Scaling crossover of the generalized Jeffreys-type law

The generalized Jeffreys-type law is formulated as a multi-term time-fractional Jeffreys-type equation, whose dynamics exhibit rich scaling crossover phenomena entailing different diffusion mechanisms. In this work, we provide a novel physical explanation for the equation from first principles, beginning with a microscopic description based on the continuous-time random walk framework with a generalized waiting time distribution and further deriving the equation from an overdamped Langevin equation subject to a stochastic time-change (subordination). Employing the Laplace transform method, we conduct a rigorous analysis of the equation, establishing its well-posedness and providing a detailed Sobolev regularity analysis. We also develop a novel numerical scheme, termed the CIM-CLG algorithm, which achieves spectral accuracy in both time and space while substantially relaxing the temporal regularity requirements on the solution. The algorithm reduces the computational complexity to $\mathcal{O}(N)$ in time and $\mathcal{O}(M\log M)$ in space and is fully parallelizable. Detailed implementation guidelines and new technical error estimates are provided. Extensive numerical experiments in 1D and 2D settings validate the efficiency, robustness, and accuracy of the proposed method. By integrating stochastic modeling, mathematical analysis, and numerical computation, this work advances the understanding of the generalized Jeffreys-type law and offers a mathematically rigorous and computationally efficient framework for tackling complex nonlocal problems.