Tempered Time Fractional ADE

• TTFADE is a model incorporating tempered fractional derivatives to accurately capture transitional anomalous transport with finite memory.
• It uses advanced finite difference techniques, including graded-mesh L1 schemes and fast sum-of-exponentials methods, to achieve high numerical accuracy.
• The framework effectively fits experimental data in disordered media, such as transient currents in boron, demonstrating its practical importance in physics and engineering.

The tempered time fractional advection-dispersion equation (TTFADE) governs anomalous transport processes in which the temporal memory is truncated, combining fractional kinetics with exponential damping of long-time effects. TTFADE generalizes the standard time-fractional advection-dispersion equation (FADE) by replacing the Caputo derivative with its tempered analog, enabling superior modeling fidelity for experimental observations that exhibit transitions in scaling regimes or have a physically-motivated cutoff in memory. The equation appears in the mathematical modeling of dispersive transport in disordered materials, hydrology, physics of anomalous diffusion, and other fields sensitive to subdiffusive behavior and nonlocal memory (Huang et al., 17 Dec 2025, Morgado et al., 2018).

1. Mathematical Formulation and Caputo-Tempered Derivative

The TTFADE in one spatial dimension is given by

$$\frac{\partial u}{\partial t}(x,t) + {}_0^C D_t^{\alpha,\lambda} u(x,t) = D\frac{\partial^2 u}{\partial x^2}(x,t) - v \frac{\partial u}{\partial x}(x,t) + f(x,t), \qquad (x,t) \in \Omega \times (0,T],$$

where $\Omega = (0,L)$, $T > 0$, $0<\alpha<1$ is the fractional order, $\lambda > 0$ is the tempering parameter, $D$ is the diffusion coefficient, $v$ the drift velocity, and $f$ a source term.

The Caputo-tempered fractional derivative ${}_0^C D_t^{\alpha,\lambda} u(t)$ is defined as: $${}_0^C D_t^{\alpha,\lambda} u(t) = \frac{e^{-\lambda t}}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} \frac{d}{ds}\left[e^{\lambda s} u(s)\right] ds,$$ which reduces to the standard Caputo derivative when $\lambda=0$. This operator models power-law memory truncated by an exponential factor, mitigating the heavy-tail effect as $t \rightarrow \infty$ (Huang et al., 17 Dec 2025, Morgado et al., 2018).

2. Initial and Boundary Conditions, Regularity Issues

Solutions to TTFADE typically exhibit weak singularities at $t=0$. The standard problem is supplemented by:

• Initial condition: $u(x,0) = \varphi(x)$, for $x \in \Omega$.
• Boundary conditions: For homogeneous Dirichlet, $u(0,t) = u(L,t) = 0$ for $t \in [0,T]$.

A salient feature, inherited from the fractional derivative, is the lack of smoothness at the initial time. For $u(x,t) \simeq C(x) + O(t^\delta)$ with $1 < \delta < 2$, the Caputo-tempered derivative behaves as $O(t^{\delta-\alpha})$ near $t=0$. Accurate numerical simulations in such regimes employ graded temporal meshes $t_n = T(n/N)^r$ for $r \geq 1$, which cluster points near $t=0$ to resolve the singularity (Huang et al., 17 Dec 2025, Morgado et al., 2018).

3. Numerical Discretization and Fast Algorithms

Advanced finite difference schemes have been developed for efficient and accurate discretization of TTFADE, addressing both computational cost and singular behavior at the initial time. Two principal methodologies are established:

• Graded-mesh L1 schemes: The L1 rule on nonuniform meshes is applied to the Caputo-tempered derivative. Local truncation errors of $O(\tau_j^{\min\{r\alpha,2-\alpha\}})$ are proven, with global temporal accuracy $O(N^{-\min\{2-\alpha, r\alpha\}})$ when $r=(2-\alpha)/\alpha$ (Morgado et al., 2018).
• Fast finite difference via sum-of-exponentials: For second-order accuracy in time, the Caputo-tempered operator is split at each half-step into a history and local part. Beylkin-Monzón's sum-of-exponentials approximation approximates the kernel $(t_{n+1/2}-s)^{-1-\alpha}$ by a sum $$\sum_{\ell=1}^{N_{exp}} \omega_\ell e^{-s_\ell (t_{n+1/2}-s)}$$. History variables $U_{his,\ell}^n$ are updated recursively, reducing the history cost per step from $O(n)$ to $O(1)$ per exponential (Huang et al., 17 Dec 2025).

A Crank–Nicolson/central difference scheme in space and time leads to a tridiagonal system per time step, which can be solved in $O(M)$ by the Thomas algorithm, where $M$ is the spatial grid size.

Discretization Scheme	Time Order	Space Order	Complexity
Graded-mesh L1 finite difference	$(2-\alpha)$	2	$O(NM)$
Fast sum-of-exponentials + CN	2	2	$O(NM + N_{exp}N)$

Each $$N_{exp} = O(\log(1/\varepsilon) \cdot \log \log (1/\varepsilon))$$ for a fixed kernel approximation tolerance $\varepsilon$ (Huang et al., 17 Dec 2025).

4. Theoretical Results: Solvability, Stability, and Convergence

Scheme stability and convergence are established via discrete energy methods:

• Unique solvability: The algebraic system at each time step is strictly diagonally dominant with nonnegative eigenvalues, ensuring uniqueness.
• Unconditional $L^2$ stability: Discrete Fourier analysis demonstrates $\|U^n - V^n\|_{\ell^2} \leq \|U^0 - V^0\|_{\ell^2}$, and the inclusion of the tempering parameter $\lambda$ does not compromise this property (Huang et al., 17 Dec 2025).
• Convergence: For sufficiently smooth solutions and graded parameter $r$ satisfying $r(\delta-1)>2$, the schemes achieve

$$\|u^n - U^n\|_{\ell^2} \leq C[N^{-2} + h^2 + \varepsilon], \quad n = 0, \dots, N,$$

(see (Huang et al., 17 Dec 2025)) and

$$\max_{i,l}|u(x_i,t_l)-U_i^l| = O(n^{-\min\{2-\alpha, r\alpha\}} + h^2)$$

for the L1 scheme (Morgado et al., 2018).

5. Numerical Experiments and Model Calibration

Quantitative performance of the schemes is established through problems with known analytical solutions and physical applications:

• Pure fractional derivative test—e.g., $u(t)=t^\delta$ with $\delta=1.5$. Experiments confirm observed order $\approx 2$ in $N$ for $\alpha=0.1$, $0.3$, $0.5$.
• Full TTFADE solution test—e.g., $$u_{exact}(x,t)=e^{-\lambda t}(t^\delta+1)x^2(1-x)^2$$, for $\delta=1.8$, $\lambda=1$. Error tables confirm order $2$ in both $N$ and $h$.
• Modeling transient currents in amorphous boron: TTFADE with parameters $\alpha=0.66$, $\lambda=1.0$, $v=0.38$, $D=2.7 \times 10^{-3}$ fitted to experimental time-of-flight data yields excellent matching across both pre- and post-transit regimes. The non-tempered ($\lambda = 0$) model is unable to reproduce the two-slope decay observed in measured transients (Morgado et al., 2018). This highlights the critical role of tempering in modeling physical processes with transitional dynamics.

6. Interpretation of the Tempering Parameter and Memory Effects

The tempering parameter $\lambda$ exponentially suppresses the contribution of remote memory in the fractional kernel. As $\lambda$ increases, history effects decay more quickly, leading to accelerated decay in the kernel coefficients and reducing the required number of exponentials $N_{exp}$ for fixed error tolerance. Physically, this truncation reflects environments where memory becomes effectively finite, allowing TTFADE to describe transitions from nonlocal subdiffusion to more classical transport at longer times. In practical applications such as disordered media, varying $\lambda$ allows fitting of empirically observed transitions between power-law regimes, a property unattainable with untempered fractional models (Huang et al., 17 Dec 2025, Morgado et al., 2018).

7. Applications and Outlook

TTFADE is instrumental in modeling systems where anomalous transport is modulated by finite memory, typical in complex media, porous flow, and disordered materials. Applications to time-of-flight transient current experiments in boron demonstrate the efficacy of TTFADE for capturing phenomena unattainable with non-tempered FADEs (Morgado et al., 2018). The numerical advances—particularly fast sum-of-exponentials history handling—enable practical large-scale simulation while retaining second-order accuracy under weak initial data singularities (Huang et al., 17 Dec 2025).

Research directions include extension of the model to higher dimensions, nonhomogeneous boundary conditions, nonlinear forcing, and coupling with spatial fractional operators. The flexibility conferred by the tempering parameter continues to drive new physical insights and computational efficiencies.