Exact solution to a class of problems for the Burgers' equation on bounded intervals

Kwassi Anani; Mensah Folly-Gbetoula

arxiv: 2409.19240 · v2 · pith:CYKKQBZ5new · submitted 2024-09-28 · 🧮 math.AP

Exact solution to a class of problems for the Burgers' equation on bounded intervals

Kwassi Anani , Mensah Folly-Gbetoula This is my paper

Pith reviewed 2026-05-23 21:22 UTC · model grok-4.3

classification 🧮 math.AP

keywords Burgers' equationHopf-Cole transformationLaplace transformsinverse Laplace transformsbounded intervalsDirichlet boundary conditionsexact solutionsreaction-diffusion equations

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The pith

The Hopf-Cole transformation combined with inverse Laplace transforms provides an exact solution to Burgers' equation on bounded intervals.

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

This paper shows how to obtain an exact solution for Burgers' equation with Dirichlet boundary conditions on any bounded interval. The approach starts with the Hopf-Cole transformation to convert the nonlinear equation into a linear reaction-diffusion equation. An operational solution for the linear problem is then inverted from the Laplace domain to recover the time-domain solution for the original equation. This method works even when analytic inversion is difficult because numerical Laplace inversion algorithms are efficient and always available. Readers interested in exact solutions for nonlinear PDEs would find this useful as it offers closed-form expressions in the Laplace domain and serves as a benchmark for other methods.

Core claim

By applying the Hopf-Cole transformation to the Burgers' equation with fixed Dirichlet boundary conditions, the problem reduces to a linear reaction-diffusion equation with constant coefficients. The exact operational solution for such linear equations is used to express the transformed variable in the Laplace domain. The solution to the original Burgers' equation is then obtained implicitly through inverse Laplace transforms, which can be analytic via Mellin transforms when possible or numerical otherwise. Tests on two cases confirm agreement with known exact solutions.

What carries the argument

The Hopf-Cole transformation that linearizes the Burgers' equation, together with the inverse Laplace transform applied to the operational solution of the resulting linear PDE.

If this is right

Analytic inverses can be obtained in closed form using Mellin transforms when they exist.
Numerical inverses in the time domain are always possible using efficient algorithms regardless of expression complexity.
The results coincide with classical exact solutions in illustration tests.
Closed-form expressions in the Laplace domain provide an innovative alternative to series expressions or numerical methods.
The inverse Laplace transform approach is more computationally efficient and serves as a reference for numerical and semi-analytical methods.

Where Pith is reading between the lines

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

The method might apply to other nonlinear equations that can be linearized by similar transformations.
Greater use of Laplace-domain closed forms could lead to new ways of analyzing solution behavior without time-stepping.
If the operational solution applies broadly, this could standardize exact solutions for a class of viscous flow problems on finite domains.

Load-bearing premise

The operational solution established for linear reaction-diffusion equations applies directly to the equation obtained after the Hopf-Cole transformation of Burgers' equation without further restrictions from the boundaries or nonlinearity.

What would settle it

A calculation showing that the inverse Laplace transform solution, when plugged back into the Burgers' equation, fails to satisfy it for a specific bounded interval and initial condition where an independent exact solution is known.

Figures

Figures reproduced from arXiv: 2409.19240 by Kwassi Anani, Mensah Folly-Gbetoula.

**Figure 1.** Figure 1: Solutions w(x, t) at different times in Example 3.1: (a) a 2 = 1, σ = 2, (b) a 2 = 0.1, σ = 2 Its derivative with respect to x is deduced as: Ux(x, p) = − u (0, 0) sin (π x) π 3 (σ + 1) (π 2a 2 + p) . Consequently, the exact solution in the time domain of the first problem in consideration, is recovered by using the inverse Laplace operator L −1 as in formula (27) as: w(x, t) = −2a 2L −1{Ux(x, p)} L−1{U(x,… view at source ↗

**Figure 2.** Figure 2: Solutions w(x, t) at different times in Example 3.2: (a) a 2 = 1, (b) a 2 = 0.1 decreases, as shown in Figures 2(a), where a 2 = 1, and 2(b), where a 2 = 0.1. In brief, the numerical inverse Laplace procedure, by using the exact implicit formula (27), is notable for its precision, consistency, and computational efficiency. 4. Conclusion This study has enabled the precise calculation of an exact solution fr… view at source ↗

read the original abstract

Burgers' equation with fixed Dirichlet boundary conditions is considered on generic bounded intervals. By using the Hopf-Cole transformation and the exact operational solution recently established for linear reaction-diffusion equations with constant coefficients, an exact solution in the time domain is implicitly derived by means of inverse Laplace transforms. Analytic inverses, whenever they exist, can be obtained in closed form using Mellin transforms. However, highly efficient algorithms are available, and numerical inverses in the time domain are always possible, regardless of the complexity of the Laplace domain expressions. Two illustration tests show that the results coincide well with those of classical exact solutions. Compared to the solutions obtained with series expressions or by numerical methods, closed-form expressions, even in the Laplace domain, represent an innovative alternative and new perspectives can be envisaged. The exact solution via the inverse Laplace transform is shown to be more computationally efficient and thus provides a reference point for numerical and semi-analytical methods.

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 implicit Laplace-domain route to Burgers solutions on intervals via Hopf-Cole plus a prior linear operational method, but the Robin BCs created by the transform need explicit checking against that prior result.

read the letter

The main thing here is that the authors linearize Burgers with fixed Dirichlet conditions via Hopf-Cole, feed the resulting heat equation into a recently cited operational Laplace solution for constant-coefficient linear reaction-diffusion, and then invert numerically to get time-domain values. Two example cases match known exact solutions, and they argue the approach is faster than series or direct numerics while still supplying a reference point. Analytic inverses are possible via Mellin transforms when the Laplace expression allows it. This combination applied specifically to bounded intervals is new, even though each step draws on existing tools. The efficiency claim for numerical inversion is reasonable and could be useful in practice. The tests provide some reassurance that the outputs are at least plausible. The soft spot is the boundary-condition transfer. Hopf-Cole maps constant Dirichlet data on u to Robin conditions on the heat variable. If the cited operational solution was worked out only for Dirichlet or Neumann cases, the s-domain spatial ODE and the inversion formula do not transfer automatically. The abstract gives no sign that this compatibility was verified or that the prior expressions were adapted. Without explicit Laplace-domain formulas or a derivation walk-through, it is difficult to judge whether the central claim holds. The two tests are too narrow to catch a mismatch in the linear step. This work is for people who already use Laplace or operational methods on linear PDEs and want a reference technique for the nonlinear bounded-domain case. A reader looking for an alternative to series solutions might extract value if the Robin issue is resolved. It deserves peer review so that the BC compatibility and the inversion step can be examined directly.

Referee Report

2 major / 2 minor

Summary. The manuscript claims an exact solution method for Burgers' equation with fixed Dirichlet boundary conditions on bounded intervals. It applies the Hopf-Cole transformation to obtain a linear reaction-diffusion equation, invokes a recently established operational solution for such linear equations (with constant coefficients) in the Laplace domain, and obtains the time-domain solution via inverse Laplace transforms (analytic via Mellin when possible, or numerical otherwise). Two illustration tests are said to agree with classical exact solutions, and the approach is positioned as more efficient than series or purely numerical methods.

Significance. If the transfer of the prior operational solution is valid under the transformed boundary conditions, the method would supply a useful alternative representation (Laplace-domain closed form plus inversion) for a class of Burgers problems, with potential computational advantages and new perspectives for semi-analytical work. The explicit mention of Mellin-transform inversion when feasible and the availability of numerical inversion algorithms are positive features.

major comments (2)

[§2 (transformation and operational solution)] The Hopf-Cole transformation with constant Dirichlet data u(0,t)=A and u(L,t)=B produces Robin boundary conditions φ_x(0,t) = (-A/(2ν))φ(0,t) and φ_x(L,t) = (B/(2ν))φ(L,t) for the linear problem. The manuscript invokes the prior operational solution without an explicit check that this result (and the associated s-domain spatial ODE plus inversion formula) remains valid under Robin conditions rather than the Dirichlet/Neumann cases that may have been assumed in the cited work. This transfer is load-bearing for the central claim of an exact implicit solution.
[Abstract, §4] Abstract and §4 (numerical illustrations): the claim that the two tests “coincide well” with classical exact solutions is stated without reported error norms, explicit Laplace-domain expressions for the test cases, or comparison tables. This leaves the verification of the method at a qualitative level and weakens the supporting evidence for the overall procedure.

minor comments (2)

[§2] Notation for the transformed variable and the precise statement of the prior operational result should be restated in the manuscript (rather than only cited) so that the boundary-condition compatibility can be verified directly.
[Abstract] The abstract refers to “analytic inverses, whenever they exist” via Mellin transforms; a brief remark on the conditions under which the Laplace-domain expression admits a closed-form Mellin inversion would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify and strengthen the presentation of our results. We address each major comment below.

read point-by-point responses

Referee: [§2 (transformation and operational solution)] The Hopf-Cole transformation with constant Dirichlet data u(0,t)=A and u(L,t)=B produces Robin boundary conditions φ_x(0,t) = (-A/(2ν))φ(0,t) and φ_x(L,t) = (B/(2ν))φ(L,t) for the linear problem. The manuscript invokes the prior operational solution without an explicit check that this result (and the associated s-domain spatial ODE plus inversion formula) remains valid under Robin conditions rather than the Dirichlet/Neumann cases that may have been assumed in the cited work. This transfer is load-bearing for the central claim of an exact implicit solution.

Authors: The operational solution cited in the manuscript is formulated for linear reaction-diffusion equations with constant coefficients; the s-domain spatial ODE is obtained by direct Laplace transformation of the PDE, after which the two integration constants are fixed by the two boundary conditions (whatever their type). The Robin conditions produced by the Hopf-Cole map are linear and enter the algebraic system for those constants in exactly the same manner as Dirichlet or Neumann data. Nevertheless, we agree that an explicit verification improves transparency. In the revised manuscript we will add a short derivation in §2 that substitutes the specific Robin coefficients into the s-domain solution and confirms that the subsequent inversion formula remains unchanged. revision: yes
Referee: [Abstract, §4] Abstract and §4 (numerical illustrations): the claim that the two tests “coincide well” with classical exact solutions is stated without reported error norms, explicit Laplace-domain expressions for the test cases, or comparison tables. This leaves the verification of the method at a qualitative level and weakens the supporting evidence for the overall procedure.

Authors: We accept that quantitative support is desirable. The revised §4 will display the explicit Laplace-domain expressions for both test problems, report L² error norms between the numerically inverted solutions and the known exact solutions at several times, and include a comparison table. The abstract will be updated to note that the agreement has been quantified. These additions keep the solution implicit (via inverse Laplace transform) while making the validation rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external prior result via standard transformation

full rationale

The paper reduces Burgers' equation via the Hopf-Cole transformation to a linear reaction-diffusion problem, then invokes a cited prior operational solution (for constant-coefficient linear equations) to obtain a Laplace-domain expression whose inverse yields the claimed exact solution. No equation or step within the manuscript defines the target solution in terms of itself, renames a fitted quantity as a prediction, or reduces the central claim to a self-citation chain that is unverified by construction. The prior result is treated as given external input; illustration tests compare against independent classical solutions. This is a self-contained application of existing methods with no load-bearing internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard domain assumptions: that Hopf-Cole linearizes Burgers' equation under the given Dirichlet conditions, and that the cited operational solution for linear reaction-diffusion equations extends without modification to the transformed problem. No free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Hopf-Cole transformation converts Burgers' equation into a linear reaction-diffusion equation with constant coefficients under fixed Dirichlet boundary conditions
Invoked at the start of the derivation; standard but requires the boundary conditions to remain compatible after transformation.
domain assumption An exact operational solution exists for linear reaction-diffusion equations with constant coefficients and can be used directly after the Hopf-Cole step
Cited as 'recently established'; the paper treats this as given without re-deriving it.

pith-pipeline@v0.9.0 · 5690 in / 1412 out tokens · 19198 ms · 2026-05-23T21:22:00.408208+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Anani, K.: Analytical approximations in short times of exact oper - ational solutions to reaction-diﬀusion problems on bounded interva ls. Appl. Appl. Math. 19(1), 1–27 (2023) 13

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work page 2006

[1] [1]

Ba¸ shan, A.: Nonlinear dynamics of the Burgers’ equation and nu - merical experiments. Math. Sci., Springer. 16(2), 183–205 (2022 ). https://doi.org/10.1007/s40096-021-00410-8 12

work page doi:10.1007/s40096-021-00410-8 2022

[2] [2]

Pramana - J

Bonkile, M.P., Awasthi, A., Lakshmi, C., Mukundan, V., Aswin, V.S.: A systematic literature review of Burgers’ equation with recent adv ances. Pramana - J. Phys. 90, 1–21 (2018)

work page 2018

[3] [3]

Hopf, E.: The partial diﬀerential equation ut + uux = µuxx. Comm. Pure Appl. Math. 3, 201–230 (1950). http://dx.doi.org/10.1002/cpa.3160030302

work page doi:10.1002/cpa.3160030302 1950

[4] [4]

Cole, J.D.: On a quasilinear parabolic equations occurring in aerodynamics. Quart. Appl. Math. 9(3), 225–236 (1951). https://www.jstor.org/stable/43633894

work page arXiv 1951

[5] [5]

Benton, E.R., Platzman, G.W.: A table of solutions of the one- dimensional Burgers’ equations. Quart. Appl. Math. 30(2), 195– 212 (1972). https://doi.org/10.1090/qam/306736

work page doi:10.1090/qam/306736 1972

[6] [6]

Dhawan, S., Kapoor, S., Kumar, S., Rawat, S.: Contemporary rev iew of techniques for the solution of nonlinear Burgers equation. J. Co mput. Sci., 3(5), 405–419. (2012). https://doi.org/10.1016/j.jocs.201 2.06.003

work page doi:10.1016/j.jocs.201 2012

[7] [7]

Naghipour, A., Manaﬁan, J.: Application of the Laplace Adomian de- composition and implicit methods for solving Burgers’ equation. TWMS J. Pure Appl. Math. 6(1), 68–77 (2015)

work page 2015

[8] [8]

Zeidan, D., Chau, C.K., Lu, T.T., Zheng, W.Q.: Mathematical studies of the solution of Burgers’ equations by Adomian decompo- sition method. Math. Methods Appl. Sci. 43(5), 2171–2188 (2020) . https://doi.org/10.1002/mma.5982

work page doi:10.1002/mma.5982 2020

[9] [9]

Lal, D., Yadav, M.: Approximate analytical solution of one dimension al nonlinear Burger’s equation using Homotopy Perturbation method. J. Algebr. Stat. 13(3), 5462–5469 (2022)

work page 2022

[10] [10]

Biazar, J., Aminikhah, H.: Exact and numerical solutions for non- linear Burger’s equation by VIM. Math. Comput. Modelling 49(7), 1394–14 00 (2009). https://doi.org/10.1016/j.mcm.2008.12.006

work page doi:10.1016/j.mcm.2008.12.006 2009

[11] [11]

Anani, K.: Analytical approximations in short times of exact oper - ational solutions to reaction-diﬀusion problems on bounded interva ls. Appl. Appl. Math. 19(1), 1–27 (2023) 13

work page 2023

[12] [12]

In: He rron I.H., Foster, M.R

Herron, I.H., Foster, M.R.: Laplace transform methods. In: He rron I.H., Foster, M.R. (eds.) Partial Diﬀerential Equations in Fluid Dynamics, p p. 148-182. Cambridge University Press, Cambridge (2008)

work page 2008

[13] [13]

In: Debnath, L., Bhatta, D

Debnath, L., Bhatta, D.: Laplace transforms and their basic pr oper- ties. In: Debnath, L., Bhatta, D. (eds.) Integral Transforms an d Their Applications, pp. 143–196. CRC press, Boca Raton (2014)

work page 2014

[14] [14]

de Hoog, F.R., Knight J.H., Stokes, A.N.: An improved method for numerical inversion of Laplace transforms. SIAM J. Sci. Stat. Com put. 3(3), 357–366 (1982). https://doi.org/10.1137/0903022

work page doi:10.1137/0903022 1982

[15] [15]

Wood, W.L.: An exact solution for Burgers’ equation. Commun. N umer. Methods Eng. 22(7), 797–798 (2006). https://doi.org/10.1002/ cnm.850 14

work page 2006