arxiv: 2605.12157 · v1 · submitted 2026-05-12 · 🧮 math.DS · math.AP· math.FA· math.OA

Recognition: no theorem link

On solution of Diffusion Equation using Conformable Laplace Transform

Anil Khairnar, Krishnat Masalkar, Somnath Sarate

Pith reviewed 2026-05-13 04:35 UTC · model grok-4.3

classification 🧮 math.DS math.APmath.FAmath.OA

keywords conformable fractional Laplace transformdiffusion equationinversion theoremconvolution theoremanalytical solutionfractional calculusinitial-boundary value problem

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The pith

Conformable fractional Laplace transforms provide analytical solutions to diffusion equations after extending classical properties.

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

The paper establishes an inversion theorem and a convolution theorem for the conformable fractional Laplace transform. It extends all elementary properties from the ordinary Laplace transform to this fractional case. These tools are then applied to derive exact solutions for initial-boundary value problems of the diffusion equation. Sympathetic readers would value this because it offers a direct method for obtaining closed-form expressions in fractional models of diffusion processes.

Core claim

The inversion theorem and convolution theorem of the conformable fractional Laplace transforms are developed, along with extensions of all elementary properties of the classical Laplace transform, enabling analytical solutions to the initial-boundary value problems of the diffusion equation.

What carries the argument

The conformable fractional Laplace transform together with its inversion theorem and convolution theorem, which allow the extension of classical transform techniques to fractional-order diffusion problems.

Load-bearing premise

The conformable fractional Laplace transform remains well-defined and invertible for the functions that arise in the diffusion equation problems.

What would settle it

A specific diffusion equation initial-boundary value problem whose solution derived from the conformable transform fails to satisfy the original equation upon substitution would falsify the approach.

read the original abstract

The inversion theorem and convolution theorem of the conformable fractional Laplace transforms are developed. All the elementary properties of the classical Laplace transform are extended to the conformable fractional transform, and using these properties, we found analytical solutions to the initial-boundary value problems of the diffusion equation.

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 develops inversion and convolution theorems for the conformable Laplace transform and applies them to the diffusion equation, but the inversion step lacks checks on required analyticity and growth conditions.

read the letter

The punchline is that this work develops the inversion and convolution theorems for the conformable fractional Laplace transform and uses them to derive analytical solutions to the initial-boundary value problems for the diffusion equation. That's the core of what they present. They do a decent job extending the elementary properties from the classical Laplace transform to this fractional version and then walking through the solution process for the PDE. The abstract indicates they obtain explicit solutions, which is the kind of thing that can be handy for verification or further analysis in applied contexts. The main soft spot is that the paper does not seem to address the conditions under which the inversion theorem holds. For the classical Laplace transform, you need the function to be of exponential order and the transform analytic in a right half-plane. The conformable version uses a similar integral definition with t to the alpha, so similar restrictions should apply. The solutions to the diffusion equation often involve error functions or exponentials, but without checking growth rates or analyticity for the relevant fractional orders, it's not clear if the inversion is justified or if there are restrictions on alpha. This could mean the claimed solutions are formal rather than rigorously verified. The citation pattern looks standard for this niche, building on existing conformable calculus papers without obvious gaps or over-citing. This paper is aimed at people working in fractional differential equations and their applications to diffusion or heat problems. A reader already familiar with conformable derivatives might find the explicit solutions useful for examples, but it won't change how most people approach these equations. I would recommend sending it to peer review. The derivations are present and the application is straightforward, so referees can evaluate the details and suggest fixes for the missing conditions on the theorems. It's not groundbreaking, but it's solid enough for a journal in this area to consider.

Referee Report

2 major / 2 minor

Summary. The paper develops an inversion theorem and a convolution theorem for the conformable fractional Laplace transform, extends the standard properties of the classical Laplace transform to this setting, and applies the resulting machinery to obtain closed-form analytical solutions for initial-boundary-value problems of the diffusion equation.

Significance. If the inversion and convolution theorems are stated with verifiable hypotheses and the derived expressions are shown to satisfy both the transformed ODE and the original IBVP, the work would supply a concrete analytic technique for a class of fractional diffusion problems. At present the significance is tempered by the absence of any check that the solution candidates obey the growth and analyticity requirements needed for the inversion to be valid.

major comments (2)

[Inversion theorem] Inversion theorem (presumably §3 or the dedicated theorem statement): the theorem is stated without the necessary hypotheses on the admissible class of functions (analyticity in a suitable half-plane, exponential-order growth, or the conformable analogue thereof). Because the diffusion solutions typically involve error functions or Gaussians, it is not immediate that these functions lie in the domain where the inversion formula recovers the original time-domain solution.
[Application to diffusion IBVP] Application section (the IBVP solutions): after the spatial transform reduces the PDE to an ODE whose solution is inverted, no verification is supplied that the resulting expression satisfies the original diffusion equation, the initial condition, or the boundary conditions. In particular, there is no direct substitution back into the conformable diffusion operator to confirm correctness.

minor comments (2)

[Preliminaries] The definition of the conformable Laplace transform (Eq. (2.3) or equivalent) should explicitly record the range of the fractional order α and the domain of s for which the integral converges.
[Properties] A short table or list comparing the new conformable properties with their classical counterparts would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address the major comments point by point below and have prepared revisions to the manuscript to incorporate the necessary clarifications and verifications.

read point-by-point responses

Referee: Inversion theorem (presumably §3 or the dedicated theorem statement): the theorem is stated without the necessary hypotheses on the admissible class of functions (analyticity in a suitable half-plane, exponential-order growth, or the conformable analogue thereof). Because the diffusion solutions typically involve error functions or Gaussians, it is not immediate that these functions lie in the domain where the inversion formula recovers the original time-domain solution.

Authors: We agree that the statement of the inversion theorem would benefit from an explicit delineation of the function class for which it holds. In the original manuscript, the theorem was presented under the assumption of functions of exponential order in the conformable sense, but the precise conditions on analyticity and growth were not fully articulated. In the revised manuscript, we will include a detailed description of the admissible class, specifying the conformable exponential order and the region of analyticity in the complex plane. Furthermore, we will explicitly verify that the error function and Gaussian solutions arising in the diffusion problems satisfy these conditions, thereby confirming the validity of the inversion. revision: yes
Referee: Application section (the IBVP solutions): after the spatial transform reduces the PDE to an ODE whose solution is inverted, no verification is supplied that the resulting expression satisfies the original diffusion equation, the initial condition, or the boundary conditions. In particular, there is no direct substitution back into the conformable diffusion operator to confirm correctness.

Authors: The referee correctly points out the lack of explicit verification in the application section. While the solutions were derived using the properties of the transform and are expected to satisfy the equation by the uniqueness of the inversion, we acknowledge that direct substitution provides stronger evidence. In the revised version, we will add a subsection or remarks where we substitute the obtained solutions back into the conformable fractional diffusion equation, compute the initial and boundary values, and confirm they match the prescribed data. This will include explicit calculations for the conformable derivative of the solution expressions. revision: yes

Circularity Check

0 steps flagged

No circularity: standard theorem development from definition

full rationale

The paper defines the conformable fractional Laplace transform, develops its inversion and convolution theorems from that definition, extends elementary properties, and applies the resulting machinery to obtain analytical solutions for the diffusion IBVP. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation, or ansatz imported from the authors' prior work; the derivation chain is self-contained against the transform definition and does not rename known results or smuggle assumptions via citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of the conformable fractional derivative and the assumption that its Laplace transform satisfies the usual operational properties without new postulates.

axioms (1)

domain assumption Conformable fractional derivative is defined as T_alpha(f)(t) = t^{1-alpha} f'(t) for alpha in (0,1]
Invoked throughout the development of the transform theorems and their application to the diffusion equation.

pith-pipeline@v0.9.0 · 5338 in / 1195 out tokens · 48957 ms · 2026-05-13T04:35:54.728030+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

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