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The Prabhakar or three parameter Mittag--Leffler function: theory and application
Roberto Garra, Roberto Garrappa
TL;DR
The paper addresses the incomplete explicit description of Prabhakar-function asymptotics for large complex arguments and the difficulty of deriving coefficients for its three parameters. It reviews the function and its fractional operators, develops complex-plane and negative-axis asymptotics, and studies nonlinear heat-conduction equations with memory, obtaining exact solutions and long-time behavior.
Problem
Explicit asymptotic behavior of the three-parameter Prabhakar function had not been described in a whole-complex-plane form, partly because its coefficients depend difficultly on three parameters.
Method
The paper reviews Prabhakar properties and operators, applies Fox-Wright asymptotic results with an algorithm for coefficient computation, and analyzes heat equations with Prabhakar derivatives.
Results
The paper explicitly represents large-argument asymptotic expansions in the whole complex plane, examines the negative semi-axis, and obtains exact solutions with long-time behavior for nonlinear heat-conduction equations.
Takeaways & Limitations
Prabhakar functions and derivatives provide a framework for analyzing anomalous dielectric models and nonlinear heat conduction with memory.
Takeaways & Limitations
Numerical evaluation of the Prabhakar function and numerical schemes for Prabhakar differential equations remain only partially developed.
Abstract
from arXiv · showhide
The Prabhakar function (namely, a three parameter Mittag-Leffler function) is investigated. This function plays a fundamental role in the description of the anomalous dielectric properties in disordered materials and heterogeneous systems manifesting simultaneous nonlocality and nonlinearity and, more generally, in models of Havriliak-Negami type. After reviewing some of the main properties of the function, the asymptotic expansion for large arguments is investigated in the whole complex plane and, with major emphasis, along the negative semi-axis. Fractional integral and derivative operators of Prabhakar type are hence considered and some nonlinear heat conduction equations with memory involving Prabhakar derivatives are studied.
1 Introduction
The paper introduces the three-parameter Prabhakar function and motivates its study through fractional calculus and anomalous dielectric models. It develops explicit large-argument asymptotics across the complex plane, with particular attention to the negative real axis, and applies Prabhakar derivatives to nonlinear heat conduction with memory.
- Motivation: The Prabhakar function is a three-parameter Mittag-Leffler function attracting increasing attention in fractional calculus and related applications.Mittag-Leffler functions play a role analogous to the exponential function in integer-order calculus.
- Research gap: Explicit asymptotic behavior in the whole complex plane remains under-explored because deriving coefficients is difficult when three parameters are involved.Earlier results were available only in limited cases, including expansions on the negative real axis.
- Contributions: The paper applies Fox-Wright asymptotic results to represent large-argument expansions of the Prabhakar function throughout the complex plane.It also provides an algorithmic procedure for computing expansion coefficients and examines the negative semi-axis in greater depth.
- Applications: The function is used to describe relaxation and response in anomalous dielectrics and Havriliak-Negami models involving simultaneous nonlocality and nonlinearity.Applications also include probability, stochastic processes, anisotropic systems, fractional viscoelasticity, and fractional boundary-value problems.
- Applications: Prabhakar-based integral and derivative operators are used to study nonlinear heat-conduction equations with memory.The paper obtains exact solutions and analyzes their behavior as t →∞.
2 Properties of the Prabhakar function
The paper reviews analytical, transform-based, monotonicity, and operator properties of the Prabhakar function. It relates the function to standard Mittag-Leffler and Fox-Wright functions and introduces associated fractional integral and derivative operators.
- Parameter reductions: When γ = 0, the Prabhakar function reduces to the standard two-parameter Mittag-Leffler function.For positive integer γ, it can also be expressed using values of the standard Mittag-Leffler function.
- Parameter reductions: When γ = −j with j ∈N, the Prabhakar function becomes a polynomial of degree j.
- Transforms and calculus: The Laplace-transform representation is useful for practical numerical computation after numerical inversion, although a direct transform of the Prabhakar function is not known.
- Transforms and calculus: Integration, differentiation, parameter-reduction, and higher-order derivative formulas can be obtained through Laplace transforms or term-by-term operations.
- Complete monotonicity: Complete monotonicity requires derivatives satisfying (−1)^k f^(k)(t) ≥0 and is relevant to monotone energy decay in isolated systems.The property has been studied extensively for Prabhakar functions used in dielectric models.
- Relationship with Fox-Wright functions: The Prabhakar function is a special case of the generalized Fox-Wright function, linking its properties to a broader multiparameter function class.
- Fractional operators: Prabhakar-type fractional integrals and derivatives are represented through convolution integrals, with practical applications often restricted to αγ ∈ (0, 1) and m = 1.A Caputo-type regularization and Grünwald-Letnikov characterization have also been studied.
3 Asymptotic expansion for large arguments
The paper derives explicit large-argument asymptotic expansions for the Prabhakar function across the complex plane, with special analysis of the negative real semi-axis. It also provides an algorithm for computing parameter-dependent coefficients and examines how many terms are needed in different regimes.
- Whole complex plane: The expansion strategy transfers asymptotic results for Fox–Wright functions to the Prabhakar function.The paper builds on prior Fox–Wright analyses, including Paris’s coefficient algorithm.
- Whole complex plane: Explicit asymptotic representations are obtained for large arguments throughout the complex plane, with sector-dependent choices for exponential contributions.The expansions use functions F and H with signs selected according to whether z lies in the upper or lower half-plane.
- Expansion on the negative semi-axis: For α > 2, the number of required terms is controlled by P, with P chosen so that 2P + 1 is the smallest qualifying odd integer.The paper illustrates this term count as a function of α in Figure 1.
- Whole complex plane: The coefficients c_k depend on α, β, and γ, and their complexity increases with k.An algorithm enables numerical evaluation of any number of coefficients, while the first few are listed explicitly.
- Expansion on the negative semi-axis: Along the negative semi-axis, the expansion combines algebraic terms with exponentially structured contributions whose relevance depends on α.For α < 2, exponentially decaying terms can be neglected asymptotically; at α = 2, the exponential in C0(t) contributes a constant.
- Expansion on the negative semi-axis: As r increases, the cosine factor decreases, so the first few functions G_r(t) can accurately describe the behavior of E^γ_α,β(−t).This explains why only a limited number of terms may be needed in the negative-axis expansion.
4 Nonlinear heat conduction equations with memory involving Prabhakar derivatives.
The paper introduces nonlinear heat-conduction models with memory by combining Caputo-type Prabhakar derivatives with temperature-dependent thermal coefficients. Using a generalized separating-variable method, it obtains particular exact solutions and analyzes their long-time behavior.
- The paper extends Prabhakar derivatives to nonlinear heat-conduction equations with memory, beyond earlier essentially linear diffusion and relaxation models.
- The models replace the first time derivative in the energy balance with a Caputo-type Prabhakar derivative and incorporate power-law or exponential temperature dependence in the thermal coefficient.The formulation also includes a linear heat-loss term in one model.
- A generalized separating-variable method is used to study a new nonlinear heat-propagation model with memory and derive exact particular solutions.The resulting nonlinear problem is reduced to a linear fractional differential equation after substituting the separating-variable ansatz.
- Under αγ ∈ (0, 1), Proposition 8 establishes a particular solution for the power-law thermal-conduction equation, with its constant determined by the initial condition.The derivation assumes f(0) = 1, corresponding to an initial temperature distribution of the stated form.
- For the exponential thermal-coefficient case, Proposition 9 gives a particular solution under β, ν > 0 and x > C.
- The analyzed solutions converge to ϕ0(β, λ, γ) as time increases, while logarithmic-scale plots expose their power-law decay.Numerical approximations of the first asymptotic coefficients are used to plot the large-t behavior as parameters vary.
5 Concluding remarks
The paper combines asymptotic analysis of the Prabhakar function with Prabhakar-type fractional operators and nonlinear heat-conduction models, while identifying coefficient evaluation and numerical computation as ongoing challenges.
- 5 Concluding remarks: The study analyzes the Prabhakar function in the whole complex plane and on the negative semi-axis, then applies Prabhakar derivatives to heat-conduction equations.The negative semi-axis receives particular attention, and the equations involve memory.
- 5 Concluding remarks: The paper describes fractional derivatives of Prabhakar type as useful for fitting experimental data through their two real powers.The authors connect this flexibility to more reliable models.
- 5 Concluding remarks: Future work should address numerical evaluation of the Prabhakar function and numerical schemes for differential equations with Prabhakar-type operators.Existing numerical evaluation has been performed only partially.
- 5 Concluding remarks: The asymptotic-expansion coefficients c_j depend on α, β, and γ, with analytic forms practical only for the first coefficients and numerical evaluation needed thereafter.This coefficient-evaluation difficulty is identified as the main obstacle when using the expansions.
- 5 Concluding remarks: An algorithmic procedure generates any number of coefficients c_j by combining auxiliary asymptotic expansions and a recursive relationship.The procedure matches terms in the relevant expansions to obtain the coefficients.
A.1 Expansion for the reciprocal of the rising factorial
This appendix section derives the reciprocal rising-factorial expansion using gamma-function ratio asymptotics and recursively computable coefficients.
- A.1 Expansion for the reciprocal of the rising factorial: The reciprocal of the rising factorial (αs + ψ)j is expanded using a result for ratios of two gamma functions.The resulting coefficients F_j,k can be recursively evaluated.
- A.1 Expansion for the reciprocal of the rising factorial: A generalized definition of binomial coefficients is required because their first argument is negative.The arguments n and k are integers.
- A.1 Expansion for the reciprocal of the rising factorial: The expansion coefficients D_j,k are obtained from the recursively evaluated coefficients and summarized through their first values in Table 1.Table 1 reports the first coefficients in the expansion of 1/(αs + ψ)j.
A.2 Asymptotic expansion for R(s)
The section constructs the asymptotic expansion of R(s) by expanding component factors, multiplying their power series, and recursively deriving the resulting coefficients.
- A.2 Asymptotic expansion for R(s): For |s| > |b/a|, the logarithmic Taylor expansion supplies an asymptotic representation used in the construction of R(s).The derivation proceeds through standard algebraic manipulations.
- A.2 Asymptotic expansion for R(s): The coefficients of e(as; b) require separate treatments for b outside {0, 1}, b = 1, and b = 0.The b = 0 case gives e(as; b) = 1, with e_0 = 1 and e_k = 0 for k ≥ 1.
- A.2 Asymptotic expansion for R(s): Miller’s formula evaluates the coefficients of powers of the unitary formal power series f(s; a, b)^n.This power-series step contributes to the coefficient construction for the expansion.
- A.2 Asymptotic expansion for R(s): The expansion of R(s) is assembled from expansions of e(s; γ), e(αs; 1−γ +β), e(αs; β), and e(s; 1).These expansions are assumed to hold under reasonable conditions.
- A.2 Asymptotic expansion for R(s): The coefficients R_k of the product of four power series are obtained by a convolution formula over the four coefficient sets.The coefficient sets are denoted {a_k}, {b_k}, {c_k}, and {d_k}.
- A.2 Asymptotic expansion for R(s): The first three coefficients in the expansion of R(s) are given explicitly, including the polynomial P2(x) = 3x4 −10x3 + 9x2.The explicit coefficient representation provides initial closed-form terms.
A.3 Expansion of Υ(s)
The expansion of Υ(s) uses scaled-gamma asymptotics, reciprocal expansions, and power-series products to compute its coefficients explicitly.
- A.3 Expansion of Υ(s): Υ(s) is expanded using asymptotic formulas for the scaled gamma function and its reciprocal as |s| →∞ with |arg(s)| < ϵ.The coefficients γ_k come from the Stirling expansion and can be recursively evaluated.
- A.3 Expansion of Υ(s): The resulting coefficients are obtained by applying the product formula to the four power series defining Υ(s).The construction follows after manipulating the scaled-gamma expansions.
- A.3 Expansion of Υ(s): Table 2 lists initial coefficients for the scaled gamma function, its reciprocal, and Υ(s), as functions of the expansion parameters.The listed values allow explicit evaluation of the first coefficients in Υ(s).