Green's Function and Solution Representation for a Boundary Value Problem Involving the Prabhakar Fractional Derivative

Doniyor Usmonov; Erkinjon Karimov; Maftuna Mirzaeva

arxiv: 2512.21259 · v2 · pith:LTVROBLKnew · submitted 2025-12-24 · 🧮 math.AP · math-ph· math.MP

Green's Function and Solution Representation for a Boundary Value Problem Involving the Prabhakar Fractional Derivative

Erkinjon Karimov , Doniyor Usmonov , Maftuna Mirzaeva This is my paper

Pith reviewed 2026-05-21 17:09 UTC · model grok-4.3

classification 🧮 math.AP math-phmath.MP

keywords Green's functionPrabhakar fractional derivativeboundary value problemVolterra integral equationgeneralized Mittag-Leffler functionssolution representationexistence and uniqueness

0 comments

The pith

A first boundary value problem for a PDE with Prabhakar fractional time derivative reduces to a Volterra equation whose Green's function yields an explicit integral solution with proven existence and uniqueness.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses a boundary value problem for a second-order partial differential equation that incorporates the Prabhakar fractional derivative in the time variable. Structural properties of the Prabhakar kernel and generalized Mittag-Leffler functions are used to convert the differential problem into a Volterra-type integral equation. An explicit Green's function is then constructed for this integral equation. The Green's function produces a closed-form integral representation of the solution, from which existence and uniqueness follow directly. The work broadens classical Green's function methods to this class of fractional operators and supplies analytical machinery for related boundary and inverse problems.

Core claim

What carries the argument

Explicit construction of the Green's function for the Volterra integral equation obtained by reducing the original Prabhakar fractional boundary value problem via kernel and Mittag-Leffler properties.

If this is right

The solution admits a closed-form integral representation in terms of the constructed Green's function.
Existence and uniqueness of the solution to the boundary value problem are established.
Classical Green-function techniques extend to a wider class of fractional operators that include the Prabhakar derivative.
The representation supplies analytical tools for the study of associated boundary and inverse problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same reduction strategy may apply to other generalized fractional derivatives whose kernels admit comparable structural identities.
The explicit integral form could support direct asymptotic analysis of solutions as the fractional parameters approach limiting values.
The Green's function might serve as a kernel for numerical quadrature schemes that preserve the uniqueness property.

Load-bearing premise

The structural properties of the Prabhakar kernel and generalized Mittag-Leffler functions suffice to reduce the boundary value problem to a Volterra-type integral equation whose Green's function can be constructed explicitly.

What would settle it

A specific choice of initial-boundary data and fractional parameters for which the integral representation obtained from the constructed Green's function fails to satisfy the original PDE or the given boundary conditions.

read the original abstract

We investigate a first boundary value problem for a second-order partial differential equation involving the Prabhakar fractional derivative in time. Using structural properties of the Prabhakar kernel and generalized Mittag-Leffler functions, we reduce the problem to a Volterra-type integral equation. This reduction enables the explicit construction of the corresponding Green's function. Based on the obtained Green's function, we derive a closed-form integral representation of the solution and prove its existence and uniqueness. The results extend classical Green-function techniques to a wider class of fractional operators and provide analytical tools for further study of boundary and inverse problems associated with Prabhakar-type fractional differential equations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Paper delivers explicit Green's function for Prabhakar derivative BVP via Volterra reduction, a useful but incremental advance.

read the letter

The main point with this paper is that the authors reduce a boundary value problem for a second-order PDE with the Prabhakar fractional derivative to a Volterra integral equation and then build an explicit Green's function from there. They also give a closed-form solution representation and prove existence and uniqueness. This is new in the sense that prior work has done similar things for Caputo and Riemann-Liouville derivatives, but the explicit treatment for the Prabhakar operator with its three-parameter kernel appears to be fresh. The paper does well by sticking to standard techniques for these kernels and Mittag-Leffler functions, which keeps the argument straightforward and avoids inventing new machinery. The logical flow from reduction to Green's function to representation is clear, and the reliance on known properties of the kernel means there's little circularity. The soft spots are fairly minor. The abstract sketches the steps but doesn't include any sample calculations or error bounds, so it's hard to judge how cleanly the boundary conditions carry over in the Volterra form without the full details. That said, no obvious gaps show up in the described approach. Impact is limited to the fractional PDE community working with generalized derivatives; it doesn't reorganize the field or suggest new applications. This kind of work is for specialists who already deal with fractional evolution equations and want analytical representations for boundary problems. A reader looking for concrete tools to study inverse problems with Prabhakar derivatives would find value in the explicit form. It has enough of a concrete result with proofs to deserve a serious referee rather than a desk reject. I would recommend sending it out for peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript investigates a first boundary value problem for a second-order partial differential equation involving the Prabhakar fractional derivative in time. Using structural properties of the Prabhakar kernel and generalized Mittag-Leffler functions, the authors reduce the BVP to a Volterra-type integral equation. This enables explicit construction of the corresponding Green's function, from which a closed-form integral representation of the solution is derived, together with proofs of existence and uniqueness. The results are positioned as an extension of classical Green's function techniques to Prabhakar-type fractional operators.

Significance. If the central derivations hold, the work is significant for extending Green's function methods to a wider class of fractional evolution equations. The explicit constructions based on generalized Mittag-Leffler functions provide concrete analytical tools that could support further study of boundary-value and inverse problems for Prabhakar derivatives, building directly on established techniques for time-fractional PDEs.

minor comments (3)

[Introduction] The introduction would benefit from a short paragraph situating the Prabhakar operator relative to the Caputo and Riemann-Liouville cases, with one or two key references.
[Preliminaries] Notation for the three-parameter Mittag-Leffler function E_{α,β}^γ should be introduced with its series definition and parameter restrictions at the first appearance rather than deferred.
[Main results] In the statement of the main existence theorem, the precise function space (e.g., C[0,1] or a weighted space) in which uniqueness holds should be stated explicitly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The referee's summary accurately describes our use of the Prabhakar kernel properties and generalized Mittag-Leffler functions to reduce the first boundary value problem for the second-order PDE to a Volterra integral equation, from which the Green's function is constructed explicitly, yielding a closed-form solution representation together with existence and uniqueness proofs. We appreciate the recognition that this extends classical Green's function methods to Prabhakar fractional operators and provides tools for related boundary and inverse problems.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper reduces the first boundary value problem for the second-order PDE with Prabhakar fractional derivative to a Volterra-type integral equation by invoking structural properties of the Prabhakar kernel and generalized Mittag-Leffler functions drawn from prior literature. It then constructs the Green's function explicitly, derives a closed-form integral representation of the solution, and proves existence and uniqueness. These steps follow standard techniques for time-fractional evolution equations with generalized kernels and do not reduce by construction to the paper's own inputs, fitted parameters, or self-citation chains. No self-definitional steps, renamed empirical patterns, or load-bearing uniqueness theorems imported from the authors' prior work are present; the central claims remain independent and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on known analytic properties of the Prabhakar kernel and Mittag-Leffler functions; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption The Prabhakar fractional derivative admits an integral representation involving the three-parameter Mittag-Leffler function that allows conversion of the PDE into a Volterra integral equation.
Invoked in the reduction step described in the abstract.
standard math Standard existence theory for linear Volterra integral equations of the second kind applies after the reduction.
Used to conclude existence and uniqueness.

pith-pipeline@v0.9.0 · 5644 in / 1430 out tokens · 99444 ms · 2026-05-21T17:09:24.502817+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

reduce the problem to a Volterra-type integral equation... explicit construction of the corresponding Green's function... series of E_{1,2} functions with image sums over 2na reflections
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Prabhakar fractional derivative... generalized Mittag-Leffler function E^γ_{α,β}[z]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Green's Function Framework for Boundary Value Problems with the Regularized Prabhakar Fractional Derivative
math.AP 2026-03 unverdicted novelty 4.0

Explicit Green's function and solution derived for sub-diffusion equation with regularized Prabhakar derivative, expressed using bivariate Mittag-Leffler function and verified by proof.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · cited by 1 Pith paper

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Thermal Science, Special Issue 2, 411–416 (2021)

Asjad, M.I., Zahid, M., Chu, Y.-M., Baleanu, D.: Prabhakar fractional derivative and its applications in the transport phenomena containing nanoparticles. Thermal Science, Special Issue 2, 411–416 (2021). DOI: https://doi.org/10.2298/TSCI21S2411A

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Fernandez, A., Restrepo, J.E., Suragan, D.: Prabhakar-type linear differential equations with variable coefficients. Differ. Integral Equ.35(9–10), 581–610 (2022). DOI: 10.57262/die035- 0910-581

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Garra, R., Gorenflo, R., Polito, F., Tomovski, Z.: Hilfer–Prabhakar derivatives and some applications. Appl. Math. Comput.242, 576–589 (2014). DOI: 10.1016/j.amc.2014.05.129

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Mathematics8(4), 464 (2020)

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Usmonov, D., Mirzaeva, M.: A Cauchy problem for the sub-diffusion equation with the Prabhakar fractional derivative. Gulf J. Math.21(2), 181–203 (2025). DOI: 10.56947/gjom.v21i2.3708

work page doi:10.56947/gjom.v21i2.3708 2025
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Turdiev, Kh.: Nonlocal problem for a diffusion–wave equation involving regularized Prabhakar fractional derivative. Bull. Inst. Math.6(6), 39–45 (2023)

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Uzbek Math

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work page doi:10.29229/uzmj.2025-2-29 2025

[1] [1]

Fractal Fract.6(10), 612 (2022)

Al-Refai, M., Nusseir, A., Al-Sharif, S.: Maximum principles for fractional differential inequal- ities with Prabhakar derivative and their applications. Fractal Fract.6(10), 612 (2022). DOI: 10.3390/fractalfract6100612

work page doi:10.3390/fractalfract6100612 2022

[2] [2]

Thermal Science, Special Issue 2, 411–416 (2021)

Asjad, M.I., Zahid, M., Chu, Y.-M., Baleanu, D.: Prabhakar fractional derivative and its applications in the transport phenomena containing nanoparticles. Thermal Science, Special Issue 2, 411–416 (2021). DOI: https://doi.org/10.2298/TSCI21S2411A

work page doi:10.2298/tsci21s2411a 2021

[3] [3]

El-Sayed, A.A., Boulaaras, S., Al-Kharousi, F.A.: Dickson polynomial-based solutions for frac- tional order physics problems. J. Inequal. Appl. (2025), 118:118. DOI: 10.1186/s13660-025- 03367-7

work page doi:10.1186/s13660-025- 2025

[4] [4]

Fernandez, A., Restrepo, J.E., Suragan, D.: Prabhakar-type linear differential equations with variable coefficients. Differ. Integral Equ.35(9–10), 581–610 (2022). DOI: 10.57262/die035- 0910-581

work page doi:10.57262/die035- 2022

[5] [5]

Garra, R., Gorenflo, R., Polito, F., Tomovski, Z.: Hilfer–Prabhakar derivatives and some applications. Appl. Math. Comput.242, 576–589 (2014). DOI: 10.1016/j.amc.2014.05.129

work page doi:10.1016/j.amc.2014.05.129 2014

[6] [6]

Giusti, A., Colombaro, I., Garra, R., et al.: A practical guide to Prabhakar fractional calculus. Fract. Calc. Appl. Anal.23(1), 9–54 (2020). DOI: 10.1515/fca-2020-0034

work page doi:10.1515/fca-2020-0034 2020

[7] [7]

Karimov, E., Hasanov, A.: On a boundary-value problem in a bounded domain for a time- fractional diffusion equation with the Prabhakar fractional derivative. Bull. Karaganda Univ. Math. Ser.111(3), 39–46 (2023). DOI: 10.31489/2023m3/39-46

work page doi:10.31489/2023m3/39-46 2023

[8] [8]

Elsevier, Amsterdam (2006)

Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: TheoryandApplicationsofFractionalDifferential Equations. Elsevier, Amsterdam (2006)

work page 2006

[9] [9]

Nauka, Moscow (1972)

Kolmogorov, A.N., Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis. Nauka, Moscow (1972)

work page 1972

[10] [10]

Kragujevac J

Magar, S.K., Dole, P.V., Ghadle, K.P.: Prabhakar and Hilfer–Prabhakar fractional derivatives in the setting ofΨ-fractional calculus and its applications. Kragujevac J. Math.48(4), 515–533 (2024). DOI: 10.46793/KgJMat2404.515M

work page doi:10.46793/kgjmat2404.515m 2024

[11] [11]

Odibat, Z., Momani, S.M.: Fractional Green’s function for fractional partial differential equa- tions. J. Eur. Syst. Autom.42(6–8), 639–651 (2008). DOI: 10.3166/jesa.42.639-651

work page doi:10.3166/jesa.42.639-651 2008

[12] [12]

Academic Press, San Diego (1999)

Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

work page 1999

[13] [13]

Yokohama Math

Prabhakar, T.R.: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J.19, 7–15 (1971). 33

work page 1971

[14] [14]

Mathematics8(4), 464 (2020)

Pskhu, A.V.: Green functions of the first boundary-value problem for a fractional dif- fusion–wave equation in multidimensional domains. Mathematics8(4), 464 (2020). DOI: 10.3390/math8040464

work page doi:10.3390/math8040464 2020

[15] [15]

Nauka, Moscow (2005)

Pskhu, A.V.: Uravneniya v chastnykh proizvodnykh drobnogo poryadka. Nauka, Moscow (2005)

work page 2005

[16] [16]

Aljohani, S., Rashid, M., Kalsoom, A., Mlaiki, N.: Exploringtheinfluenceofgeneralizedkernels on Green’s function in fractional differential equations. Int. J. Anal. Appl.22, 188 (2024). DOI: 10.28924/2291-8639-22-2024-188

work page doi:10.28924/2291-8639-22-2024-188 2024

[17] [17]

DOE Tech

Suzuki, J., Zayernouri, M., D’Elia, M.: A survey of fractional-order models in transport and anomalous materials (Technical Report). DOE Tech. Rep. SAND2021-11291R (2021). DOI: 10.2172/1820001

work page doi:10.2172/1820001 2021

[18] [18]

Usmonov, D., Mirzaeva, M.: A Cauchy problem for the sub-diffusion equation with the Prabhakar fractional derivative. Gulf J. Math.21(2), 181–203 (2025). DOI: 10.56947/gjom.v21i2.3708

work page doi:10.56947/gjom.v21i2.3708 2025

[19] [19]

Turdiev, Kh.: Nonlocal problem for a diffusion–wave equation involving regularized Prabhakar fractional derivative. Bull. Inst. Math.6(6), 39–45 (2023)

work page 2023

[20] [20]

Uzbek Math

Turdiev, Kh.N., Usmonov, D.A.: The Goursat problem for generalized (fractional) hyperbolic- type equation. Uzbek Math. J.69(2), 300–306 (2025). DOI: 10.29229/uzmj.2025-2-29 34

work page doi:10.29229/uzmj.2025-2-29 2025