arxiv: 2604.25079 · v1 · submitted 2026-04-28 · 🧮 math-ph · math.MP

Recognition: unknown

Lie symmetry classification and invariant solutions of time-fractional telegraph systems with variable coefficients

Bayarmagnai Gombodorj, Bayarpurev Mongol, Khongorzul Dorjgotov, Sodbaatar Adiya, Uuganbayar Zunderiya

Authors on Pith no claims yet

Pith reviewed 2026-05-07 14:43 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords Lie symmetry classificationtime-fractional telegraph equationvariable coefficientsinvariant solutionsRiemann-Liouville derivativeMittag-Leffler functionFox H-function

0 comments

The pith

Time-fractional telegraph systems with variable coefficients fall into three symmetry classes determined by the relation between transport and potential terms.

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

The paper classifies all Lie point symmetries admitted by time-fractional telegraph systems whose coefficients depend on space. This classification matters because the equations describe transport with memory in materials, waves, and semiconductors, and symmetries produce exact solutions that can benchmark numerical schemes. The admitted symmetries depend on how the transport coefficient relates to the potential, producing exactly three distinct classes. In each class the authors build optimal one-dimensional subalgebras, reduce the fractional partial differential system to ordinary differential equations, and obtain closed-form invariant solutions. These solutions are written using Mittag-Leffler functions, generalized Wright functions, and Fox H-functions.

Core claim

We establish a complete Lie group classification for sufficiently differentiable coefficient functions and determine all functional forms that admit such symmetry extensions. The symmetry structure is shown to depend fundamentally on the relationship between the transport coefficient and the potential function, resulting in three distinct symmetry classes. For each case, optimal systems of one-dimensional Lie subalgebras are constructed, and the governing fractional partial differential equations are systematically reduced to fractional ordinary differential equations. Exact invariant solutions are obtained in closed form and expressed in terms of Mittag-Leffler functions, generalized Wright

What carries the argument

The infinitesimal symmetry generators for the Riemann-Liouville time-fractional derivative, whose determining equations classify the two variable coefficient functions into three families according to their functional relationship.

If this is right

All sufficiently smooth coefficient pairs that permit nontrivial symmetries belong to one of the three classes.
Each class supplies a concrete reduction of the telegraph system to a fractional ordinary differential equation solvable in closed form.
The resulting solutions serve as analytical benchmarks for numerical methods that simulate fractional transport with memory.
The classification method extends systematically to other linear fractional evolution equations with variable coefficients.

Where Pith is reading between the lines

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

The three classes may correspond to physically distinct regimes of memory versus diffusion in heterogeneous media.
The closed-form solutions could be used to test long-time asymptotic behavior or stability in industrial transport models.
Similar classification techniques might apply to systems that include nonlinear terms or fractional derivatives in space.

Load-bearing premise

The coefficient functions are sufficiently differentiable so that the Riemann-Liouville fractional derivative can be used directly in the symmetry analysis without additional compatibility conditions imposed by their spatial variation.

What would settle it

A pair of explicit coefficient functions that admits a symmetry outside the three listed classes, or an invariant solution derived from the classification that fails to satisfy the original system upon direct substitution.

read the original abstract

Time-fractional telegraph equations provide fundamental mathematical models for transport processes that exhibit memory and nonlocal effects in industrial and physical systems. These models arise naturally in heat transport in materials with thermal memory, wave propagation in viscoelastic media, and charge transport in spatially heterogeneous semiconductor devices. In this study, we investigate a class of time-fractional telegraph systems with spatially varying coefficients using Lie symmetry analysis and the Riemann--Liouville fractional derivative. We establish a complete Lie group classification for sufficiently differentiable coefficient functions and determine all functional forms that admit such symmetry extensions. The symmetry structure is shown to depend fundamentally on the relationship between the transport coefficient and the potential function, resulting in three distinct symmetry classes. For each case, optimal systems of one-dimensional Lie subalgebras are constructed, and the governing fractional partial differential equations are systematically reduced to fractional ordinary differential equations. Exact invariant solutions are obtained in closed form and expressed in terms of Mittag--Leffler functions, generalized Wright functions, and Fox $H$-functions. These analytical solutions provide valuable insights into fractional telegraph-type transport phenomena and serve as important benchmarks for validating numerical methods in industrial transport modeling and fractional evolution systems.

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

This paper classifies Lie symmetries for a time-fractional telegraph system with variable coefficients into three cases and produces invariant solutions in Mittag-Leffler and related functions.

read the letter

The core contribution is a Lie symmetry classification for time-fractional telegraph equations with spatially varying coefficients. The authors split the admitted symmetries into three classes according to the functional relation between the transport coefficient and the potential, then build optimal one-dimensional subalgebras, reduce the system to fractional ODEs, and write down closed-form invariant solutions using Mittag-Leffler, generalized Wright, and Fox H-functions. That classification and the explicit reductions are the parts that are actually new for this particular system. The work follows the standard prolongation procedure for Riemann-Liouville derivatives and applies it systematically to the variable-coefficient case, which is useful when exact solutions are needed for benchmarking numerical schemes in transport models with memory. The reductions themselves look clean for the cases they treat. The main soft spot is whether the prolongation fully accounts for the integral terms that appear when the coefficients vary in space. The determining equations can impose extra compatibility conditions on the coefficient functions that are not always automatic, and the abstract claim of a complete classification for “sufficiently differentiable” coefficients leaves open whether every reported form survives those conditions without further restrictions. The paper also does not show direct substitution checks that the derived solutions satisfy the original fractional system in each class. Readers working on symmetry methods for fractional PDEs or on exact solutions for heterogeneous transport will find the three classes and the solution forms worth looking at. The paper is solid enough on its own terms to merit a serious referee rather than a desk rejection, though reviewers will likely press for explicit verification of the compatibility conditions and back-substitution of the solutions.

Referee Report

2 major / 2 minor

Summary. The manuscript applies Lie symmetry analysis to a system of time-fractional telegraph equations with spatially varying coefficients, employing the Riemann-Liouville derivative. It asserts a complete classification of admissible coefficient functions into exactly three symmetry classes determined by the relation between the transport coefficient and the potential function. Optimal systems of one-dimensional subalgebras are constructed for each class, the system is reduced to fractional ODEs, and closed-form invariant solutions are obtained in terms of Mittag-Leffler, generalized Wright, and Fox H-functions.

Significance. If the classification is exhaustive and the derived solutions satisfy the original system, the results extend Lie methods to variable-coefficient fractional PDEs and supply analytical benchmarks for memory-dependent transport models arising in viscoelasticity, heat conduction with memory, and heterogeneous semiconductors.

major comments (2)

[§3] §3 (Lie symmetry classification and determining equations): The invariance condition is obtained by applying the prolonged generator to the system. For the Riemann-Liouville time-fractional derivative the prolongation contains an integral operator; when this acts on terms multiplied by the spatially varying transport and potential coefficients, additional PDE constraints on those coefficients generally appear. The manuscript does not display the full set of determining equations or demonstrate that these extra compatibility conditions have been solved simultaneously with the symmetry equations, so the completeness of the three-class division cannot be confirmed.
[§4] §4 (Reduction to fractional ODEs and invariant solutions): The reported closed-form solutions (expressed via Mittag-Leffler, Wright, and Fox H-functions) are obtained after reduction by the listed subalgebras. Because the original system has variable coefficients, it must be verified explicitly that each solution satisfies the fractional system after the variable coefficients are restored; the manuscript provides the functional forms but does not include a direct substitution check or residual computation for the variable-coefficient case.

minor comments (2)

[Preliminaries] The precise regularity assumptions on the coefficient functions (beyond 'sufficiently differentiable') should be stated explicitly in the preliminaries section, together with any compatibility conditions required for the Riemann-Liouville operator to be well-defined on the solution space.
[Introduction] Notation for the telegraph system (the exact form of the two coupled equations, the symbols used for the transport and potential coefficients) should be introduced once in the introduction and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and valuable suggestions. We have carefully considered the major comments and will revise the manuscript accordingly to enhance clarity and rigor. Below we provide point-by-point responses.

read point-by-point responses

Referee: [§3] §3 (Lie symmetry classification and determining equations): The invariance condition is obtained by applying the prolonged generator to the system. For the Riemann-Liouville time-fractional derivative the prolongation contains an integral operator; when this acts on terms multiplied by the spatially varying transport and potential coefficients, additional PDE constraints on those coefficients generally appear. The manuscript does not display the full set of determining equations or demonstrate that these extra compatibility conditions have been solved simultaneously with the symmetry equations, so the completeness of the three-class division cannot be confirmed.

Authors: We thank the referee for pointing this out. Upon re-examination, we realize that while the determining equations were derived and solved internally to obtain the three classes, the full system was not explicitly presented in the paper. In the revised manuscript, we will add a subsection or appendix detailing the complete set of determining equations, including those arising from the integral terms in the prolongation. We will show the step-by-step solution process that leads to the three symmetry classes depending on the relation between the transport coefficient and the potential function. This will confirm that all compatibility conditions have been accounted for and that the classification is exhaustive. revision: yes
Referee: [§4] §4 (Reduction to fractional ODEs and invariant solutions): The reported closed-form solutions (expressed via Mittag-Leffler, Wright, and Fox H-functions) are obtained after reduction by the listed subalgebras. Because the original system has variable coefficients, it must be verified explicitly that each solution satisfies the fractional system after the variable coefficients are restored; the manuscript provides the functional forms but does not include a direct substitution check or residual computation for the variable-coefficient case.

Authors: We agree with the referee that explicit verification strengthens the results. In the revised version, we will include direct substitution of each invariant solution into the original time-fractional telegraph system, restoring the variable coefficients. We will demonstrate that the fractional derivatives and other terms cancel appropriately, yielding zero residuals for the solutions in each symmetry class. This verification will be presented for representative cases from the three classes. revision: yes

Circularity Check

0 steps flagged

No circularity: standard determining-equation classification of coefficient forms

full rationale

The paper applies the prolonged infinitesimal generator to the time-fractional telegraph system (Riemann-Liouville derivative) and solves the resulting over-determined system of PDEs for the infinitesimals together with the admissible functional forms of the spatially varying transport and potential coefficients. The three symmetry classes emerge directly from the algebraic and differential constraints obtained in that step; no parameter is fitted to data and then relabeled as a prediction, no self-definition equates an output to its own input, and no load-bearing uniqueness theorem is imported solely via self-citation. The subsequent reductions to fractional ODEs and closed-form solutions in Mittag-Leffler/Wright/H-functions follow the classical invariant-solution procedure and remain independent of the classification step. The derivation 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 paper rests on standard background results from Lie group theory and fractional calculus; no free parameters, ad-hoc axioms, or new postulated entities are introduced in the abstract.

axioms (2)

standard math Properties of the Riemann-Liouville fractional derivative under Lie transformations
Invoked for the symmetry analysis of time-fractional equations.
standard math Existence of Lie point symmetries for systems of PDEs
Basis for the classification procedure.

pith-pipeline@v0.9.0 · 5531 in / 1331 out tokens · 55552 ms · 2026-05-07T14:43:47.501542+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references

[1]

A stochastic model related to the telegrapher’s equation.Rocky Mountain Journal of Math- ematics, 4(3):497–509, 1974

Mark Kac. A stochastic model related to the telegrapher’s equation.Rocky Mountain Journal of Math- ematics, 4(3):497–509, 1974

1974
[2]

Time-fractional telegraph equations and telegraph processes with brownian time.Probability Theory and Related Fields, 128(1):141–160, 2004

Enzo Orsingher and Luisa Beghin. Time-fractional telegraph equations and telegraph processes with brownian time.Probability Theory and Related Fields, 128(1):141–160, 2004

2004
[3]

Sahadevan

George Bluman, Temuerchaolu, and R. Sahadevan. Local and nonlocal symmetries for nonlinear tele- graph equations.Journal of Mathematical Physics, 46(2):023505, 2005

2005
[4]

Conservation laws for nonlinear telegraph equations.Journal of Mathematical Analysis and Applications, 310(2):459–476, 2005

George Bluman and Temuerchaolu. Conservation laws for nonlinear telegraph equations.Journal of Mathematical Analysis and Applications, 310(2):459–476, 2005

2005
[5]

Comparing symmetries and conservation laws of nonlinear telegraph equations.Journal of Mathematical Physics, 46(7):073513, 2005

George Bluman and Temuerchaolu. Comparing symmetries and conservation laws of nonlinear telegraph equations.Journal of Mathematical Physics, 46(7):073513, 2005. 17

2005
[6]

Academic Press, San Diego, 1999

Igor Podlubny.Fractional Differential Equations, volume 198 ofMathematics in Science and Engineer- ing. Academic Press, San Diego, 1999

1999
[7]

Miller and Bertram Ross.An Introduction to the Fractional Calculus and Fractional Differ- ential Equations

Kenneth S. Miller and Bertram Ross.An Introduction to the Fractional Calculus and Fractional Differ- ential Equations. John Wiley & Sons, New York, 1993

1993
[8]

Imperial College Press, London, 2010

Francesco Mainardi.Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Math- ematical Models. Imperial College Press, London, 2010

2010
[9]

The random walk’s guide to anomalous diffusion: a fractional dynamics approach.Physics Reports, 339(1):1–77, 2000

Ralf Metzler and Joseph Klafter. The random walk’s guide to anomalous diffusion: a fractional dynamics approach.Physics Reports, 339(1):1–77, 2000

2000
[10]

Montroll

Harvey Scher and Elliott W. Montroll. Anomalous transit-time dispersion in amorphous solids.Physical Review B, 12(6):2455–2477, 1975

1975
[11]

Henry and Stephen L

Bruce I. Henry and Stephen L. Wearne. Fractional reaction–diffusion.Physica A: Statistical Mechanics and its Applications, 276(3–4):448–455, 2000

2000
[12]

Bluman and Sukeyuki Kumei.Symmetries and Differential Equations, volume 81 ofApplied Mathematical Sciences

George W. Bluman and Sukeyuki Kumei.Symmetries and Differential Equations, volume 81 ofApplied Mathematical Sciences. Springer-Verlag, New York, 1989

1989
[13]

Olver.Applications of Lie Groups to Differential Equations, volume 107 ofGraduate Texts in Mathematics

Peter J. Olver.Applications of Lie Groups to Differential Equations, volume 107 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 2nd edition, 1993

1993
[14]

Gazizov, Alexander A

Rafail K. Gazizov, Alexander A. Kasatkin, and Stanislav Yu. Lukashchuk. Continuous transformation groups of fractional differential equations.Symmetry, Integrability and Geometry: Methods and Appli- cations (SIGMA), 3:002, 2007

2007
[15]

Lukashchuk and A

Stanislav Yu. Lukashchuk and A. V. Makunin. Group classification of nonlinear time-fractional diffusion equation with a source term.Applied Mathematics and Computation, 257:335–343, 2015

2015
[16]

Lie symmetries and group classification of a class of time-fractional evolution systems.Journal of Mathematical Physics, 56(12):123504, 2015

Qing Huang and Shoufeng Shen. Lie symmetries and group classification of a class of time-fractional evolution systems.Journal of Mathematical Physics, 56(12):123504, 2015

2015
[17]

Lie symmetry analysis of a class of time-fractional nonlinear evolution systems.Applied Mathematics and Computation, 329:105–117, 2018

Khongorzul Dorjgotov, Hiroyuki Ochiai, and Uuganbayar Zunderiya. Lie symmetry analysis of a class of time-fractional nonlinear evolution systems.Applied Mathematics and Computation, 329:105–117, 2018

2018
[18]

Dumitru Baleanu, Shama Sajjad, Waqas Jamshed, and M. Safdar. Lie symmetry analysis of fractional differential equations.Fractals, 30(04):2250075, 2022

2022
[19]

Ying-Hui Wang, Hou-Dao Deng, and Wei-Guo Cheng. Lie symmetry analysis, optimal system and exact solutions of the (2+1)-dimensional time-fractional kadomtsev-petviashvili equation.Fractional Calculus and Applied Analysis, 26(1):240–264, 2023

2023
[20]

The fractional derivative with respect to another function and its application to lie symmetry analysis.Chaos, Solitons & Fractals, 170:113335, 2023

Yusuf Gurefe. The fractional derivative with respect to another function and its application to lie symmetry analysis.Chaos, Solitons & Fractals, 170:113335, 2023

2023
[21]

Patera and P

J. Patera and P. Winternitz. Subalgebras of real three- and four-dimensional lie algebras.Journal of Mathematical Physics, 18(7):1449–1455, 1977

1977
[22]

On solutions of linear fractional differential equations and systems thereof.Fractional Calculus and Applied Analysis, 22(2):479–494, 2019

Khongorzul Dorjgotov, Hiroyuki Ochiai, and Uuganbayar Zunderiya. On solutions of linear fractional differential equations and systems thereof.Fractional Calculus and Applied Analysis, 22(2):479–494, 2019

2019
[23]

Gl¨ ockle and Theo F

Walter G. Gl¨ ockle and Theo F. Nonnenmacher. Fox function representation of non-debye relaxation processes.Journal of Statistical Physics, 71(3-4):741–757, 1993

1993
[24]

Fractional calculus operators of special functions? the result is well predictable! Chaos, Solitons & Fractals, 102:2–15, 2017

Virginia Kiryakova. Fractional calculus operators of special functions? the result is well predictable! Chaos, Solitons & Fractals, 102:2–15, 2017

2017
[25]

A. M. Mathai, Ram Kishore Saxena, and Hans J. Haubold.The H-Function: Theory and Applications. Springer-Verlag, New York, 2010

2010
[26]

Wright functions as scale-invariant solutions of the diffusion-wave equation.Journal of Computational and Applied Mathematics, 118(1-2):175–191, 2000

Rudolf Gorenflo, Yuri Luchko, and Francesco Mainardi. Wright functions as scale-invariant solutions of the diffusion-wave equation.Journal of Computational and Applied Mathematics, 118(1-2):175–191, 2000

2000
[27]

Anatoly A. Kilbas. Fractional calculus of the generalized wright function.Fractional Calculus and Ap- plied Analysis, 8(2):113–126, 2005

2005
[28]

Exact solutions to a class of time- fractional evolution systems with variable coefficients.Journal of Mathematical Physics, 59(8):081504, 2018

Khongorzul Dorjgotov, Hiroyuki Ochiai, and Uuganbayar Zunderiya. Exact solutions to a class of time- fractional evolution systems with variable coefficients.Journal of Mathematical Physics, 59(8):081504, 2018. 18 National Intelligence University and Department of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia Email address:sodoomath@g...

2018