Tempered Time Fractional ADE (TTFADE)

• TTFADE is a tempered time-fractional advection-dispersion equation that models anomalous transport by incorporating an exponentially truncated Caputo derivative.
• The numerical methodology employs graded temporal meshes, second-order spatial finite differences, and fast sum-of-exponentials to efficiently resolve the initial singularity and nonlocal history.
• TTFADE is applied to simulate time-of-flight measurements in semiconductors, capturing the transition between classical diffusion and fractional subdiffusion with calibrated parameters.

The tempered time-fractional advection-dispersion equation (TTFADE) is a non-classical evolution equation describing anomalous transport, where memory effects are governed by a tempered (exponentially truncated) time-fractional derivative of Caputo type. The TTFADE arises in models for dispersive transport in disordered materials, notably in the analysis of transient currents in time-of-flight (ToF) experiments for semiconductors and dielectrics. Its key features are the interpolation between classical diffusive and fractional subdiffusive regimes, and the inclusion of a tempering parameter, which imposes finite moments on the waiting time distributions of carriers (Morgado et al., 2018, Huang et al., 17 Dec 2025).

1. Mathematical Formulation

The canonical form of the TTFADE—considered on a spatial domain $\Omega = (0, L)$ and time interval $t \in (0, T]$—is: $$\mathbb{D}_t^{\alpha,\lambda} u(x,t) = -v\,\frac{\partial u(x,t)}{\partial x} + D\,\frac{\partial^2 u(x,t)}{\partial x^2} + f(x,t),$$ with $0 < \alpha < 1$, $\lambda \geq 0$, drift velocity $v > 0$, dispersion coefficient $D > 0$, and a general forcing term $f(x,t)$. The initial and Dirichlet boundary conditions are: $u(x,0) = g(x),\quad 0 < x < L;$

$u(0,t) = u(L,t) = 0,\quad 0 < t \leq T.$

The boundary conditions model perfect carrier extraction (sinks) at electrodes, and $g(x)$ typically describes a localized initial carrier pulse (Morgado et al., 2018, Huang et al., 17 Dec 2025).

2. Tempered Caputo Fractional Derivative

The operator $\mathbb{D}_t^{\alpha,\lambda}$ is the tempered Caputo derivative, defined as: $$\mathbb{D}_t^{\alpha,\lambda} y(t) = e^{-\lambda t} D_t^\alpha\bigl(e^{\lambda t}y(t)\bigr) = \frac{e^{-\lambda t}}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} \frac{d}{ds}\bigl(e^{\lambda s} y(s)\bigr) ds.$$ Here, $D_t^\alpha$ denotes the standard Caputo derivative: $$D_t^\alpha y(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} y'(s)\,ds.$$ Key properties include linearity, reduction to the Caputo derivative for $\lambda=0$, and exponential tempering of the nonlocal kernel, which ensures finite moments in the underlying stochastic transport interpretation (Morgado et al., 2018, Huang et al., 17 Dec 2025).

3. Initial-Boundary Value Problem and Solution Regularity

The TTFADE typically models carrier transport from a sharp initial pulse (e.g., Gaussian localized near $x=0$) with absorbing boundaries. Due to the singularity in the fractional kernel, solutions to TTFADE often are not $C^2$ in time at $t=0$. Rigorous analysis proceeds under the regularity assumption: $$|\partial_t^l u(x,t)| \leq C(1 + t^{\delta - l}),\quad l=0,1,2,\quad 1 < \delta < 2,$$ which reflects weak initial temporal singularities common in subdiffusive and tempered-fractional dynamics (Huang et al., 17 Dec 2025).

4. Numerical Discretization Strategies

4.1 Spatial and Temporal Discretization

Standard approaches employ second-order centered finite differences in space: $$\frac{\partial u}{\partial x}(x_i,t) \approx \frac{U_{i+1}^l - U_{i-1}^l}{2h},\qquad \frac{\partial^2 u}{\partial x^2}(x_i,t) \approx \frac{U_{i+1}^l - 2U_i^l + U_{i-1}^l}{h^2},$$ on a uniform mesh $x_i = ih,\, i=0,\ldots,K$, $h=L/K$ (Morgado et al., 2018, Huang et al., 17 Dec 2025).

In time, a graded mesh

$$t_n = T (n/N)^r, \quad r \geq \max\{3, 2/(\delta - 1)\},$$

clusters time steps near $t=0$ to resolve singular initial layers arising from the fractional kernel (Huang et al., 17 Dec 2025).

4.2 Time Discretization for Tempered Caputo Term

Classical L1-type discretizations suffer accuracy loss near $t=0$. Enhanced schemes—using graded meshes and nonuniform step sizes—restore optimal convergence, with coefficients $a_{j,l}$ incorporating both the mesh grading and the singular nature of the kernel: $$D_t^\alpha y(t_\ell) \approx \tilde D^\alpha y_\ell = \frac{1}{\Gamma(2-\alpha)} \sum_{j=0}^{\ell-1} \tau_j^{-\alpha} a_{j,\ell}\, (y_{j+1}-y_j),$$ with $a_{j,\ell}$ explicitly constructed to reflect the graded structure (Morgado et al., 2018).

The “de-tempering” transformation $y(x,t) = e^{\lambda t} u(x,t)$ recasts the TTFADE as a standard time-fractional equation for $y(x,t)$, facilitating reuse of Caputo-based discretizations. The complete scheme yields for each spatial node and time level a nonlinear system, efficiently solvable under CFL-type grid constraints.

4.3 Fast Sum-of-Exponentials (SOE) Approach

To address the computational bottleneck of the historical fractional term (cost $O(N^2)$ for $N$ steps), the kernel $(t-s)^{-1-\alpha}$ is approximated by a sum of $N_{\exp}$ exponentials,

$$t^{-1-\alpha} \approx \sum_{\ell=1}^{N_{\exp}} \omega_\ell e^{-s_\ell t},$$

with $$N_{\exp} = O(\log(1/\epsilon)\log\log(1/\epsilon) + ...)$$ for error $\epsilon$. This allows recursive update of “history sums” and reduces the total complexity to $O(N \log N)$ for fixed accuracy (Huang et al., 17 Dec 2025).

5. Theoretical Properties: Stability and Convergence

The schemes constructed by both (Morgado et al., 2018) and (Huang et al., 17 Dec 2025) provide rigorous guarantees:

• Stability: The fully discrete schemes are stable under a maximum norm or $L_2$ norm, with perturbations in initial data not amplified through time, given mesh constraints (e.g., $h < 2D/v$ for the maximum norm scheme).
• Convergence: Provided the solution regularity holds (as above), global errors in the maximum or $L_2$ norm are

$O(\tau^2 + h^2)$

with time grading and SOE accuracy balanced. For the graded mesh scheme, selecting $r=(2-\alpha)/\alpha$ yields temporal accuracy of order $2-\alpha$ (Morgado et al., 2018, Huang et al., 17 Dec 2025).

Observed numerical errors confirm the theoretical rates. For example, with graded mesh and Caputo approximation, temporal experimental order of convergence (EOC) values of 1.36 (for $\alpha=0.5$) and 1.59 (for $\alpha=0.25$) versus much lower orders for uniform meshes were reported (Morgado et al., 2018). The fast SOE-based scheme achieves second-order accuracy for both $u(t)=t^\delta$ and full TTFADE test cases (Huang et al., 17 Dec 2025).

6. Computational Efficiency

The naive history-sum update for the Caputo derivative incurs $O(N^2)$ work. The SOE strategy, by compressing the historical memory to a handful of exponentials, reduces both work and storage per step to $O(\log N)$ for fixed error, so the total cost is $O(N \log N)$. This computational advantage enables large-scale simulation for high-fidelity models (Huang et al., 17 Dec 2025).

7. Applications in Time-of-Flight Measurements

The TTFADE framework is deployed to analyze time-of-flight (ToF) transient current measurements in disordered semiconducting materials. The predicted current is

$$I(t) = \frac{1}{L} \int_0^L j(x', t)\,dx' = -q\,\frac{d}{dt} \int_0^L (L - x) u(x, t)\,dx,$$

where $q$ is carrier charge (Morgado et al., 2018). Fitting TTFADE parameters $(\alpha, \lambda, v, D)$ to ToF data yields excellent agreement with experimental currents, capturing two distinct pre- and post-transit power-law decay regimes $t^{-1+\alpha}$ and $t^{-1-\alpha}$. For amorphous boron, suitable fits were reported at $\alpha=0.66$, $\lambda = 1.0 t_T^{-1}$, $v = 0.38 L^{-1} t_T^{-\alpha}$, and $D = 2.7 \times 10^{-3} L^{-2} t_T^{-\alpha}$, with the initial distribution localized near one electrode as a Gaussian pulse (Morgado et al., 2018).

8. Summary and Practical Guidelines

The TTFADE generalizes standard time-fractional advection-diffusion models by incorporating an exponential tempering parameter $\lambda$, enabling the modeling of truncated waiting-time distributions (finite-moment transport). Its practical implementation includes:

• Second-order centered finite differences in space ($h$-accuracy $O(h^2)$).
• Graded temporal meshes for resolving $t = 0$ singularity (select $r$ for desired time accuracy).
• De-tempering transformations to leverage standard Caputo approximations.
• Fast SOE algorithms for efficient nonlocal kernel summation.
• Calibration of $\alpha$ and $\lambda$ from measured power-law regimes; $v$ and $D$ fitted from time-of-flight and pulse broadening data.

TTFADE and its associated fast, accurate numerical schemes are suited for simulating anomalous transport dynamics in disordered solids, with established convergence, stability, and computational efficiency (Morgado et al., 2018, Huang et al., 17 Dec 2025).