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arxiv: 2604.21267 · v1 · submitted 2026-04-23 · 🧮 math-ph · math.MP

On invariant solutions of linear time-fractional diffusion-wave equations with variable coefficients

Sodbaatar Adiya , Khongorzul Dorjgotov , Bayarmagnai Gombodorj , Hiroyuki Ochiai , Uuganbayar Zunderiya This is my paper

Pith reviewed 2026-05-08 13:52 UTC · model grok-4.3

classification 🧮 math-ph math.MP
keywords Lie symmetry analysistime-fractional diffusion-wave equationinvariant solutionsMittag-Leffler functiongeneralized Wright functionFox H-functionvariable coefficients
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The pith

Lie symmetry analysis yields invariant solutions for time-fractional diffusion-wave equations with variable coefficients in Mittag-Leffler and Fox H-functions.

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

The paper applies Lie symmetry analysis to a class of linear time-fractional diffusion-wave equations that feature variable coefficients. It first determines the infinitesimal symmetries that depend on those coefficients, then uses the symmetries to reduce the partial differential equations. The reductions produce exact invariant solutions expressed through Mittag-Leffler functions, generalized Wright functions, and Fox H-functions. A reader would care because these fractional models capture anomalous diffusion and wave behavior in physical systems, and closed-form solutions make scaling laws and long-time asymptotics directly visible.

Core claim

For a certain class of time-fractional diffusion-wave equations with variable coefficients, the infinitesimal symmetries with respect to the coefficients can be found systematically; these symmetries reduce the original equations and deliver exact invariant solutions written in terms of the Mittag-Leffler function, the generalized Wright function, and the Fox H-function.

What carries the argument

Infinitesimal symmetries determined with respect to the variable coefficients, which reduce the time-fractional PDE to an ordinary fractional equation solvable by the named special functions.

Load-bearing premise

The chosen class of equations with variable coefficients admits non-trivial infinitesimal symmetries that can be determined from the coefficient forms.

What would settle it

Explicit computation of the symmetry algebra for a concrete choice of variable coefficients that yields only trivial symmetries or fails to produce the claimed special-function solutions would refute the general claim.

read the original abstract

We study invariant solutions of a certain class of time-fractional diffusion-wave equations with variable coefficients via Lie symmetry analysis. In physics, the fractional diffusion equation describes transport dynamics that are governed by anomalous diffusion while the fractional wave equation describes oscillations and wave propagation in various physical systems. In order to obtain exact invariant solutions of these equations, we firstly determine infinitesimal symmetries with respect to the variable coefficients of the equations. With the help of these symmetries, we then find solutions in terms of Mittag-Leffler functions, generalized Wright functions and Fox H-functions.

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

0 major / 3 minor

Summary. The manuscript applies Lie symmetry analysis to a certain class of linear time-fractional diffusion-wave equations with variable coefficients. It determines the infinitesimal symmetries (which depend on the specific form of the variable coefficients), reduces the PDEs to fractional ODEs via invariant solutions, and expresses the resulting solutions in terms of Mittag-Leffler functions, generalized Wright functions, and Fox H-functions.

Significance. If the symmetry calculations and reductions are correct, the work provides exact solutions for a family of fractional PDEs relevant to anomalous diffusion and wave propagation in non-homogeneous media. Extending Lie methods to variable-coefficient fractional equations is a useful contribution, as the special-function solutions permit direct study of long-time asymptotics and other properties without numerical approximation.

minor comments (3)
  1. [Abstract / Introduction] The abstract and introduction refer to 'a certain class' of equations but do not explicitly state the admissible forms of the variable coefficients (e.g., power-law, exponential, or other functional dependence). This should be stated at the outset, ideally with the precise PDE written in §1 or §2.
  2. [Abstract] The range of the fractional order (typically 0 < α ≤ 2 for diffusion-wave equations) is not specified in the abstract or early sections; this is needed to clarify whether the analysis covers both sub-diffusive and wave-like regimes.
  3. [Results / Conclusion] No verification (e.g., direct substitution of the constructed solutions back into the original PDE or numerical checks for special cases) is mentioned in the provided summary; if present in the full text, a brief statement or table in the results section would strengthen the claims.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for reviewing our manuscript and recommending minor revision. The provided summary correctly captures the application of Lie symmetry analysis to time-fractional diffusion-wave equations with variable coefficients, the determination of infinitesimal symmetries, and the expression of invariant solutions via Mittag-Leffler, generalized Wright, and Fox H-functions. We appreciate the positive evaluation of the work's relevance to anomalous diffusion and wave propagation in non-homogeneous media.

Circularity Check

0 steps flagged

No circularity: standard Lie symmetry reduction applied to a restricted class of equations

full rationale

The paper applies the classical Lie point symmetry method to a deliberately chosen subclass of linear time-fractional diffusion-wave equations whose variable coefficients are selected precisely so that the determining equations for the infinitesimal generators admit non-trivial solutions. The symmetries are computed directly from the PDE, the invariant solutions are obtained by reducing the PDE to fractional ODEs, and the resulting ODEs are solved in closed form using Mittag-Leffler, Wright, and Fox H-functions. No parameter is fitted to data, no result is renamed as a prediction, and no load-bearing step relies on a self-citation whose content is itself unverified. The derivation chain is therefore self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions of fractional calculus and Lie group theory for PDEs; no free parameters, new entities, or ad-hoc axioms are indicated in the abstract.

axioms (2)
  • domain assumption Time-fractional derivatives are defined via Caputo or Riemann-Liouville operators.
    Standard definition required for the diffusion-wave equation class.
  • domain assumption Lie symmetry analysis extends to fractional-order differential equations.
    The paper determines infinitesimal symmetries with respect to variable coefficients.

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Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

  1. [1]

    Baleanu and A.M

    D. Baleanu and A.M. Lopes, editors,Handook of Fractional Calculus with Applications: Applications in Engineering, Life and Social Sciences, Part A.De Gruyter (2019)

    work page 2019

  2. [2]

    Baleanu and A.M

    D. Baleanu and A.M. Lopes, editors,Handook of Fractional Calculus with Applications: Applications in Engineering, Life and Social Sciences, Part B.De Gruyter (2019)

    work page 2019

  3. [3]

    Bluman, A

    G. Bluman, A. Cheviakov and S. Anco,Applications of Symmetry Methods to Partial Differential Equa- tions.Springer, New York,168(2010)

    work page 2010

  4. [4]

    Bluman and S

    G. Bluman and S. Kumei,On invariance properties of the wave equation.J. Math. Phys.,28(2)(1987), 307–318

    work page 1987

  5. [5]

    Buckwar and Yu

    E. Buckwar and Yu. Luchko,Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations.J. Math. Anal. Appl.,227(1998), 81–97

    work page 1998

  6. [6]

    Dorjgotov, H

    D. Dorjgotov, H. Ochiai and U. Zunderiya,On solutions of linear fractional differential equations and systems thereof.Fract. Calc. Appl. Anal.,22(2019), 479–494

    work page 2019

  7. [7]

    Gazizov, A.A

    R.K. Gazizov, A.A. Kasatkin and S.Yu. Lukashchuk,Symmetry properties of fractional diffusion equa- tions.Phys. Scr.,136(2009), 014016

    work page 2009

  8. [8]

    Glockle and T.F

    W.G. Glockle and T.F. Nonnenmacher,Fox function representation of Non-debye relaxation processes. J. Statist. Phys.,71(3/4)(1999), 741–757

    work page 1999

  9. [9]

    Gorenflo, Yu

    R. Gorenflo, Yu. Luchko and F. Mainardi,Wright functions as scale-invariant solutions of the diffusion- wave equation.J. Comp. Appl. Math.,118(2000), 175-191

    work page 2000

  10. [10]

    G¨ ungor,Equivalence and symmetries for variable coefficient linear heat type equations

    F. G¨ ungor,Equivalence and symmetries for variable coefficient linear heat type equations. I.J. Math. Phys.,59(5)(2018), 051507

    work page 2018

  11. [11]

    G¨ ungor,Equivalence and symmetries for variable coefficient linear heat type equations

    F. G¨ ungor,Equivalence and symmetries for variable coefficient linear heat type equations. II.J. Math. Phys.,59(6)(2018), 061507

    work page 2018

  12. [12]

    Kilbas,Fractional calculus of the generalized Wright function.Fract

    A.A. Kilbas,Fractional calculus of the generalized Wright function.Fract. Calc. Appl. Anal.,8(2)(2005), 113–126. 10

    work page 2005

  13. [13]

    Kilbas, H.M

    A.A. Kilbas, H.M. Srivastava and J.J. Trujillo,Theory and Applications of Fractional Differential Equa- tions.Elsevier Science, Amsterdam (2006)

    work page 2006

  14. [14]

    Kiryakova,Fractional calculus operators of special functions? The result is well predictable!.Chaos, Solitons and Fractals,102(2017), 2–15

    V. Kiryakova,Fractional calculus operators of special functions? The result is well predictable!.Chaos, Solitons and Fractals,102(2017), 2–15

    work page 2017

  15. [15]

    Lie, ¨Uber die integration durch bestimmte integrale von einer klasse linear partieller differentialgle- ichung.Arch

    S. Lie, ¨Uber die integration durch bestimmte integrale von einer klasse linear partieller differentialgle- ichung.Arch. Math.,6(3)(1881), 328–368

    work page

  16. [16]

    Mainardi,Fractional diffusive waves in viscoelastic solids.In Nonlinear Waves in Solids, ASME/AMR 137(1995), Fairfield, 93–97

    F. Mainardi,Fractional diffusive waves in viscoelastic solids.In Nonlinear Waves in Solids, ASME/AMR 137(1995), Fairfield, 93–97

    work page 1995

  17. [17]

    Mainardi and P

    F. Mainardi and P. Paradisi,Fractional diffusive waves.J. Comput. Acoustics,9(4)(2001), 1417–1436

    work page 2001

  18. [18]

    Mathai and R.K

    A.M. Mathai and R.K. Sahena,The H-function.Springer, New York (2010)

    work page 2010

  19. [19]

    Metzler, W.G

    R. Metzler, W.G. Glockle and T.F. Nonnenmacher,Fractional model equation for anomalous diffusion. Physica A,211(1994), 13–24

    work page 1994

  20. [20]

    Metzler and J

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

    work page 2000

  21. [21]

    Nigmatullin,To the theoretical explanation of the universal response.Phys

    R.R. Nigmatullin,To the theoretical explanation of the universal response.Phys. Stat. Sol. (b), 123(2)(1984), 739–745

    work page 1984

  22. [22]

    Nigmatullin,The realization of the generalized transfer equation in a medium with fractal geometry

    R.R. Nigmatullin,The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. (b),133(1)(1986), 425–430

    work page 1986

  23. [23]

    Oldham and J

    K.B. Oldham and J. Spanier,The Fractional Calculus.Academic Press, New York-London (1974)

    work page 1974

  24. [24]

    Ovsyannikov,Group Analysis of Differential Equations.Academic Press, New York (1982)

    L.V. Ovsyannikov,Group Analysis of Differential Equations.Academic Press, New York (1982)

    work page 1982

  25. [25]

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

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

    work page 1999

  26. [26]

    Pskhu and S.Sh

    A.V. Pskhu and S.Sh. Rekhviashvili,Retarded potentials in fractional electrodynamics.Mosc. Univ. Phys. Bull.,75(2020), 316–319

    work page 2020

  27. [27]

    Sakamoto and M

    K. Sakamoto and M. Yamamoto,Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems.J. Math. Anal. Appl.,382(1)(2011), 426–447

    work page 2011

  28. [28]

    H. Sun, Y. Zhang, D. Baleanu, W. Chen and Y. Chen.A new collection of real world applications of fractional calculus in science and engineering.Commun. Nonlinear Sci. Numer. Simul.,64(2018), 213–231

    work page 2018

  29. [29]

    Tarasov, editor,Handook of Fractional Calculus with Applications: Applications in Physics, Part A.De Gruyter (2019)

    V.E. Tarasov, editor,Handook of Fractional Calculus with Applications: Applications in Physics, Part A.De Gruyter (2019)

    work page 2019

  30. [30]

    Tarasov, editor,Handook of Fractional Calculus with Applications: Applications in Physics, Part B.De Gruyter (2019)

    V.E. Tarasov, editor,Handook of Fractional Calculus with Applications: Applications in Physics, Part B.De Gruyter (2019). Department of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia Email address:sodoomath@gmail.com Department of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia Email address:pilpalpil@gmail.com Dep...

    work page 2019