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arxiv: 2605.02779 · v1 · submitted 2026-05-04 · 🧮 math.AP

An Inverse Source Problem For a Time-Fractional Mixed Wave-Diffusion-Wave Equation in a Cylindrical Domain

Erkinjon Karimov , Muzaffar Toshpulatov This is my paper

Pith reviewed 2026-05-08 17:49 UTC · model grok-4.3

classification 🧮 math.AP
keywords inverse source problemtime-fractional derivativemixed wave-diffusion equationcylindrical domainFourier-Bessel seriesexistence of solutionvariable-order derivativeseparation of variables
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The pith

Existence of a solution to the inverse source problem is proved for a time-fractional mixed wave-diffusion-wave equation in a cylindrical domain.

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

The paper studies an inverse source problem for an equation that mixes wave and diffusion behavior through a fractional time derivative whose order varies with time. The setting is a cylindrical domain, so the radial dependence is handled by expanding in a series of Bessel functions. Separation of variables produces an infinite Fourier-Bessel series for the unknown source and solution; the authors then prove that this series converges uniformly under the given conditions on the derivative order and the boundary data. Uniform convergence directly implies that a solution exists. The result matters because it supplies a rigorous foundation for using such variable-order models to describe processes that shift from diffusive to wave-like dynamics inside cylindrical geometries.

Core claim

By applying separation of variables and the properties of Bessel functions, the solution is written as a Fourier-Bessel series. The paper then establishes uniform convergence of this infinite series for the time-dependent variable-order fractional derivative and the imposed boundary conditions. This convergence supplies the rigorous proof that a solution to the inverse source problem exists.

What carries the argument

The Fourier-Bessel series obtained via separation of variables, whose uniform convergence is verified to establish existence of the solution.

If this is right

  • The inverse source problem admits a solution that can be expressed as a uniformly convergent Fourier-Bessel series.
  • The variable-order fractional derivative allows the model to transition between wave-like and diffusive regimes in time.
  • The separation-of-variables approach works for the cylindrical geometry and the stated boundary conditions.

Where Pith is reading between the lines

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

  • The same series technique could supply error bounds when the infinite expansion is truncated for numerical recovery of the source.
  • Convergence arguments of this type may extend to inverse problems with other variable-order operators that retain cylindrical symmetry.

Load-bearing premise

The uniform convergence of the resulting infinite Fourier-Bessel series holds for the given time-dependent variable-order fractional derivative and boundary conditions in the cylindrical domain.

What would settle it

A concrete choice of source function, initial data, boundary values, and variable-order function for which the associated Fourier-Bessel series diverges would show that the claimed existence fails.

Figures

Figures reproduced from arXiv: 2605.02779 by Erkinjon Karimov, Muzaffar Toshpulatov.

Figure 1
Figure 1. Figure 1: Mixed domain Ω. Direct problem. Find a regular solution of (1) in Ω that satisfies the following - initial condition u(0, r) = φ(r), 0 ≤ r ≤ 1, (3) - boundary conditions h r ∂u(t, r) ∂r i r=0 = 0, u(t, r)|r=1 = 0, 0 ≤ t ≤ T, (4) - transmitting conditions lim t→T1+0 P CD α2,β2,γ2,δ T1t u(t, r) = lim t→T1−0 ut(t, r), 0 ≤ r ≤ 1, (5) lim t→T2+0 ut(t, r) = lim t→T2−0 P CD α2,β2,γ2,δ T1t u(t, r), 0 ≤ r ≤ 1. (6) … view at source ↗
read the original abstract

This paper addresses the inverse source problem for a mixed-type fractional wave-diffusion-wave equation posed in a cylindrical domain. The governing equation involves a time-dependent variable-order fractional derivative, which enables the model to effectively capture temporal transitions between wave-like and diffusive behaviors. The solution is constructed in the form of a Fourier-Bessel series. By employing the method of separation of variables together with fundamental properties of Bessel functions, we analyze the uniform convergence of the resulting infinite series. This analysis ultimately leads to a rigorous proof of the existence of a solution.

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.

Referee Report

1 major / 1 minor

Summary. The manuscript addresses an inverse source problem for a time-fractional mixed wave-diffusion-wave equation with time-dependent variable order posed in a cylindrical domain. The solution is constructed via separation of variables as a Fourier-Bessel series; the authors analyze its uniform convergence using properties of Bessel functions and conclude that this yields a rigorous existence proof.

Significance. If the claimed uniform convergence holds with suitable estimates, the result would be a useful contribution to existence theory for inverse problems involving variable-order fractional operators that interpolate between wave and diffusion regimes in cylindrical geometries. The separation-of-variables approach is standard and appropriate for the geometry, but the variable-order feature requires non-standard bounds.

major comments (1)
  1. [Convergence analysis after separation of variables] The analysis of uniform convergence of the Fourier-Bessel series (following separation of variables): the manuscript asserts uniform convergence on the cylinder but supplies no explicit a-priori estimates on the modal time functions that solve the non-autonomous fractional ODEs with time-dependent order α(t). Such estimates must be uniform in the Bessel eigenvalues λ_n and valid across the mixed wave-diffusion regime to justify the Weierstrass M-test for all t ∈ [0,T]; without them the existence claim remains unverified.
minor comments (1)
  1. [Abstract] The abstract could more precisely state the assumptions on the variable order α(t) and the regularity required of the source term.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. The concern regarding the explicit a-priori estimates for the modal time functions is valid and will be addressed in the revision.

read point-by-point responses

  1. Referee: The analysis of uniform convergence of the Fourier-Bessel series (following separation of variables): the manuscript asserts uniform convergence on the cylinder but supplies no explicit a-priori estimates on the modal time functions that solve the non-autonomous fractional ODEs with time-dependent order α(t). Such estimates must be uniform in the Bessel eigenvalues λ_n and valid across the mixed wave-diffusion regime to justify the Weierstrass M-test for all t ∈ [0,T]; without them the existence claim remains unverified.

    Authors: We agree that the convergence analysis requires more explicit a-priori estimates on the modal solutions to the non-autonomous variable-order fractional ODEs. In the revised manuscript we will insert a dedicated subsection deriving these bounds via the equivalent integral equation formulation of the Caputo-type variable-order derivative together with a fractional Gronwall inequality that accounts for the continuous dependence of α(t) on t. The resulting estimates will be shown to be uniform with respect to the Bessel eigenvalues λ_n (by tracking the dependence through the source term and initial data) and to hold uniformly across the regime 0 < α(t) < 2, thereby justifying the Weierstrass M-test and the claimed uniform convergence on the cylinder for all t ∈ [0,T]. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses standard separation of variables and Bessel properties

full rationale

The paper constructs the solution via separation of variables into a Fourier-Bessel series, then claims to establish uniform convergence using fundamental properties of Bessel functions to prove existence. This chain relies on external mathematical tools (Bessel function theory and series convergence criteria) rather than defining the result in terms of itself, fitting parameters to the target quantity, or importing uniqueness via self-citation. No load-bearing step reduces by construction to the inputs; the variable-order fractional ODEs are handled as part of the modal analysis without self-referential closure. The skeptic concern about explicit estimates for non-autonomous fractional time equations pertains to proof completeness, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard mathematical properties of Bessel functions and separation of variables without introducing new free parameters, axioms beyond domain assumptions, or invented entities.

axioms (1)
  • standard math Bessel functions satisfy the required orthogonality and convergence properties for the Fourier-Bessel series in the cylindrical domain
    Invoked to establish uniform convergence of the series solution

pith-pipeline@v0.9.0 · 5391 in / 1084 out tokens · 28560 ms · 2026-05-08T17:49:04.996153+00:00 · methodology

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Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    The random walk’s guide to anomalous diffusion: A fractional dynamics approach

    Metzler R., Klafter J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Physics Reports, 2000, 339(1), pp.1-77

    work page 2000

  2. [2]

    L., Chechkin A

    Sandev T., Tomovski Ž., Dubbeldam J. L., Chechkin A. Generalized diffusion-wave equation with memory kernel. Journal of Physics A: Mathematical and Theoretical, 2019, 52(1), 015201

    work page 2019

  3. [3]

    Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, 2010

    Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, 2010

    work page 2010

  4. [4]

    Analytical modeling of non-Fickian wave-diffusion of gas in heterogeneous media

    Rasmussen M., Civan F. Analytical modeling of non-Fickian wave-diffusion of gas in heterogeneous media. Applied Mathematical Modelling, 2015, 39(2), pp. 862-880

    work page 2015

  5. [5]

    Inverse source problem for an equation of mixed parabolic-hyperbolic type with the time fractional derivative in a cylindrical domain

    Durdiev D. Inverse source problem for an equation of mixed parabolic-hyperbolic type with the time fractional derivative in a cylindrical domain. Vestnik Samara state technical university, 2022, 26(2), pp.355-367

    work page 2022

  6. [6]

    Fractional-order mixed partial differential equation: Inverse source problem related to the wave-diffusion process

    Karimov E., Toshpulatov M. Fractional-order mixed partial differential equation: Inverse source problem related to the wave-diffusion process. Bulletin of the Institute of Mathematics, 2026, 9(1), pp. 85-100

    work page 2026

  7. [7]

    A singular integral equation with a generalized Mittag-Leffler function in the kernel

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

    work page 1971

  8. [8]

    Giusti A., Colombaro I., Garra R. et al. A Practical Guide to Prabhakar Fractional Calculus. Fract. Calc. Appl. Anal., 2020, 23(1), pp.9–54

    work page 2020

  9. [9]

    Fractional diffusion-telegraph equations and their associated stochastic solutions

    D’Ovidio M., Polito F. Fractional diffusion-telegraph equations and their associated stochastic solutions. Theory Probab. Appl., 2018, 62(4), pp.552-574. [arXiv: 1307.1696 (2013)]

    work page arXiv 2018

  10. [10]

    Mixed wave-diffusion-wave equation: Solvability of an initial-boundary problem

    Karimov E., Toshpulatov M. Mixed wave-diffusion-wave equation: Solvability of an initial-boundary problem. Gulf Journal of Mathematics, 2025, 20(1), pp.120-139

    work page 2025

  11. [11]

    Introduction to Bessel functions,Dover publications, 1958

    Bowman F. Introduction to Bessel functions,Dover publications, 1958. 24 An inverse source problem for a time-fractional wave-diffusion-wave eqiation in ...A PREPRINT

    work page 1958

  12. [12]

    The inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain

    Sabitov K.B., Safin E.M. The inverse problem for a mixed-type parabolic-hyperbolic equation in a rectangular domain. Russ. Math. (Iz. VUZ), 2010, 54(4), pp.48-54

    work page 2010

  13. [13]

    Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition

    Sabitov K.B., Sidorov S.N. Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition. Russ. Math. (Iz. VUZ), 2015, 59(1), pp.39-50

    work page 2015

  14. [14]

    Inverse source problems for time-fractional mixed parabolic-hyperbolic type equations

    Feng P., Karimov E.T. Inverse source problems for time-fractional mixed parabolic-hyperbolic type equations. Journal of Inverse and Ill-posed problems, 2015, 23(4), pp.339-353

    work page 2015

  15. [15]

    Uniqueness of inverse source non-local problem for fractional order mixed type equation

    Salakhitdinov M.S., Karimov E.T. Uniqueness of inverse source non-local problem for fractional order mixed type equation. Euroasian Mathematical Journal, 2016, 1, pp.74-83

    work page 2016

  16. [16]

    An inverse source non-local problem for a mixed type equation with a Caputo fractional differential operator

    Karimov E., Al-Salti N., Kerbal S. An inverse source non-local problem for a mixed type equation with a Caputo fractional differential operator. East Asian Journal of Applied Mathematics, 2017, 7(2), pp.417-438

    work page 2017

  17. [17]

    Inverse Boundary Value Problem for a Fractional Differential Equations of Mixed Type with Integral Redefinition Conditions

    Yuldashev T.K., Kadirkulov B.J. Inverse Boundary Value Problem for a Fractional Differential Equations of Mixed Type with Integral Redefinition Conditions. Lobachevskii J. Math., 2021, 42, pp.649–662

    work page 2021

  18. [18]

    Inverse Problems for Time-Fractional Mixed Equation Involving the Caputo Fractional Derivative

    Karimov E., Tokmagambetov N., Khasanov Sh. Inverse Problems for Time-Fractional Mixed Equation Involving the Caputo Fractional Derivative. Trends in Mathematics. Extended Abstracts 2021/2022 Research Perspectives Ghent Analysis and PDE Center, 2024. 2, pp.167-172

    work page 2021

  19. [19]

    Inverse source problem for an equation of mixed parabolic-hyperbolic type with the time fractional derivative in a cylindrical domain

    Durdiev D.K. Inverse source problem for an equation of mixed parabolic-hyperbolic type with the time fractional derivative in a cylindrical domain. Vestn. Samar. Tekh. Univ., Ser. Fiz.-Mat. Nauki, 2022, 26, pp.355-367

    work page 2022

  20. [20]

    Abdullaev O. Kh. Inverse problems for parabolic-hyperbolic equation with non-linear load. Bull. Inst. Math., 2022, 5(3), pp. 184-194

    work page 2022

  21. [21]

    On an inverse problem for a nonlinear loaded parabolic-hyperbolic equation of fractional order

    Abdullaev O.Kh., Sobirjonov A.Q. On an inverse problem for a nonlinear loaded parabolic-hyperbolic equation of fractional order. Uzbek Mathematical Journal, 2025, 69(1), pp.5-13

    work page 2025

  22. [22]

    A Mittag-Leffler-type function of two variables

    Garg M., Manohar P, Kalla S. A Mittag-Leffler-type function of two variables. Integral transforms and Special functions. 2013. V ol.24, Issue 11, pp. 934-944

    work page 2013

  23. [23]

    On a mixed equation involving Prabhakar fractional order integral-differential operators

    Karimov E., Tokmagambetov N., Toshpulatov M. On a mixed equation involving Prabhakar fractional order integral-differential operators. Trends in Mathematics. Extended Abstracts 2021/2022 Research Perspectives Ghent Analysis and PDE Center, 2024. 2, pp.221-230

    work page 2021

  24. [24]

    Mixed partial differential equation: Forward problem linked with the wave-diffusion process

    Karimov E., Toshpulatov M. Mixed partial differential equation: Forward problem linked with the wave-diffusion process. Georgian Mathematical Journal. https://doi.org/10.1515/gmj-2025-2075

    work page doi:10.1515/gmj-2025-2075 2025

  25. [25]

    Direct and inverse problems with a dynamical condition for the sub-diffusion equation involving the Hilfer-Prabhakar integral-differential operator

    Karimov E., Kerbal S., Turdiev Kh. Direct and inverse problems with a dynamical condition for the sub-diffusion equation involving the Hilfer-Prabhakar integral-differential operator. Rendiconti del Circolo Matematico di Palermo Series 2, 2026, 75(1), Article No 35, 15p

    work page 2026

  26. [26]

    Partial differential equations of fractional-order, 2005

    Pskhu A. Partial differential equations of fractional-order, 2005. 25

    work page 2005