Caputo-Tempered Derivative Overview

• Caputo-Tempered derivative is a modified fractional derivative that uses an exponentially truncated kernel to bridge anomalous (power-law) and normal (Gaussian) diffusion.
• Its formulation employs Laplace and Fourier transforms, ensuring a rigorous connection between fractional calculus and classical diffusion models.
• The derivative enables finite-variance modeling in fractional Cattaneo equations, mitigating unrealistic long-range jumps in transport phenomena.

The Caputo-Tempered derivative is a modification of the classical Caputo space-fractional derivative, in which the underlying convolution kernel is exponentially truncated. This construction systematically incorporates an exponential tempering parameter $\lambda>0$ into the fractional calculus framework, interpolating between strictly power-law memory and short-range exponential decay. As formalized in the context of space-fractional transport equations, notably the Cattaneo (telegrapher’s) equation, the Caputo-Tempered derivative enables the description of phenomena exhibiting a crossover from anomalous (fractional) to normal (Gaussian) diffusion, thereby providing a physically realistic mechanism suppressing unrealistically long-range jumps present in pure Caputo-based models (Beghin et al., 2022).

1. Formal Definition and Integral Representations

Let $f\in C^1([0,\infty))$ with $f(0)$ finite, $0<\alpha<1$, and $\lambda>0$. The Caputo-Tempered derivative of order $\alpha$ and tempering $\lambda$ is defined as

$$D_{x}^{\alpha,\lambda}\,f(x) := e^{-\lambda x}\;D_{x}^{C,\alpha}\bigl[e^{\lambda\,x}\,f(x)\bigr] = \frac{e^{-\lambda x}}{\Gamma(1-\alpha)} \int_{0}^{x} (x-y)^{-\alpha} \frac{d}{dy}\bigl(e^{\lambda y}f(y)\bigr)\,dy.$$

The tempered fractional integral of order $\gamma>0$ is

$$I_x^{\gamma,\lambda}f(x) = \frac{1}{\Gamma(\gamma)} \int_0^x (x-y)^{\gamma-1}\,e^{-\lambda(x-y)}\,f(y)\,dy,$$

providing the alternative forms

$$D_{x}^{\alpha,\lambda}f(x) = I_x^{1-\alpha,\lambda}[f'](x) = \frac{d}{dx} I_x^{1-\alpha,\lambda}f(x).$$

An explicit representation reflecting the tempering property is

$$D_{x}^{\alpha,\lambda}f(x) = \frac{\lambda^\alpha}{\Gamma(1-\alpha)}\int_{0}^{x} \frac{f(x)-f(y)}{e^{\lambda(x-y)}\,(x-y)^{\alpha+1}}\,dy + \lambda^\alpha f(x),$$

where the exponential $e^{-\lambda(x-y)}$ truncates the algebraic kernel.

2. Laplace and Fourier Transform Symbols

Under the one-sided Laplace transform,

$$\mathcal L\{f\}(s) = \int_0^\infty e^{-s x}f(x)\,dx,$$

the Caputo-Tempered derivative satisfies

$$\mathcal L\{D_{x}^{\alpha,\lambda}f(x)\}(s) = (s+\lambda)^\alpha\,\mathcal L\{f\}(s) - (s+\lambda)^{\alpha-1}f(0).$$

If the domain is extended to all real $x$ with vanishing boundary terms at $\pm\infty$, the Fourier transform yields

$$\mathcal F\{D_{x}^{\alpha,\lambda}f(x)\}(k) = \left[(\lambda-ik)^{\alpha} - \lambda^{\alpha}\right]\widehat f(k),$$

where $\widehat f(k)$ is the Fourier transform of $f$. The transform symbol reduces to the classical form $(-ik)^{\alpha}$ in the limit $\lambda\to0$.

3. Application in the Time-Fractional Cattaneo Equation

The Caputo-Tempered derivative is employed within a generalized form of the Cattaneo (telegrapher's) equation, a canonical model for heat transfer and wave propagation. In this context, the second spatial derivative in the classical model

$$\partial^2_t u + 2 k\,\partial_t u = c^2 \partial_x^2 u$$

is replaced by $D_{x}^{\alpha,\lambda}$, and the time derivative is fractionalized using the Caputo derivative of order $\beta\in(0,\tfrac12)$. The equation reads

$$D_{t}^{2\beta}u(x,t) + 2k\,D_{t}^{\beta}u(x,t) = D_{x}^{\alpha,\lambda}\,u(x,t), \quad x \in \mathbb{R},\; t>0,$$

subject to the initial and boundary conditions

$$u(x,0) = \delta(x), \quad D_{t}^{\beta}u(x,t)|_{t=0} = 0, \quad \lim_{|x|\to\infty} u(x,t) = 0.$$

With the kernel $e^{-\lambda(x-y)}$, the spatial flux law acquires an exponentially truncated memory, providing a continuous interpolation between ballistic (wave-like) and diffusive transport regimes.

4. Solution: Characteristic Function and Stochastic Process

The Fourier transform in $x$ and Laplace transform in $t$ reduce the time-fractional Cattaneo equation to a solvable algebraic form. Define

$$\widehat u(k,t) = \mathcal F\{u(\cdot,t)\}(k), \quad \widetilde{\widehat u}(k,s) = \mathcal L_t\{\widehat u(k,\cdot)\}(s).$$

Given the symbol $$\Psi_{\alpha,\lambda}(k)=(\lambda-ik)^\alpha-\lambda^\alpha$$ and Caputo time-fractional symbol $s^{\beta}$, one obtains

$$\widetilde{\widehat u}(k,s) = \frac{s^{2\beta-1} + 2k\,s^{\beta-1}}{s^{2\beta} + 2k\,s^{\beta} + \Psi_{\alpha,\lambda}(k)}.$$

Factoring the denominator with $$r_{1,2}(k) = -k \pm \sqrt{k^2 - \Psi_{\alpha,\lambda}(k)}$$, the solution for the characteristic function is

$$\widehat u(k,t) = \frac{1}{2}\left(1 + \frac{k}{\sqrt{k^2-\Psi_{\alpha,\lambda}(k)}}\right) E_{\beta,1}(r_1\,t^\beta) + \frac{1}{2}\left(1 - \frac{k}{\sqrt{k^2-\Psi_{\alpha,\lambda}(k)}}\right) E_{\beta,1}(r_2\,t^\beta),$$

where $E_{\beta,1}$ is the Mittag–Leffler function. $\widehat u(k,t)$ serves as the characteristic function of the associated random motion $W(t)$,

$$W(t) = B\left(T_{\alpha,\lambda}\circ L_\beta(t)\right),$$

with $T_{\alpha,\lambda}$ a tempered stable subordinator (generator $\Psi_{\alpha,\lambda}(k)$) and $L_\beta$ its inverse of index $\beta$. This construction yields finite moments for all orders and reproduces the two-term Mittag–Leffler mixture in closed form.

5. Tempering vs. Classical Caputo Fractional Derivative

A principal distinction arises between the tempered ($\lambda>0$) and un-tempered ($\lambda=0$) cases:

• With $\lambda=0$, $D_x^{\alpha,0}$ reduces to the standard Caputo or Riemann–Liouville space-fractional derivative, possessing algebraically decaying kernel $(x-y)^{-1-\alpha}$ and symbol $(-ik)^{\alpha}$, admitting long-range “jumps” in underlying Lévy processes.
• For $\lambda>0$, the exponential cutoff $e^{-\lambda(x-y)}$ suppresses the kernel’s tail, producing the transform symbol

$$\Psi_{\alpha,\lambda}(k) = (\lambda-ik)^{\alpha} - \lambda^{\alpha},$$

leading to exponential damping at large $|x|$ or $|k|$. This tempering imparts finite moments of all orders and causes a continuous transition from fractional to standard Gaussian behavior with increasing scale.

• In the Cattaneo equation, tempering tunes the transition between ballistic and diffusive dynamics, mitigating the issue of infinite variance and unrealistic long-distance propagation characteristic of pure power-law kernels.

6. Physical Interpretation and Modeling Implications

The Caputo-Tempered derivative modifies the nonlocal spatial memory inherent in fractional models, enabling finite-variance stochastic descriptions and reconciling anomalous transport with classical diffusion over large domains. Within the time-fractional Cattaneo equation, the derivative yields a solution characterized by a two-term Mittag–Leffler mixture for the process characteristic function. This formulation precisely interpolates between the limiting regimes—wave-like, ballistic motion at short time or length scales, and normal diffusion at longer times or distances. The exponential truncation, both in real-space convolution and transform domains, is essential for capturing crossover phenomena and constraining anomalous propagation in physical models (Beghin et al., 2022).