The inverse Mittag–Leffler function has been used to model the logarithmic growth of the mean squared displacement in anomalous diffusion, the restricted mobility of membrane proteins, and the slow viscoelastic creep observed in glasses. These applications are hindered because the inverse Mittag–Leffler function has no explicit form and cannot be approximated by existing methods in the domain \(x \in (0,+\infty )\). This study proposes a conversion method to compute the inverse Mittag–Leffler function in terms of the Mittag–Leffler function. The new method uses the one- and two-parameter Mittag–Leffler function to compute the inverse Mittag–Leffler function in the target domain. We apply this method to fit data collected in studies of: (i) the ultraslow mobility of beta-barrel proteins in bacterial membranes, (ii) the ultraslow creep observed in high strength self-compacting concrete, and (iii) the ultraslow relaxation seen in various glasses. The results show that the inverse Mittag–Leffler function can capture ultraslow dynamics in all three cases. This method may also be extended to other generalized logarithmic laws.

反 Mittag-Leffler 函数已用于模拟反常扩散中均方位移的对数增长、膜蛋白的受限流动性以及在玻璃中观察到的缓慢粘弹性蠕变。这些应用受到阻碍，因为反 Mittag-Leffler 函数没有明确的形式，并且不能用域\(x \in (0,+\infty )\)中的现有方法来近似. 本研究提出了一种根据 Mittag-Leffler 函数计算逆 Mittag-Leffler 函数的转换方法。新方法使用单参数和双参数 Mittag-Leffler 函数来计算目标域中的反 Mittag-Leffler 函数。我们应用这种方法来拟合在以下研究中收集的数据：(i) β-桶蛋白在细菌膜中的超慢流动性，(ii) 在高强度自密实混凝土中观察到的超慢蠕变，以及 (iii) 观察到的超慢松弛在各种眼镜。结果表明，逆 Mittag-Leffler 函数可以在所有三种情况下捕获超慢动力学。这种方法也可以扩展到其他广义对数定律。