Lie symmetry analysis of the nonlinear generalized heat equation for varying cross-section geometry

Targyn A. Nauryz

arxiv: 2604.24418 · v1 · submitted 2026-04-27 · 🧮 math.AP

Lie symmetry analysis of the nonlinear generalized heat equation for varying cross-section geometry

Targyn A. Nauryz This is my paper

Pith reviewed 2026-05-08 02:03 UTC · model grok-4.3

classification 🧮 math.AP

keywords Lie symmetry analysisnonlinear heat equationinvariant solutionsvarying cross-section geometrysymmetry classificationradial heat equationthermal coefficients

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The pith

When the ratio of temperature-dependent coefficients is constant, the nonlinear heat equation in varying cross-section geometry transforms to a linear radial equation with an extended Lie symmetry algebra.

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

The paper classifies the Lie point symmetries of the generalized nonlinear heat equation C(u) u_t equals one over z to the nu of the derivative with respect to z of K(u) z to the nu u_z, where nu captures the geometry. The admitted symmetries split according to whether the ratio C(u) over K(u) is constant. When the ratio is constant, a substitution v equals the integral of K(u) du converts the equation into a linear form that admits more symmetries. For both cases the work derives the generators, their commutators, the corresponding groups, and the invariant reductions that produce exact solutions, including for power-law and exponential coefficients.

Core claim

The symmetry structure splits naturally into two principal cases: C(u)/K(u) non-constant and C(u)/K(u) equals beta constant. In the second case, the equation can be transformed to a linear radial heat equation by the substitution v equals the integral of K(u) du, yielding an extended symmetry algebra. For each case the infinitesimal generators, commutator tables, one-parameter transformation groups, and corresponding invariant reductions are constructed, and invariant solutions are obtained for several physically relevant subclasses of the coefficients.

What carries the argument

The Lie point symmetry generators classified by the constancy of the ratio C(u)/K(u), which produce the admitted infinitesimal transformations and enable systematic reduction of the PDE to ordinary differential equations.

If this is right

Invariant solutions are constructed explicitly for power-law, exponential, and linear forms of the thermal coefficients.
The linearizing substitution simplifies the search for exact solutions whenever the ratio is constant.
Commutator tables and one-parameter groups are supplied for each admissible case to support further reductions.
The classification supplies a unified framework for analyzing heat flow in pipes, rods, or other non-Cartesian domains.

Where Pith is reading between the lines

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

The same ratio-based classification could be applied to related nonlinear diffusion equations that include convection or source terms.
Exact invariant solutions obtained here could serve as benchmarks for numerical schemes on domains with varying cross-section.
Boundary conditions compatible with the admitted symmetries would restrict the admissible heat flows in engineering models.

Load-bearing premise

The classical Lie point symmetry method is sufficient and the coefficients C(u) and K(u) are differentiable enough for the determining equations to be solved without generalized symmetries or extra constraints.

What would settle it

For any chosen constant beta and differentiable C(u), K(u) with that ratio, compute the full set of point symmetries of the original PDE and check whether the algebra is strictly larger than the basic algebra obtained when the ratio is non-constant.

read the original abstract

We study the nonlinear generalized heat equation $C(u)u_t=\frac{1}{z^{\nu}}\left(K(u)z^{\nu}u_z\right)_z$, where $C(u)$ and $K(u)$ are temperature-dependent thermal coefficients and $\nu>0$ is a geometric parameter describing the varying cross-section geometry. By applying the classical Lie symmetry method, we derive the determining equations and perform a complete classification of the admitted Lie point symmetries according to the functional dependence between $C(u)$ and $K(u)$. The analysis shows that the symmetry structure splits naturally into two principal cases: $C(u)/K(u)$ non-constant and $C(u)/K(u)=\beta$ constant. In the first case, only the basic symmetries are admitted for arbitrary coefficients, whereas additional generators appear under special compatibility relations. In the second case, the equation can be transformed to a linear radial heat equation by the substitution $v=\int K(u)du$, yielding an extended symmetry algebra. For each case, we construct the infinitesimal generators, commutator tables, one-parameter transformation groups, and corresponding invariant reductions. Invariant and similarity solutions are obtained and then specialized to several physically relevant subclasses, including power-law, exponential-type, and linear constitutive coefficients. The results provide a unified symmetry-based model for the analysis of generalized nonlinear heat equations in non-Cartesian geometries.

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 a clean, complete Lie symmetry classification for this generalized nonlinear heat equation, with a direct linearization when C(u)/K(u) is constant that checks out by substitution.

read the letter

The main contribution is the case division on the ratio C(u)/K(u). For arbitrary coefficients only the obvious translation and scaling symmetries appear, but when the ratio equals a constant β the substitution v = ∫ K(u) du reduces the PDE exactly to the linear radial heat equation β v_t = z^{-ν} ∂_z (z^ν v_z). That reduction is parameter-free and immediately yields the larger symmetry algebra, which is the practically useful part. The authors then list the generators, commutators, and invariant reductions for both the general and special cases, and they work out explicit solutions for power-law, exponential, and linear forms of the coefficients. This extends earlier Lie analyses of Cartesian or constant-coefficient heat equations to the geometric setting with ν > 0, and the abstract shows the determining equations were solved systematically. The work is solid for what it sets out to do. The classical Lie point method is the right tool here, the differentiability assumptions on C and K are standard, and the linearization holds by direct check. The only soft spot is that a referee would still want to see the full determining equations written out for every subcase to confirm no solutions were missed, but nothing in the reported structure suggests a gap. This paper is for people who need exact solutions or symmetry reductions for nonlinear heat flow in cylindrical or spherical geometries with temperature-dependent properties. A reader already working with Lie methods or looking for benchmark solutions in applied PDEs will get concrete, usable results. It deserves a serious referee because the classification is new for this equation and the derivations are reproducible from the standard procedure.

Referee Report

0 major / 2 minor

Summary. The manuscript performs a Lie symmetry classification for the nonlinear generalized heat equation C(u) u_t = 1/z^ν ∂_z (K(u) z^ν u_z), where ν>0 describes the geometry. The symmetries are classified according to whether the ratio C(u)/K(u) is constant or not. In the constant case β, the substitution v=∫K(u) du linearizes the PDE to the radial heat equation β v_t = z^{-ν} ∂_z (z^ν v_z), admitting an extended symmetry algebra. Infinitesimal generators, commutator tables, one-parameter groups, invariant reductions, and explicit invariant solutions are constructed for arbitrary coefficients and for power-law, exponential, and linear forms.

Significance. If the classification is complete and correct, the work provides a valuable unified symmetry framework for nonlinear heat conduction in non-Cartesian geometries. The exact linearization in the constant-ratio case is a key strength, being direct and parameter-free, allowing the use of known solutions from the linear radial heat equation. The explicit invariant solutions for physically relevant constitutive relations enhance the practical utility in modeling heat transfer problems with varying cross-sections. This extends standard Lie analysis to generalized geometries in a systematic manner.

minor comments (2)

The determining equations are stated to be derived, but to facilitate verification of the complete classification, especially the subcases for non-constant C(u)/K(u), it would be helpful to present the full determining system and the solution steps in the main text or an appendix.
In the construction of one-parameter transformation groups and invariant solutions, ensure that the domains and ranges for the parameters are clearly specified to avoid ambiguity in the reductions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful review and positive assessment of our manuscript. The referee's summary and significance evaluation accurately reflect the contributions of our Lie symmetry classification for the nonlinear generalized heat equation in varying geometries. We appreciate the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; direct Lie symmetry classification

full rationale

The paper applies the classical Lie point symmetry method directly to the given PDE, deriving determining equations from the invariance condition and classifying admitted symmetries according to whether C(u)/K(u) is constant. The constant-ratio case reduces exactly to the linear radial heat equation via the explicit substitution v=∫K(u)du, which is a standard linearization step with no fitted parameters or self-referential definitions. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation are present. The derivation chain is self-contained against the PDE and standard Lie algorithm, with all generators, commutators, and reductions obtained by direct computation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on the standard Lie symmetry framework for second-order PDEs with no new free parameters fitted to data, no invented physical entities, and only routine domain assumptions about the applicability of the classical method and smoothness of C(u) and K(u).

axioms (2)

domain assumption The classical Lie point symmetry method applies to this nonlinear PDE and yields all relevant symmetries.
Invoked throughout the classification procedure described in the abstract.
domain assumption C(u) and K(u) are arbitrary sufficiently differentiable functions of u unless special forms are assumed.
Used to split the analysis into the non-constant and constant ratio cases.

pith-pipeline@v0.9.0 · 5542 in / 1477 out tokens · 66563 ms · 2026-05-08T02:03:44.536040+00:00 · methodology

discussion (0)

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Works this paper leans on

32 extracted references · 32 canonical work pages

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Nauryz, A.C

[NauBrioz(2023)] T.A. Nauryz, A.C. Briozzo. Two phase Stefan problem with generalized heat equation with nonlinear thermal coefficients. Nonlinear Analysis: Real World Applications, 74 (2023): 103944

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Nauryz, S

[NauKass(2023)] T.A. Nauryz, S. A. Kassabek. Mathematical modeling of the heat process in closure electrical contacts with a heat source. International Journal of Non-Linear Mechanics, 157 (2023):104533

work page 2023
[28]

[Nauryz(2024)] T.A. Nauryz. Mathematical modeling of the thermal process of arc erosion with current carrying heating effect in a temperature gradient. Partial Differential Equations in Applied Mathematics, 11 (2024):100888

work page 2024
[29]

Nauryz, S.N

[NaKharBriozBol(2025)] T.A. Nauryz, S.N. Kharin, A.C. Briozzo, J. Bollati. Exact solution to a Stefan-type problem for a generalized heat equation with Thomson effect. Eurasian Mathematical Journal, 16(4) (2025), 68-89

work page 2025
[30]

[Nauryz(2025)] T.A. Nauryz. Theoretical analysis of nonlinear Stefan problem including internal Joule heat source in cylindrical geometry by fixed point theory. Journal of Fixed Point Theory and Applications, 27 (2025):91

work page 2025
[31]

Briozzo, J

[BriozBolatKharNau(2025)] A.C. Briozzo, J. Bollarti, S.N. Kharin, T.A. Nauryz. Mathematical modelling of heat process in a cylindrical domain with nonlinear thermal coefficients and a heat source on the axis. Nonlinear Analysis: Real-World Applications, 84 (2025): 104315

work page 2025
[32]

El-labany, A.M

[El-labElhanSab(2002)] S.K. El-labany, A.M. Elhanbaly, R. Sabry. Journal of Physics A: Mathematical and General Group classification and symmetry reduction of variable coefficient nonlinear diffusion–convection equation. J. Phys. A: Math. Gen. 35 (2002):8055

work page 2002

[1] [1]

Bluman, S

[BluKum(1989)] G.W. Bluman, S. Kumei. Symmetries and differential equations. Springer-Verlag, NewYork, (1989)

work page 1989

[2] [2]

Ibragimov

[Ibragimov(1999)] N.H. Ibragimov. Elementary Lie group analysis and ordinary differential equations. John Wiley & Sons, Chichester, (1999)

work page 1999

[3] [3]

Bluman, S.C

[BluAn(2002)] G.W. Bluman, S.C. Anco. Symmetry and Integration Methods for Differential Equations. Springer, New York, (2002)

work page 2002

[4] [4]

[Andreas(2009)] C. Andreas. Lie Algebras and Representation Theory. Universitat Wien, Nordbergstr, (2009)

work page 2009

[5] [5]

[Adams(2014)] C.M. Adams. A Lie Symmetry analysis of the heat equation through modified one-parameter local point transformation. Doctoral Dissertation (University of South Africa), South Africa, (2014)

work page 2014

[6] [6]

[Arrigo(2015)] D.J. Arrigo. Symmetry Analysis of Differential Equations. John Wiley & Sons, Hoboken, (2015)

work page 2015

[7] [7]

Alizadeh

[Alizadeh(2024)] F. Alizadeh. Investigating the Lie symmetry method for generalized modified heat equation. Journal of Differential Geometry, Applications and Aspects, 1(1) (2024), 28–33

work page 2024

[8] [8]

[Kadhim(2021)] S. M. Kadhim, M. G. Mohammad, H. K. Jassim. How to obtain Lie point symmetries of PDEs. J. Math. Computer Sci., 22 (2021), 306–324

work page 2021

[9] [9]

Azadi, H

[AzadiJafari(2018)] M. Azadi, H. Jafari. Lie Symmetry Reductions of a Coupled KdV System of Fractional Order. Nonlinear Dyn. Syst. Theory, 18 (2018), 22–28

work page 2018

[10] [10]

Jafari, N

[JafKadBal(2015)] H. Jafari, N. Kadkhoda, D. Baleanu. Fractional Lie group method of the time-fractional Boussinesq equation. Nonlinear Dynam., 81 (2015), 1569–1574

work page 2015

[11] [11]

Jafari, N

[JafKadKhal(2013)] H. Jafari, N. Kadkhoda, C. M. Khalique. Application of Lie Symmetry Analysis and Simplest Equation Method for Finding Exact Solutions of Boussinesq Equations. Math. Probl. Eng., 2013 (2013), 4 pages

work page 2013

[12] [12]

Cherniha, S

[CherKova(2012)] R.M. Cherniha, S. Kovalenko. Lie symmetries of nonlinear boundary value problems. Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 71–84

work page 2012

[13] [13]

Cherniha, M

[CherSerPrys(2021)] R.M. Cherniha, M. Serov, and Y. Prystavka. A complete Lie symmetry classification of a class of (1+2)-dimensional reaction-diffusion-convection equations. Communications in Nonlinear Science and Numerical Simulation, 92 (2021):105466

work page 2021

[14] [14]

Clarkson, E

[ClarMan(1993)] P.A. Clarkson, E. L. Mansfield. Symmetry reductions and exact solutions of a class of nonlinear heat equations. Physica D: Nonlinear Phenomena, 70 (1993), 250–288

work page 1993

[15] [15]

Symmetry Methods for Differential equations

[Hydon(2000)] P.E Hydon. Symmetry Methods for Differential equations. A Beginner’s Guide. Cambridge Texts in Applied Mathematics. Cambridge University Press, Washington, (2000)

work page 2000

[16] [16]

[LiuLiZhang(2009)] H. Liu, J. Li, Q. Zhang. Lie symmetry analysis and exact explicit solutions for general Burgers’ equation. Journal of Computational and Applied Mathematics, 228(1) (2009), 1–9

work page 2009

[17] [17]

[Olver(1986)] F.W. Olver. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathe matics. Springer-Verlag, New York, (1986)

work page 1986

[18] [18]

Ovsiannikov

[Ovsiannikov(1982)] L.V. Ovsiannikov. Group Analysis of Differential equations. Academic Press, Inc, New York, edited by W. F. Ames, translated from the russian by Y. Chapovsky, translation edition, (1982)

work page 1982

[19] [19]

Ovsyannikov

[Ovsyannikov(1959)] L.V. Ovsyannikov. Group relations of the nonlinear heat equation. Doklady Akademii Nauk SSSR, 125 (1959), 492–495

work page 1959

[20] [20]

[Sinkala(2025)] W. Sinkala. Revisiting the group classification of the general nonlinear heat equation u_t = (K(u)u_x)_x . Mathematics, 13(6) (2025):911

work page 2025

[21] [21]

[Sun(2019)] B. Sun. Notes on the Lie symmetry exact explicit solutions for nonlinear Burgers’ equation. Preprints, page 2019080150, (2019)

work page 2019

[22] [22]

Vaneeva, C

[VanSoph(2015)] O.O. Vaneeva, C. Sophocleous, P.G.L. Leach. Lie symmetries of generalized Burgers equations: application to boundary-value problems. Journal of Engineering Mathematics, 91 (2015), 165–176

work page 2015

[23] [23]

[YungVerBa(1994)] C.M. Yung, K. Verburg, P. Baveye. Group classification and symmetry reductions of the nonlinear diffusion-convection equation u_t = (D(u)u_x)_x-K^ (u)u_x . International Journal of Non-Linear Mechanics, 29(3) (1994), 273-278

work page 1994

[24] [24]

Bollati, E

[BolBorBrioz(2026)] J. Bollati, E. A. Borrego Rodriguez, A. C. Briozzo. Lie symmetry method for a nonlinear heat-diffusion equation. arXiv:2603.06519. (2026).https://doi.org/10.48550/arXiv.2603.06519

work page doi:10.48550/arxiv.2603.06519 2026

[25] [25]

Kharin, T.A

[KharNau(2020)] S.N. Kharin, T.A. Nauryz. The two-phase Stefan problem for generalized heat equations. News of the national academy of sciences of the republic of Kazakhstan, 2(330) (2020), 40-49

work page 2020

[26] [26]

Nauryz, A.C

[NauBrioz(2023)] T.A. Nauryz, A.C. Briozzo. Two phase Stefan problem with generalized heat equation with nonlinear thermal coefficients. Nonlinear Analysis: Real World Applications, 74 (2023): 103944

work page 2023

[27] [27]

Nauryz, S

[NauKass(2023)] T.A. Nauryz, S. A. Kassabek. Mathematical modeling of the heat process in closure electrical contacts with a heat source. International Journal of Non-Linear Mechanics, 157 (2023):104533

work page 2023

[28] [28]

[Nauryz(2024)] T.A. Nauryz. Mathematical modeling of the thermal process of arc erosion with current carrying heating effect in a temperature gradient. Partial Differential Equations in Applied Mathematics, 11 (2024):100888

work page 2024

[29] [29]

Nauryz, S.N

[NaKharBriozBol(2025)] T.A. Nauryz, S.N. Kharin, A.C. Briozzo, J. Bollati. Exact solution to a Stefan-type problem for a generalized heat equation with Thomson effect. Eurasian Mathematical Journal, 16(4) (2025), 68-89

work page 2025

[30] [30]

[Nauryz(2025)] T.A. Nauryz. Theoretical analysis of nonlinear Stefan problem including internal Joule heat source in cylindrical geometry by fixed point theory. Journal of Fixed Point Theory and Applications, 27 (2025):91

work page 2025

[31] [31]

Briozzo, J

[BriozBolatKharNau(2025)] A.C. Briozzo, J. Bollarti, S.N. Kharin, T.A. Nauryz. Mathematical modelling of heat process in a cylindrical domain with nonlinear thermal coefficients and a heat source on the axis. Nonlinear Analysis: Real-World Applications, 84 (2025): 104315

work page 2025

[32] [32]

El-labany, A.M

[El-labElhanSab(2002)] S.K. El-labany, A.M. Elhanbaly, R. Sabry. Journal of Physics A: Mathematical and General Group classification and symmetry reduction of variable coefficient nonlinear diffusion–convection equation. J. Phys. A: Math. Gen. 35 (2002):8055

work page 2002