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

$f\in C^1([0,\infty))$0

An explicit representation reflecting the tempering property is

$f\in C^1([0,\infty))$1

where the exponential $f\in C^1([0,\infty))$2 truncates the algebraic kernel.

2. Laplace and Fourier Transform Symbols

Under the one-sided Laplace transform,

$f\in C^1([0,\infty))$3

the Caputo-Tempered derivative satisfies

$f\in C^1([0,\infty))$4

If the domain is extended to all real $f\in C^1([0,\infty))$5 with vanishing boundary terms at $f\in C^1([0,\infty))$6, the Fourier transform yields

$f\in C^1([0,\infty))$7

where $f\in C^1([0,\infty))$8 is the Fourier transform of $f\in C^1([0,\infty))$9. The transform symbol reduces to the classical form $f(0)$0 in the limit $f(0)$1.

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

$f(0)$2

is replaced by $f(0)$3, and the time derivative is fractionalized using the Caputo derivative of order $f(0)$4. The equation reads

$f(0)$5

subject to the initial and boundary conditions

$f(0)$6

With the kernel $f(0)$7, 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 $f(0)$8 and Laplace transform in $f(0)$9 reduce the time-fractional Cattaneo equation to a solvable algebraic form. Define

$0<\alpha<1$0

Given the symbol $0<\alpha<1$1 and Caputo time-fractional symbol $0<\alpha<1$2, one obtains

$0<\alpha<1$3

Factoring the denominator with $0<\alpha<1$4, the solution for the characteristic function is

$0<\alpha<1$5

where $0<\alpha<1$6 is the Mittag–Leffler function. $0<\alpha<1$7 serves as the characteristic function of the associated random motion $0<\alpha<1$8,

$0<\alpha<1$9

with $\lambda>0$0 a tempered stable subordinator (generator $\lambda>0$1) and $\lambda>0$2 its inverse of index $\lambda>0$3. 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$4) and un-tempered ($\lambda>0$5) cases:

• With $\lambda>0$6, $\lambda>0$7 reduces to the standard Caputo or Riemann–Liouville space-fractional derivative, possessing algebraically decaying kernel $\lambda>0$8 and symbol $\lambda>0$9, admitting long-range “jumps” in underlying Lévy processes.
• For $\alpha$0, the exponential cutoff $\alpha$1 suppresses the kernel’s tail, producing the transform symbol

$\alpha$2

leading to exponential damping at large $\alpha$3 or $\alpha$4. 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).