Scaling crossover of the generalized Jeffreys-type law

TL;DR

The paper derives a generalized Jeffreys-type diffusion law (MT-TFJE) from microscopic CTRW and time-change Langevin dynamics to capture scaling crossover diffusion. It establishes a rigorous solution theory with well-posedness and Sobolev regularity via Laplace-transform techniques and introduces a novel CIM-CLG algorithm that achieves spatio-temporal spectral accuracy with $\mathcal{O}(N)$ time and $\mathcal{O}(M\log M)$ space, while tolerating low temporal regularity. The method is backed by detailed error analyses and validated through extensive 1D and 2D numerical experiments, demonstrating efficiency, robustness, and high accuracy for nonlocal multi-term fractional diffusion models. This framework advances nonlocal diffusion modeling by unifying stochastic microdynamics, analytic well-posedness, and scalable spectral numerics, with potential impact on complex transport phenomena where crossover diffusion appears. The combination of first-principles modeling, rigorous analysis, and highly efficient computation provides a rigorous and practical toolkit for tackling MT-TFDEs in bounded domains.

Abstract

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.

Figures (7)

• Figure 1: Two-dimensional trajectories illustrating anomalous diffusion within the CTRW framework. The dynamics follow the generalized waiting time distribution in \ref{['eq:PDFTime']} and \ref{['eq:denoteeta']}, with waiting times distribution numerically generated via Talbot's numerical inverse Laplace transform algorithm Talbot1979. Parameters are set to $a=1$, $b=100$, with each trajectory simulated over 1000 steps. The starting point is at the origin.
• Figure 1: Illustration of auxiliary function $\mathcal{L}(x)$, the open symmetric strip $\mathcal{S}$, and its neighbourhood $\mathcal{N}_{e}$ in the $z$-plan. $(a)$ Monotonic decrease of $\mathcal{L}(x)$. $(b)$ and $(c)$ display the images of $\mathcal{S}$ under the conformal mapping $z(\phi)$ defined by \ref{['eq:hyperboliccontour']}. The upper half of $\mathcal{S}$ (with $y>0$) maps to the shaded interior region enclosed by $z(x + i\,0)$ within $\mathcal{N}{e}$; as $y$ increases, $z(\phi)$ approaches the negative real axis. Conversely, the lower half of $\mathcal{S}$ (with $y<0$) maps to the exterior shaded region outside $z(x+i\,0)$ in $\mathcal{N}_{e}$; as $y$ decreases, $z(\phi)$ converges toward a vertical line; $|y|\leq d$, where $d:=\min\{\widetilde{\alpha},\frac{\pi}{2}-\widetilde{\alpha}-\delta'\}$ and $0<\widetilde{\alpha}-d<\widetilde{\alpha}+d<\frac{\pi}{2}-\delta'$. Parameters for $(b)$ and $(c)$: $\alpha = \pi/4$, $\mu = 0.8$, and $\delta = \pi/12$.
• Figure 1: Numerical performance of the CIM illustrating its spectral accuracy, stability, and robustness. $(a)$ Error evolution with the number of quadrature points $N$ for various $\Lambda$ (semi-discrete scheme \ref{['eq:computeu']}), with $\alpha=0.50$, $\beta=0.35$, $\gamma=0.45$, $\lambda=1.50$, $a=1.00$, $b=100.00$, and $t=0.50$. $(b)$ Error under varying fractional orders, with $a=1.00$, $b=10.00$ and $\Lambda = 10.00$. $(c)$ Error across different ratios of $b/a$, with $\alpha=0.50$, $\beta=0.35$, $\gamma=0.45$, $\lambda=1.50$, and $t=0.50$.
• Figure 2: Spatial spectral accuracy convergence of the CLG method. The figure shows the exponential decay of spatial errors with increasing polynomial degree under different fractional orders, with given parameters $a=1$, $b=1000$, $t=0.5$, and $\kappa=1/4$ (low temporal regularity).
• Figure 3: Exponential convergence of the CIM in time. Errors are shown for different fractional orders with an increasing number of quadrature points $N$, with given parameters $M = 20$, $a = 1$, $b = 1000$, and $\kappa = 1/4$ (low temporal regularity).

Theorems & Definitions (30)

• Proposition 2.1: Parameter Constraints
• Proof 1
• Lemma 3.1: Resolvent estimate Arendt2011
• Lemma 3.2: Analyticity
• Proof 2
• Remark 3.3
• Lemma 3.4: Prior estimates
• Proof 3
• Theorem 3.5: Sobolev regularity
• Proof 4