Source-linked AI summary
Hilfer-Prabhakar Derivatives and Some Applications
Roberto Garra, Rudolf Gorenflo, Federico Polito, Zivorad Tomovski
TL;DR
The paper addresses how to generalize Hilfer fractional derivatives while extending the underlying integral operator beyond the Riemann–Liouville form. It defines and analyzes Hilfer–Prabhakar derivatives, then applies them to heat and free-electron-laser equations and generalized renewal processes, while identifying scope constraints for regularization and model recovery.
Problem
Existing fractional calculus uses multiple integral and derivative definitions, motivating a generalization of Hilfer derivatives based on more general integrals.
Method
The paper replaces Riemann–Liouville integrals with Prabhakar integrals, derives regularized and non-regularized operators and transforms, and applies them to mathematical-physics and renewal-process equations.
Results
The generalized operators yield analytical solutions for time-fractional heat Cauchy problems and state probabilities for generalized renewal processes, while encompassing specified classical equations as limiting cases.
Takeaways & Limitations
Hilfer–Prabhakar derivatives provide a common framework for fractional heat, free-electron-laser, and generalized renewal-process equations.
Takeaways & Limitations
Regularized formulations require absolutely continuous functions, and the original free-electron-laser equation is recovered only under specified parameter choices and limits.
Abstract
from arXiv · showhide
We present a generalization of Hilfer derivatives in which Riemann--Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Further, we show some applications of these generalized Hilfer-Prabhakar derivatives in classical equations of mathematical physics, like the heat and the free electron laser equations, and in difference-differential equations governing the dynamics of generalized renewal stochastic processes.
1. Introduction
Fractional calculus has expanded through diverse integral and derivative definitions and increasingly intersects applied science, probability, and stochastic-process research. The paper introduces Prabhakar-based generalizations and studies their mathematical-physics and probability applications.
- Fractional-calculus research increasingly develops applied models using fractional operators.
- The field includes well-studied Riemann–Liouville and Caputo operators alongside less-established definitions such as Hadamard and Marchaud derivatives.
- The Prabhakar integral extends the Riemann–Liouville kernel with a three-parameter Mittag–Leffler function.
- The Hilfer–Prabhakar derivative contains the Hilfer derivative as a special case and interpolates Prabhakar and regularized Caputo-like derivatives.
- Applications include time-fractional heat equations, free electron laser equations, and difference-differential equations for generalized renewal processes.
2. Preliminaries on fractional calculus
The paper reviews Riemann–Liouville integrals and derivatives, Caputo derivatives, and the function spaces needed to relate regularized and non-regularized operators. These preliminaries establish when Riemann–Liouville derivatives can be regularized.
- The preliminaries define the Riemann–Liouville integral for locally integrable functions on an interval.
- The Riemann–Liouville derivative is defined for functions satisfying an integrability and Sobolev-space condition.
- The Caputo derivative is introduced for functions in AC^m[a,b], with m equal to the ceiling of α.
- The reviewed operators are connected through a relation between Riemann–Liouville and Caputo derivatives on absolutely continuous function spaces.
- For functions in AC^m[a,b], a theorem expresses the Riemann–Liouville derivative in a form that permits regularization.
3. Hilfer derivatives
Hilfer’s generalized fractional derivative includes the Riemann–Liouville and Caputo derivatives as limiting cases. Its fractional-integral initial conditions lack clear physical meaning except in the Caputo case, motivating a regularized formulation.
- The Hilfer derivative generalizes operators whose specific cases include the Riemann–Liouville and Caputo derivatives.
- Setting ν=0 yields the Riemann–Liouville derivative, while ν=1 yields the Caputo derivative.
- Hilfer Cauchy problems require initial values involving a fractional integral of order (1−ν)(1−µ).
- These fractional-integral initial conditions do not have clear physical meaning unless ν=1.
- On AC^1[0,b], the Hilfer derivative coincides with the Riemann–Liouville derivative of order µ, and its regularized form is obtained accordingly.
4. Hilfer–Prabhakar derivatives
The paper replaces Riemann–Liouville integrals with Prabhakar integrals to define Hilfer–Prabhakar derivatives and analyzes their regularized forms and transforms. The construction recovers established operators under specific parameter choices.
- Hilfer–Prabhakar derivatives: The Hilfer–Prabhakar generalization substitutes Riemann–Liouville integrals with a kernel-based Prabhakar integral operator.
- Prabhakar operators: The Prabhakar integral uses a generalized Mittag–Leffler function, and its inverse is the Prabhakar derivative.
- Hilfer–Prabhakar derivatives: The Hilfer–Prabhakar derivative is defined to interpolate the Prabhakar derivative and its regularized Caputo counterpart.
- Transform properties: The regularized operator does not depend on the interpolating parameter ν, and both operators receive Laplace-transform formulas.
- Regularization: The regularized Hilfer–Prabhakar derivative is introduced for absolutely continuous functions to use initial conditions involving the function and integer-order derivatives.
5. Applications
The paper applies Hilfer–Prabhakar derivatives to generalized heat and free-electron-laser equations, and to generalized fractional Poisson and renewal processes. It derives analytical solutions, establishes convergence and positivity conditions, and gives a subordination representation for the stochastic process.
- Time-fractional heat equation: The generalized time-fractional heat equation is analyzed using both non-regularized and regularized Hilfer–Prabhakar derivatives.The paper states Cauchy-problem solutions under parameter ranges including µ ∈ (0, 1), ν ∈ [0, 1], ρ > 0, and γ ≥ 0.
- Time-fractional heat equation: Fourier–Laplace transforms and inverse transforms are used to derive the heat-equation solutions and prove absolute convergence of the resulting series.The convergence argument uses the entire-function property of the generalized Mittag–Leffler function and large-argument asymptotics.
- Fractional free electron laser equation: The fractional free electron laser equation generalizes the classical FEL equation by introducing Hilfer–Prabhakar derivatives and an inhomogeneous function f(x).The original FEL equation is recovered for γ = 0, ν = 0, µ → 1, f ≡ 0, and the specified parameter choices.
- Fractional free electron laser equation: The paper gives explicit solutions for the generalized FEL Cauchy problem and illustrates them with cases such as κ = 0 and f(x) = x^{m−1}.The solutions are obtained by Laplace transformation, inversion, and the convolution theorem.
- Fractional Poisson processes: The generalized fractional Poisson process is defined through difference-differential equations containing the regularized Hilfer–Prabhakar operator in time.Its state probabilities are represented by infinite series and integral forms, while its waiting distribution characterizes the associated renewal process.
- Fractional Poisson processes: Non-negativity of the generalized Poisson state probabilities is ensured under the parameter constraints 0 < µ⌈γ⌉/γ − rρ < 1 for r = 0, …, ⌈γ⌉ when γ ≠ 0.The proof uses complete monotonicity and closure properties of Bernstein functions; a subordination theorem also relates the process to a homogeneous Poisson process evaluated at a hitting-time process.
Appendix A. Comments on the proofs of Theorems 5.1 and 5.4
The appendix explains how inverse Laplace-transform proofs require contours satisfying several constraints, whose validity regions are illustrated for two parameter settings.
- Theorem 5.1 proof: The Bromwich contour is chosen so that all singularities lie to the left of the integration path.This enables termwise inversion of the Laplace transforms.
- Theorem 5.1 proof: The inversion abscissa must satisfy ℜ(s) > 0, |ωs^-ρ| < 1, and |Kk^2/(s^µ(1 −ωs^-ρ)^γ)| < 1.The required abscissa depends on k, which varies over R.
- Theorem 5.1 proof: Figure A.1 plots the blue validity region for |Kk^2/(s^µ(1 −ωs^-ρ)^γ)| < 1 in the complex s-plane.The plotted parameters are (K, k, µ, ω, ρ, γ) = (1, 1, 0.5, −1, 0.25, 1).
- Theorem 5.1 proof: The inverse Laplace transform can be applied term by term when the abscissa is sufficiently large, with absolute convergence for t ≥ 0.The text notes that the convergence is independent of the chosen sufficiently large abscissa.
- Theorem 5.4 proof: Figure A.2 shows the validity region for |λ(1 −v)/(s^µ(1 + φs^-ρ)^γ)| < 1 under a second parameter setting.The plotted parameters are (λ, v, µ, φ, ρ, γ) = (1, 0.5, 0.5, 1, 0.25, 1).