔，条件是MLF的近似误差足够小。