Prabhakar-Type Kernels in Fractional Calculus

Updated 13 January 2026

Prabhakar-type kernels are convolution kernels derived from the three-parameter Mittag-Leffler function that generalize Riemann–Liouville and Caputo operators for modeling memory effects.
They exhibit explicit Laplace-domain representations and semigroup convolution laws, offering analytical tractability and a rich algebraic structure.
Applications span anomalous transport, fractional viscoelasticity, nonlocal PDEs, and complex relaxation phenomena in various physical systems.

Prabhakar-Type Kernels

Prabhakar-type kernels, a generalization of the convolution kernels underlying classical fractional calculus, arise from the three-parameter Mittag-Leffler function introduced by T.R. Prabhakar (1971). These kernels define a flexible family of weakly singular, completely monotone, memory kernels that underpin the modern theory of generalized fractional integrals and derivatives. They interpolate and extend the classical Riemann–Liouville and Caputo operators, have explicit Laplace-domain representations, and admit rich algebraic and analytic structures. Such kernels are central in modeling anomalous transport, non-Debye relaxation, fractional viscoelasticity, statistical physics, and the analysis of nonlocal PDEs with memory effects.

1. Definition and Core Structure

The Prabhakar kernel is constructed from the three-parameter Mittag-Leffler function: $E_{\alpha,\beta}^{\gamma}(z) = \sum_{k=0}^{\infty}\frac{(\gamma)_k}{k! \,\Gamma(\alpha k+\beta)} z^k$ where $(\gamma)_k = \Gamma(\gamma + k)/\Gamma(\gamma)$ , and parameters satisfy $\Re(\alpha)>0$ , $\Re(\beta)>0$ , $\gamma\in\mathbb{C}$ (Giusti, 2019, Garra et al., 2017, Garrappa et al., 2020). The Prabhakar kernel itself is given by

$K_{\alpha,\beta}^{\gamma}(t;\omega) = t^{\beta-1}\, E_{\alpha,\beta}^{\gamma}(\omega t^{\alpha}),\quad t>0,$

with scaling/frequency parameter $\omega\in\mathbb{C}$ . When $\gamma=1$ and $\omega=0$ , this reduces to the Riemann–Liouville kernel $t^{\beta-1}/\Gamma(\beta)$ .

The convolution operator associated with the Prabhakar kernel,

$\left( I_{\alpha,\beta}^{\gamma,\omega} f \right)(t) = \int_{0}^{t} (t-\tau)^{\beta-1} E_{\alpha,\beta}^{\gamma}\left(\omega (t-\tau)^{\alpha}\right) f(\tau) d\tau,$

generalizes the Volterra operator of classical fractional calculus (Giusti, 2019, Giusti et al., 2017, Giusti et al., 2020).

2. Analytical and Operational Properties

2.1 Laplace Transform

A fundamental property is the closed-form Laplace transform,

$\mathcal{L}\{ t^{\beta-1} E_{\alpha,\beta}^{\gamma}(\omega t^{\alpha}) \}(s) = s^{-\beta}(1-\omega s^{-\alpha})^{-\gamma}, \quad \Re(s)>|\omega|^{1/\alpha}.$

This formula underpins the operational calculus for Prabhakar-type kernels and simplifies the analysis of fractional differential equations (Giusti, 2019, Giusti et al., 2020, Garra et al., 2017, Garrappa et al., 2020).

2.2 Convolution and Semigroup Laws

Prabhakar operators exhibit an additive two-parameter semigroup under convolution: $I_{\alpha,\beta_1}^{\gamma_1} \circ I_{\alpha,\beta_2}^{\gamma_2} = I_{\alpha,\,\beta_1+\beta_2}^{\,\gamma_1+\gamma_2},$ and their kernels satisfy $K_{\alpha,\beta_1}^{\gamma_1}*K_{\alpha,\beta_2}^{\gamma_2}=K_{\alpha,\beta_1+\beta_2}^{\gamma_1+\gamma_2}$ (Giusti et al., 2020, Pachpatte et al., 2017, Polito et al., 2015).

2.3 Complete Monotonicity

For specified parameters $0<\alpha\leq1$ , $0<\gamma\alpha\leq\beta\leq1$ , and $\omega<0$ or $\lambda\leq0$ , the kernel $t\mapsto t^{\beta-1}E_{\alpha,\beta}^{\gamma}(-|\lambda| t^\alpha)$ is completely monotone on $(0,\infty)$ , i.e., $(-1)^n d^n/dt^n K_{\alpha,\beta}^{\gamma}(t;\lambda)\geq0$ for all $n$ (Giusti, 2019, Polito et al., 2015, Garra et al., 2017, Garrappa et al., 2020). This property underlies the positivity of relaxation moduli in viscoelasticity and the well-posedness of various evolution equations.

2.4 Asymptotic Behavior

As $t\to 0^+$ , the kernel displays an initial power-law singularity,

$t^{\beta-1}E_{\alpha,\beta}^{\gamma}(z) \sim t^{\beta-1}/\Gamma(\beta).$

As $t\to\infty$ (along the negative real axis), the kernel decays algebraically: $E_{\alpha,\beta}^\gamma(-t^{\alpha}) \sim \frac{t^{-\alpha\gamma}}{\Gamma(\beta-\alpha\gamma)},$ provided $\beta \ne \alpha\gamma$ (Giusti et al., 2020, Garrappa et al., 2020, Garra et al., 2017). For more general arguments, the asymptotics involve both algebraic and exponential components with sector-dependent expansions.

3. Fractional Operators Built from Prabhakar Kernels

Prabhakar kernels induce a hierarchy of generalized fractional integrals and derivatives:

Prabhakar-type fractional integral:

$(I_{\alpha, \beta}^{\gamma, \omega} f)(t) = \int_{0}^{t} (t-\tau)^{\beta-1} E_{\alpha, \beta}^{\gamma}(\omega (t-\tau)^{\alpha}) f(\tau) d\tau.$

Prabhakar (Riemann–Liouville-type) fractional derivative (for $m=\lceil \Re \beta \rceil$ ):

$D_{\alpha, \beta, \omega}^{\gamma} f(t) = \frac{d^m}{dt^m} \int_{0}^{t} (t-\tau)^{m-\beta-1} E_{\alpha, m-\beta}^{-\gamma}(\omega (t-\tau)^{\alpha}) f(\tau)\, d\tau.$

Caputo–Prabhakar regularized derivative:

${}^{C}D_{\alpha, \beta, \omega}^{\gamma} f(t) = \int_{0}^{t}(t-\tau)^{m-\beta-1} E_{\alpha, m-\beta}^{-\gamma}(\omega (t-\tau)^{\alpha}) f^{(m)}(\tau)\, d\tau.$

Hilfer–Prabhakar and nth-level generalizations: These operators interpolate between Caputo- and Riemann–Liouville-type forms by varying the sequence and nesting of fractional integrals and derivatives, and support operational calculus with Mikusinski-type fields (Waheed et al., 24 Dec 2025, Garra et al., 2014, Polito et al., 2015).

Extensions include distributed-order Prabhakar kernels and derivatives by integrating over secondary indices ( $\mu$ ) or composing with arbitrary monotone functions ( $\phi$ ) as in generalized operators with respect to functions (Górska et al., 2022, Fernandez et al., 2022).

4. Probabilistic, Spectral, and Memory Properties

4.1 Probabilistic Mixtures and Complete Monotonicity

Prabhakar kernels admit representation as mixtures of Riemann–Liouville integrals of one-sided stable densities $f_\alpha(x|t)$ convoluted against gamma laws, leading to explicit spectral densities and confirming complete monotonicity as a Laplace transform of a positive law (Sibisi, 2023). In particular,

$x^{\beta-1} E_{\alpha,\beta}^\gamma(-\lambda x^\alpha) = \frac{1}{\Gamma(\gamma)} \int_0^\infty I_{0+}^{\beta-\alpha\gamma}[f_\alpha(\cdot|t)](x) \, t^{\gamma-1} e^{-\lambda t} dt,$

which, after integration, confirms the function's Stieltjes nature and suitability as a relaxation kernel (Górska et al., 2022, Sibisi, 2023).

4.2 Spectral Representation

Any completely monotone Prabhakar kernel can be expressed via

$K_{\alpha,\beta}^{\gamma}(t; -1) = \int_0^{\infty} e^{-r t} K_{\alpha,\beta}^{\gamma}(r)\, dr,$

with explicit formulas for the density $K_{\alpha,\beta}^{\gamma}(r)$ (Polito et al., 2015).

4.3 Fading Memory

Laplace-domain asymptotics guarantee Prabhakar kernels encode strong nonlocality and power-law memory: $\hat{k}(s)\sim s^{-1-\mu+\alpha \nu}$ yields a "fading memory" effect, with memory tails tunable by the triple $(\alpha, \beta, \gamma)$ and yielding a broad range of relaxation and anomalous transport regimes (Górska et al., 2021, Karimov et al., 11 Mar 2025).

5. Applications in Fractional Differential Equations and Models

Prabhakar kernels are central in the analysis of:

Fractional relaxation and diffusion processes: Prabhakar derivatives generalize classical models like the Cole–Cole, Davidson–Cole, and Havriliak–Negami laws, with direct correspondence between the susceptibility or relaxation modulus and the Laplace transform of Prabhakar kernels (Garra et al., 2017, Giusti et al., 2017, Giusti et al., 2020).
Memory and viscoelasticity: Inclusion of Prabhakar operators in constitutive laws allows interpolation between exponential (Debye), pure power-law, and stretched-exponential (Kohlrausch) behaviors, fitting complex measured responses in viscoelasticity (Giusti et al., 2017).
Volterra and boundary-value problems: Nonlocal and boundary-value PDEs with Caputo–Prabhakar or Riemann–Liouville–Prabhakar derivatives can be reduced systematically to Volterra integral equations with Prabhakar-type kernels, allowing for explicit construction of Green's functions and proofs of existence and uniqueness (Karimov et al., 11 Mar 2025, Karimov et al., 24 Dec 2025).
Stochastic processes and fractional Poisson/Bernoulli processes: Prabhakar kernels serve as waiting-time densities in renewal and counting processes, with discrete-time analogues converging to Prabhakar-driven processes in the scaling limit (Michelitsch et al., 2020).
Generalized Fokker–Planck equations and nonlocal diffusion: Distributional mixtures, complete monotonicity, and Stieltjes properties ensure positivity and normalization of fundamental solutions under Prabhakar memory (Górska et al., 2022).
Linear/nonlinear equations with variable coefficients: Via compositional and operational properties, explicit infinite-series and closed-form solutions can be constructed for variable- or constant-coefficient equations, often involving multivariate Mittag-Leffler functions (Fernandez et al., 2022).

6. Numerical Methods and Implementation

Prabhakar kernels in time-stepping and numerical methods are typically handled either by:

Truncated power series: When arguments are small, the defining series can be truncated, though numerical instabilities limit practical truncation sizes.
Bromwich inversion: For moderate to large arguments, contour-integration based inversion from the Laplace domain is efficient and robust (Giusti et al., 2020, Garrappa et al., 2020).
Convolution quadrature: For time-stepping applications, the kernel's Laplace symbol is embedded in convolution-quadrature weights via generating functions or integral formulas, supporting fast and stable discretizations (Garrappa et al., 2020).

7. Generalizations and Parameter Regimes

The flexibility of the Prabhakar kernel family is realized by varying $(\alpha, \beta, \gamma, \omega)$ :

Special cases: Setting $\gamma=1$ recovers the two-parameter Mittag-Leffler, $\gamma=1$ , $\beta=1$ recovers the classical Mittag-Leffler, $\omega=0$ recovers the Riemann–Liouville kernel, and $\alpha=\beta=\gamma=1$ yields the exponential kernel.
nth-level and distributed-order kernels: High-level generalizations involve nested compositions and integration over distributions of orders in $(\alpha,\beta)$ ("distributed order") (Waheed et al., 24 Dec 2025, Górska et al., 2022).
Semigroup and commutativity: Composition rules provide constructive semigroup laws, though strict independence in $\beta$ and $\gamma$ is lost for the derivatives involving integer differentiation.

The admissible parameter domain for complete monotonicity and positivity is $0<\alpha\le1$ , $0<\beta\le1$ , $0<\gamma\alpha\le\beta$ , with $\omega\le 0$ .