Green's Function Framework for Boundary Value Problems with the Regularized Prabhakar Fractional Derivative

Doniyor Usmonov; Erkinjon Karimov; Maftuna Mirzaeva

arxiv: 2603.24252 · v2 · pith:AZ6SAI3Tnew · submitted 2026-03-25 · 🧮 math.AP

Green's Function Framework for Boundary Value Problems with the Regularized Prabhakar Fractional Derivative

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

Pith reviewed 2026-05-22 11:19 UTC · model grok-4.3

classification 🧮 math.AP

keywords Green's functionregularized Prabhakar fractional derivativesub-diffusion equationbivariate Mittag-Leffler functioninitial-boundary value problemfractional differential equation

0 comments

The pith

The solution to the initial-boundary value problem for a sub-diffusion equation with the regularized Prabhakar fractional derivative is given explicitly by a Green's function in bivariate Mittag-Leffler form.

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

This paper examines the first initial-boundary value problem for a sub-diffusion equation that uses the regularized Prabhakar fractional derivative. The authors apply the superposition method to split the problem into two simpler initial-boundary value problems. They then construct an explicit solution whose corresponding Green's function takes the form of a bivariate Mittag-Leffler type function. Finally they verify that this explicit representation satisfies the original fractional equation together with the given initial and boundary conditions. A reader would care because closed-form solutions remain uncommon for fractional-order diffusion models and directly support analysis of anomalous transport.

Core claim

An explicit representation of the solution and the corresponding Green's function is obtained. The explicit form of the Green's function is expressed in terms of a bivariate Mittag-Leffler type function. It is proved that the obtained solution indeed constitutes the solution of the considered problem.

What carries the argument

Bivariate Mittag-Leffler type function that expresses the Green's function and thereby supplies the explicit solution while incorporating the fractional-order dynamics and boundary data.

If this is right

The explicit Green's function yields the solution for arbitrary initial data and source terms by direct integration.
Verification confirms that the constructed representation satisfies both the fractional sub-diffusion equation and the imposed conditions.
The same Green's function supplies the solution operator for the linear homogeneous problem under the given boundary conditions.

Where Pith is reading between the lines

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

The bivariate Mittag-Leffler representation may allow direct asymptotic analysis of long-time decay rates in the sub-diffusive regime.
The reduction technique could be tested on related fractional operators that also admit a superposition principle.
Numerical quadrature of the bivariate function against sample data would provide a practical check of computational cost for realistic domains.

Load-bearing premise

The superposition method splits the original initial-boundary value problem into two independent subproblems while preserving the action of the regularized Prabhakar derivative and the boundary conditions.

What would settle it

Substitute the derived Green's function into the regularized Prabhakar operator applied to the proposed solution and verify that the result equals the forcing term while the initial and boundary conditions are satisfied pointwise.

Figures

Figures reproduced from arXiv: 2603.24252 by Doniyor Usmonov, Erkinjon Karimov, Maftuna Mirzaeva.

**Figure 1.** Figure 1: Influence of the initial data. The next example deals with the influence of the external force. That is, we assume that there is no influence of the initial data (u(0, x) ≡ 0). Example 2. Let a = π, T = 2, φ0(t) = φ1(t) = 0, τ (x) = 0, f(t, x) = tsin x. Then according to (2.4) we get u(t, x) = Z t 0 Zπ 0 ηsin(ξ)G (t, x, η, ξ) dξdη, where G(t, x, η, ξ) = 1 2 (t − η) β1−1× × X +∞ n=−∞ X +∞ k=0 X +∞ m=0   … view at source ↗

**Figure 2.** Figure 2: Influence of the external force. 4 Conclusion In this work, we have successfully established a Green’s function framework for solving the first initial-boundary value problem for a sub-diffusion equation involving the regularized Prabhakar fractional derivative. By utilizing the superposition method, we reduced the complex non-homogeneous problem into manageable components, allowing for the derivation of … view at source ↗

read the original abstract

In this work, the first initial-boundary value problem for a sub-diffusion equation involving the regularized Prabhakar fractional derivative is studied. The problem is solved by reducing it to two initial-boundary value problems using the superposition method. An explicit representation of the solution and the corresponding Green's function is obtained. The explicit form of the Green's function is expressed in terms of a bivariate Mittag-Leffler type function. Then, it is proved that the obtained solution indeed constitutes the solution of the considered problem.

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

The paper gives an explicit Green's function in bivariate Mittag-Leffler form for the regularized Prabhakar sub-diffusion problem after a superposition split, but the non-local regularization term makes that split worth double-checking in the proof.

read the letter

The main takeaway is that this work constructs an explicit solution and Green's function for the first initial-boundary value problem involving the regularized Prabhakar fractional derivative, expressed through a bivariate Mittag-Leffler type function after reducing the original problem to two simpler ones via superposition. They then state that the result is verified to solve the full problem. That explicit form is the concrete new piece here, extending prior Mittag-Leffler techniques to this specific derivative in a boundary-value setting. The approach follows standard lines for linear fractional PDEs and delivers a closed-form representation that could be useful for further study or testing in the subfield. The abstract indicates they carry out the derivation and the verification step, which is the part that matters most for an analytical paper like this. The soft spot is the reduction itself. The regularized Prabhakar operator includes a history-dependent correction, so splitting the initial data and boundary conditions between the two subproblems has to be done so the integrals and the regularization term recombine correctly. Linearity takes care of the differential part, but the non-local piece needs explicit confirmation that no mismatch occurs. The paper claims a proof that the constructed solution works, yet without the detailed steps it is hard to see exactly how they handled the initial-condition assignment. If that part is only sketched, it leaves a gap that a referee would want to see filled. This is for readers already working on analytical methods for fractional diffusion or boundary-value problems with non-standard derivatives. Someone hunting for explicit representations in this niche would get direct value from the formula. It is grounded enough in a specific construction and verification claim to go to a serious referee who can examine the proof details and the operator splitting. I would send it for peer review.

Referee Report

1 major / 1 minor

Summary. The manuscript studies the first initial-boundary value problem for a sub-diffusion equation involving the regularized Prabhakar fractional derivative. The problem is reduced to two independent initial-boundary value problems via the superposition method. An explicit representation of the solution is derived, with the corresponding Green's function expressed in terms of a bivariate Mittag-Leffler type function. The authors then prove that this explicit solution satisfies the original problem.

Significance. If the reduction and verification steps are rigorous, the work supplies an explicit Green's function for boundary-value problems with the regularized Prabhakar derivative, extending analytical methods to this non-local operator. The closed-form expression via the bivariate Mittag-Leffler function is a concrete strength that could facilitate further analysis and computation.

major comments (1)

The superposition reduction (described in the abstract and the opening sections) is load-bearing for the central claim. Because the regularized Prabhakar derivative is a non-local integro-differential operator whose regularization term depends on the full history, the initial data must be split between the two subproblems so that both the fractional integrals and the regularization corrections add correctly. The manuscript should explicitly state the assignment of initial conditions to each subproblem and verify that the sum recovers the original initial data and that the operator applied to the sum equals the sum of the operators applied to each piece. Without this verification, it remains unclear whether the derived Green's function solves the original IBVP.

minor comments (1)

Notation for the bivariate Mittag-Leffler function should be introduced with a precise definition (including parameters and arguments) at first use to avoid ambiguity for readers unfamiliar with this extension.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive feedback on the superposition reduction. We address the major comment in detail below and will incorporate the requested clarifications in the revised version.

read point-by-point responses

Referee: The superposition reduction (described in the abstract and the opening sections) is load-bearing for the central claim. Because the regularized Prabhakar fractional derivative is a non-local integro-differential operator whose regularization term depends on the full history, the initial data must be split between the two subproblems so that both the fractional integrals and the regularization corrections add correctly. The manuscript should explicitly state the assignment of initial conditions to each subproblem and verify that the sum recovers the original initial data and that the operator applied to the sum equals the sum of the operators applied to each piece. Without this verification, it remains unclear whether the derived Green's function solves the original IBVP.

Authors: We agree that the non-local character of the regularized Prabhakar derivative requires an explicit justification of how the initial data are distributed between the two subproblems. In the revised manuscript we will insert a new paragraph immediately after the statement of the superposition method (in the section describing the reduction). We will specify that the original initial condition u(x,0)=f(x) is assigned entirely to the first subproblem while the second subproblem is given homogeneous initial data, and we will verify by direct computation that the sum of the two solutions recovers f(x) at t=0. Because the regularized Prabhakar operator is linear, the fractional derivative of the sum equals the sum of the fractional derivatives; the regularization terms, being linear functionals of the history, likewise add correctly. This explicit verification will be placed before the derivation of the Green's function and will be referenced in the final existence proof, thereby confirming that the constructed solution satisfies the original IBVP. revision: yes

Circularity Check

0 steps flagged

No circularity: standard superposition and Mittag-Leffler representation verified by direct substitution

full rationale

The derivation reduces the linear IBVP to two subproblems via superposition, constructs an explicit Green's function in bivariate Mittag-Leffler form, and verifies by substitution that the sum satisfies the original equation and conditions. No step equates a derived quantity to a fitted input or prior self-citation by construction; the Mittag-Leffler functions and linearity properties are invoked from established fractional calculus without the present paper's result being presupposed in the inputs. The approach is self-contained against external benchmarks for linear operators.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of the regularized Prabhakar derivative and Mittag-Leffler functions drawn from prior literature, with no free parameters fitted to data, no invented entities, and one domain assumption about superposition applicability.

axioms (1)

domain assumption The regularized Prabhakar fractional derivative permits direct application of the superposition principle to split the initial-boundary value problem into two independent sub-problems without altering the fractional order or boundary behavior.
This assumption is required for the reduction step that enables construction of the explicit Green's function.

pith-pipeline@v0.9.0 · 5613 in / 1344 out tokens · 32364 ms · 2026-05-22T11:19:37.094040+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.

The problem is solved by reducing it to two initial-boundary value problems using the superposition method. ... explicit form of the Green's function is expressed in terms of a bivariate Mittag-Leffler type function.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

P CD^{α,β,γ,δ}_{0t} u(t,x) - u_xx(t,x) = f(t,x) with regularized Prabhakar derivative

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.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages · 1 internal anchor

[1]

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
[2]

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
[3]

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 17

work page doi:10.2298/tsci21s2411a 2021
[4]

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

work page doi:10.1186/s13660-025- 2025
[5]

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
[6]

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
[7]

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
[8]

Journal of Computational and Applied Mathematics,

Gorska, K., Pietrzak, T., Sandev, T., Tomovski, Z.: Volterra-Prabhakar derivative of dis- tributed order and some applications. Journal of Computational and Applied Mathematics,

work page
[9]

S., Fernandez, A., Özarslan, M

Isah, S. S., Fernandez, A., Özarslan, M. A.: On bivariate fractional calculus with general univariate analytic kernels. Chaos, Solitons & Fractals, 171 (2023)

work page 2023
[10]

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
[11]

Rendiconti del Circolo Matematico di Palermo, II

Karimov, E., Kerbal, S., Turdiev, K.: Direct and inverse problems with a dynamical condi- tion for the sub-diffusion equation involving the Hilfer-Prabhakar integral-differential operator. Rendiconti del Circolo Matematico di Palermo, II. Ser., 75:35 (2026). DOI: 10.1007/s12215- 025-01351-0

work page doi:10.1007/s12215- 2026
[12]

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

Karimov E., Usmonov D., Mirzaeva M.; Green’s Function and Solution Representation for a Boundary Value Problem Involving the Prabhakar Fractional Derivative. Preprint, arXiv:2512.21259 [math.AP], 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025
[13]

Gulf Journal of Mathematics, 20(1), 190–221 (2025)

Kerbal, S., Khasanov, Sh.: On Katugampola-Prabhakar Fractional Integral- Differential Operators. Gulf Journal of Mathematics, 20(1), 190–221 (2025). DOI: https://doi.org/10.56947/gjom.v20i.2856

work page doi:10.56947/gjom.v20i.2856 2025
[14]

Elsevier, Amsterdam (2006)

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

work page 2006
[15]

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
[16]

Ren- diconti del Circolo Matematico di Palermo Series 2, 74:162 (2025)

Mamanazarov, A., Khalilov, K., Kodiraliev, A.: Direct and inverse source problems with non- local boundary conditions for a time-fractional time and space degenerate heat equation. Ren- diconti del Circolo Matematico di Palermo Series 2, 74:162 (2025). DOI: 10.1007/s12215-025- 01286-6

work page doi:10.1007/s12215-025- 2025
[17]

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 18

work page doi:10.3166/jesa.42.639-651 2008
[18]

Academic Press, San Diego (1999)

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

work page 1999
[19]

Some properties of Prabhakar-type fractional calculus opera- tors

Polito, F., Tomovski, Z.: (2016). Some properties of Prabhakar-type fractional calculus opera- tors. Fractional Differential Calculus, 6(1), 73–94 (2016)

work page 2016
[20]

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)

work page 1971
[21]

Prudnikov A.P., Brychkov Yu.A., Marichev O.I.; Integrals and Series. Vol. 3: More Special Functions. Gordon and Breach Science Publishers,–New York,–1986

work page 1986
[22]

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
[23]

Nauka, Moscow (2005)

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

work page 2005
[24]

Integral Transforms and Special Functions,33, 12, 945–965 (2022)

Rani, N., Fernandez, A.: Mikusiński’s operational calculus for Prabhakar fractional calculus. Integral Transforms and Special Functions,33, 12, 945–965 (2022)

work page 2022
[25]

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
[26]

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
[27]

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

work page doi:10.29229/uzmj.2025-2-29 2025
[28]

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 19

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

[1] [1]

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

[2] [2]

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

[3] [3]

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 17

work page doi:10.2298/tsci21s2411a 2021

[4] [4]

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

work page doi:10.1186/s13660-025- 2025

[5] [5]

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

[6] [6]

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

[7] [7]

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

[8] [8]

Journal of Computational and Applied Mathematics,

Gorska, K., Pietrzak, T., Sandev, T., Tomovski, Z.: Volterra-Prabhakar derivative of dis- tributed order and some applications. Journal of Computational and Applied Mathematics,

work page

[9] [9]

S., Fernandez, A., Özarslan, M

Isah, S. S., Fernandez, A., Özarslan, M. A.: On bivariate fractional calculus with general univariate analytic kernels. Chaos, Solitons & Fractals, 171 (2023)

work page 2023

[10] [10]

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

[11] [11]

Rendiconti del Circolo Matematico di Palermo, II

Karimov, E., Kerbal, S., Turdiev, K.: Direct and inverse problems with a dynamical condi- tion for the sub-diffusion equation involving the Hilfer-Prabhakar integral-differential operator. Rendiconti del Circolo Matematico di Palermo, II. Ser., 75:35 (2026). DOI: 10.1007/s12215- 025-01351-0

work page doi:10.1007/s12215- 2026

[12] [12]

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

Karimov E., Usmonov D., Mirzaeva M.; Green’s Function and Solution Representation for a Boundary Value Problem Involving the Prabhakar Fractional Derivative. Preprint, arXiv:2512.21259 [math.AP], 2025

work page internal anchor Pith review Pith/arXiv arXiv 2025

[13] [13]

Gulf Journal of Mathematics, 20(1), 190–221 (2025)

Kerbal, S., Khasanov, Sh.: On Katugampola-Prabhakar Fractional Integral- Differential Operators. Gulf Journal of Mathematics, 20(1), 190–221 (2025). DOI: https://doi.org/10.56947/gjom.v20i.2856

work page doi:10.56947/gjom.v20i.2856 2025

[14] [14]

Elsevier, Amsterdam (2006)

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

work page 2006

[15] [15]

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

[16] [16]

Ren- diconti del Circolo Matematico di Palermo Series 2, 74:162 (2025)

Mamanazarov, A., Khalilov, K., Kodiraliev, A.: Direct and inverse source problems with non- local boundary conditions for a time-fractional time and space degenerate heat equation. Ren- diconti del Circolo Matematico di Palermo Series 2, 74:162 (2025). DOI: 10.1007/s12215-025- 01286-6

work page doi:10.1007/s12215-025- 2025

[17] [17]

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 18

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

[18] [18]

Academic Press, San Diego (1999)

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

work page 1999

[19] [19]

Some properties of Prabhakar-type fractional calculus opera- tors

Polito, F., Tomovski, Z.: (2016). Some properties of Prabhakar-type fractional calculus opera- tors. Fractional Differential Calculus, 6(1), 73–94 (2016)

work page 2016

[20] [20]

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)

work page 1971

[21] [21]

Prudnikov A.P., Brychkov Yu.A., Marichev O.I.; Integrals and Series. Vol. 3: More Special Functions. Gordon and Breach Science Publishers,–New York,–1986

work page 1986

[22] [22]

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

[23] [23]

Nauka, Moscow (2005)

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

work page 2005

[24] [24]

Integral Transforms and Special Functions,33, 12, 945–965 (2022)

Rani, N., Fernandez, A.: Mikusiński’s operational calculus for Prabhakar fractional calculus. Integral Transforms and Special Functions,33, 12, 945–965 (2022)

work page 2022

[25] [25]

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

[26] [26]

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

[27] [27]

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

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

[28] [28]

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 19

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