Jeffreys equation and its fractional generalizations provide extensions of the classical diffusive laws of Fourier and Fick for heat and particle transport. In this work, a class of multi-term time-fractional generalizations of the classical Jeffreys equation is studied. Restrictions on the parameters are derived, which ensure that the fundamental solution to the one-dimensional Cauchy problem is a spatial probability density function evolving in time. The studied equations are recast as Volterra integral equations with kernels represented in terms of multinomial Mittag-Leffler functions. Applying operator-theoretic approach, we establish subordination results with respect to appropriate evolution equations of integer order, depending on the considered range of parameters. Analyticity of the corresponding solution operator is also discussed. The main tools in the proofs are Laplace transform and the Bernstein functions’ technique, especially, some properties of the sets of real powers of complete Bernstein functions.

杰弗里方程及其分数推广提供了傅立叶和菲克经典扩散定律对于热和粒子传输的扩展。在这项工作中，研究了经典 Jeffreys 方程的一类多项时间分数推广。导出了对参数的限制，确保一维柯西问题的根本解是随时间演化的空间概率密度函数。研究的方程被改写为 Volterra 积分方程，其核以多项 Mittag-Leffler 函数表示。应用算子理论方法，我们根据所考虑的参数范围建立关于适当的整数阶演化方程的从属结果。还讨论了相应解算子的解析性。证明的主要工具是拉普拉斯变换和伯恩斯坦函数技术，特别是完全伯恩斯坦函数的实幂集的一些性质。