Tempered Fractional Derivative

• Tempered fractional derivative is a generalization of classical fractional derivatives, modulating power-law kernels with an exponential factor to ensure finite moments.
• It is applied to model phenomena such as anomalous diffusion, nonlocal turbulence, and complex relaxation processes in physics and finance.
• Advanced numerical methods, including quasi-compact schemes and SOE approximations, enable efficient high-order solutions for tempered derivative problems.

The tempered fractional derivative is a generalization of the classical fractional derivative in which the algebraic kernel $(t-s)^{-\alpha}$ is modulated by an exponential factor, yielding integral operators of the form

$$D_{a}^{\alpha, \lambda} u(t) = e^{-\lambda t} D_{a}^{\alpha}[e^{\lambda t} u(t)]$$

and

$$D_{a}^{\alpha, \lambda} u(t) = \frac{1}{\Gamma(n - \alpha)} e^{-\lambda t} \frac{d^n}{dt^n} \int_a^t (t-s)^{n-\alpha-1} e^{\lambda s} u(s) ds, \qquad n-1 < \alpha < n,\, \lambda \geq 0,$$

where $D_{a}^{\alpha}$ is the standard Riemann–Liouville fractional derivative. The exponential tempering factor truncates the heavy tails of the underlying power-law kernel, making the model physically realistic by guaranteeing finite moments, such as the second moment in Lévy flight models. These operators are essential in modeling anomalous diffusion, nonlocal turbulence, complex relaxation phenomena, and offer well-developed analytical and numerical frameworks for both direct and inverse problems.

1. Mathematical Foundations and Operator Definitions

The canonical tempered fractional integral is given by

$$I_{a}^{\gamma, \lambda} u(t) = \frac{1}{\Gamma(\gamma)} \int_a^t e^{-\lambda (t-s)} (t-s)^{\gamma-1} u(s) ds,$$

where the exponential term introduces localization with the tempering parameter $\lambda \geq 0$. The tempered Riemann–Liouville derivative of order $\gamma$ (with $n-1 < \gamma < n$) is

$$D_{a}^{\gamma, \lambda} u(t) = e^{-\lambda t} \frac{d^n}{dt^n} \left[ e^{\lambda t} I_a^{n-\gamma, \lambda} u(t) \right].$$

A tempered Caputo derivative alternatively reads

$$^C\!D_{a}^{\gamma,\lambda} u(t) = e^{-\lambda t} I_a^{n-\gamma, \lambda} \left[ e^{\lambda t} u^{(n)}(t) \right].$$

In the spatial case, the left Riemann–Liouville tempered fractional derivative for $u(x)$ on $[a, b]$ is

$_{a}D_{x}^{\alpha, \lambda} u(x) = e^{-\lambda x} \, _{a}D_{x}^{\alpha}(e^{\lambda x} u(x)) = \frac{e^{-\lambda x}}{\Gamma(n - \alpha)} \frac{d^n}{dx^n} \int_a^x (x-s)^{n-\alpha-1} e^{\lambda s} u(s) ds.$

Corresponding Fourier or Laplace transforms replace $s \rightarrow s+\lambda$, leading to symbolizations of the form $(s+\lambda)^{\alpha}$ or $(\lambda + i\omega)^{\mu}$, providing compact representations suitable for harmonic analysis in infinite or periodic domains (Chen et al., 2017, Beghin et al., 2022).

For $0 < \gamma < 1$, the Caputo–Djrbashian tempered derivative is: $$\frac{\partial^{\gamma,\lambda}f(t)}{\partial t^{\gamma}} = \frac{1}{\Gamma(1-\gamma)} \int_0^t \exp(-\lambda(t-\tau)) (t-\tau)^{-\gamma} \frac{\partial f(\tau)}{\partial \tau} d\tau.$$

Tempered derivatives admit linearity, the semigroup property (in suitable function classes), and tight connections to convolution structures and special functions including Mittag–Leffler and hypergeometric functions (Fernandez et al., 2019).

2. Stochastic and Physical Motivation

The leading physical motivation is the continuous time random walk (CTRW), where jumps or waiting times are governed by heavy-tailed distributions. Power-law waiting time densities produce fractional derivatives but yield infinite mean sojourn times. Exponential tempering modulates these distributions: $p(t) \sim t^{-1-\alpha} e^{-\lambda t},$ ensuring all moments exist. In the space-fractional case, the corresponding Fokker–Planck equations transition from

$$\text{(non-tempered)}\quad |x|^{-1-\alpha} \rightarrow \text{(tempered)}\quad |x|^{-1-\alpha} e^{-\lambda|x|},$$

which alters the associated generator to a tempered nonlocal operator.

The inverse tempered stable subordinator is used to time-change renewal processes, e.g. in the tempered fractional Hawkes process, yielding semi-heavy tailed phenomena with finite variance and exponentially vanishing memory at long times (Gupta et al., 16 May 2024). In turbulence modeling, tempering mimics the finite propagation horizon in unbounded domains (Mehta, 2023).

The resulting stochastic processes and their governing equations exhibit semi-long range dependence and provide a more robust and physically interpretable modeling framework for transient anomalous diffusion, option pricing in finance, and relaxation in complex media.

3. Numerical Discretization and High-Order Schemes

Efficient and accurate scheme construction for tempered fractional derivatives exploits their shifted convolution structure:

• Quasi-compact finite difference: The fourth-order (in space) quasi-compact scheme combines shifted Grünwald–Letnikov operators with tempering in the weights, e.g.

$P_x^\lambda \, _{a}D_x^{\alpha,\lambda} u(x) = \mu_1 \Delta_1^{\alpha, \lambda} u(x) + \mu_0 \Delta_0^{\alpha, \lambda} u(x) + \mu_{-1} \Delta_{-1}^{\alpha, \lambda} u(x) + O(h^4),$

where each $\Delta_p^{\alpha, \lambda}$ sums over $g_k^{(\alpha)} e^{-(k-p)\lambda h}$-weighted shifted values, and $P_x^\lambda$ is a compact correction operator (Yu et al., 2014).

• Time stepping: The Jacobi predictor–corrector method (and its fast equidistributing variants) applies high-order quadrature (e.g., Jacobi–Gauss–Lobatto nodes) to the weakly singular tempered convolution, maintaining linear computational cost via judicious kernel window selection exploiting rapid exponential decay (Li et al., 2015, Deng et al., 2015).
• SOE approximation: The sum-of-exponentials technique compresses the temporally nonlocal history kernel $t^{-\alpha} e^{-\lambda t}$ into a rapidly updatable, recursively computable sum, dramatically decreasing complexity in long-time or finely discretized regime (Feng et al., 2021, Zhou et al., 2023).
• Spectral and Petrov–Galerkin methods: On unbounded or semi-infinite domains, Laguerre and generalized Laguerre functions (GLFs) exactly diagonalize the action of tempered fractional operators, making spectral convergence achievable even in highly singular or infinite-horizon problems (Chen et al., 2017).
• Discontinuous Galerkin and finite element methods: LDG schemes with Lubich-type discretizations or Ritz–Galerkin projections admit high-order, unconditionally stable, strongly error-controlled implementations for tempered PDEs (Sun et al., 2017, Deng et al., 2016).

4. Analytical Properties, Special Functions, and Theoretical Advancements

Tempered fractional calculus preserves and extends many structures of ordinary fractional calculus:

• Analytical connections: Tempered operators are expressible as exponential conjugations of standard Riemann–Liouville operators:

$$\text{Tempered operator} = e^{-\lambda t} \cdot \text{Fractional Operator} \cdot e^{\lambda t}.$$

This allows direct series expansion into classical Riemann–Liouville operators and facilitates the transfer of analytic and spectral theory (Fernandez et al., 2019, Mali et al., 2021).

• Special function representations: Tempered differintegrals acting on power functions produce combinations of confluent and standard hypergeometric functions, Appell’s function, and generalizations of the Mittag–Leffler function. These connections are used for closed-form fundamental solutions and integral representations (Fernandez et al., 2019, Gupta et al., 16 May 2024).
• Taylor expansions and integral inequalities: Fractional Taylor’s theorems are established, expressing smooth functions in terms of repeated tempered derivatives. Synchronous function inequalities generalize classical integral bounds to this context (Fernandez et al., 2019).
• Functional-analytic setting: In Sobolev-type spaces, tempered derivatives and integrals satisfy boundedness, embedding, and integration-by-parts formulas parallel to their classical counterparts, ensuring well-posedness and the validity of variational principle derivations (Ledesma et al., 2023).

5. Applications in Physics, Finance, and Machine Learning

Applications span a wide range:

• Anomalous diffusion and CTRW models: Tempering modifies the propagator, endowing models with finite mean waiting or jump lengths, and is essential in media with bounded domains or finite temporal/spatial scales (Yu et al., 2014, Feng et al., 2021).
• Stochastic integration and Lévy processes: In stochastic process theory, tempered fractional integrals and derivatives are used to define new classes of transient anomalous diffusion with finite variance and semi-long-range dependence, including TFLP and TFLP II processes (Boniece et al., 2019).
• Option pricing and financial models: Tempered time-fractional Black–Scholes models encode semi-heavy tailed fluctuations in market prices, with robust, unconditionally stable, and high-accuracy fast solvers built on compact finite difference and SOE techniques (Zhou et al., 2023).
• Turbulence modeling: Fractional and tempered f-RANS models provide nonlocal closure relations admitting finite moments and tunable horizons of influence, matching the physical limitations of turbulent flows in bounded or unbounded domains (Mehta, 2023).
• Physics-informed machine learning: Physics-Informed Neural Networks (PINNs) are extended to tempered fractional PDEs by formulating the (tempered) fractional Laplacian as an expectation and using hybrid quadrature–Monte Carlo for efficient, scalable training to extremely high dimension (Hu et al., 17 Jun 2024).
• Variational principles: Fractional tempered Euler–Lagrange equations, mountain pass solutions, and explicit Noether-type constants of motion are derived for functionals containing Caputo and Riemann–Liouville tempered derivatives, extending classical calculus of variations (Ledesma et al., 2023).

6. Future Directions and Impact

Tempered fractional derivatives are under rapid methodological development. Prospective lines of inquiry include:

• Further extension to variable-order and multi-term tempered operators for complex multiscale phenomena.
• Development of robust, high-dimensional solvers combining mesh-free and mesh-based approaches, leveraging the rapid kernel decay.
• Deeper integration into stochastic calculus for generalized processes and in modeling of biological, geophysical, or financial systems where long memory coexists with cutoffs.
• Expansion of theoretical results, including spectral theory, controllability, and regularity for tempered nonlocal equations.
• Exploiting connections to special functions for analytic inversion and asymptotic analysis, particularly in inverse and parameter identification problems (Gupta et al., 16 May 2024, Liu et al., 2019, Feng et al., 2021).

The tempered fractional derivative thus provides a highly flexible and physically relevant extension of nonlocal calculus, underpinning a broad class of equations, efficient numerical methods, and modern data-driven modeling frameworks across applied mathematics, physics, and computational science.