Solving fractional relaxation equations requires precisely characterized domains of definition for applications of fractional differential and integral operators. Determining these domains has been a longstanding problem. Applications in physics and engineering typically require extension from domains of functions to domains of distributions. In this work convolution modules are constructed for given sets of distributions that generate distributional convolution algebras. Convolutional inversion of fractional equations leads to a broad class of multinomial Mittag-Leffler type distributions. A comprehensive asymptotic analysis of these is carried out. Combined with the module construction the asymptotic analysis yields domains of distributions, that guarantee existence and uniqueness of solutions to fractional differential equations. The mathematical results are applied to anomalous dielectric relaxation in glasses. An analytic expression for the frequency dependent dielectric susceptibility is applied to broadband spectra of glycerol. This application reveals a temperature independent and universal dynamical scaling exponent.

求解分数松弛方程需要精确表征的定义域，以用于分数微分和积分算子的应用。确定这些域一直是一个长期存在的问题。物理学和工程学中的应用通常需要从函数域扩展到分布域。在这项工作中，卷积模块是为生成分布卷积代数的给定分布集构建的。分数方程的卷积反演导致了一大类多项 Mittag-Leffler 型分布。对这些进行了全面的渐近分析。结合模块构造，渐近分析产生分布域，保证分数阶微分方程解的存在性和唯一性。数学结果应用于玻璃中的异常介电弛豫。频率相关的介电磁化率的解析表达式适用于甘油的宽带光谱。此应用程序揭示了一个与温度无关且通用的动态标度指数。