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arxiv: 2604.23008 · v1 · submitted 2026-04-24 · 🧮 math.NA · cs.NA· math-ph· math.MP

Delay Modeling with Conformable and Caputo Derivatives: Analytical and Computational Insights

Yhon Flores , Michel Molina del Sol , Genly Leon , Byron Droguett , Guillermo Fern\'andez-Anaya , Yoelsy Leyva This is my paper

Pith reviewed 2026-05-08 10:23 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath-phmath.MP
keywords conformable derivativeCaputo derivativedelay differential equationsfractional calculusLaplace transformnumerical stabilityseries expansion
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The pith

Conformable derivatives deliver stable analytic-numerical agreement for fractional delay equations while Caputo versions accumulate long-term discretization errors.

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

The paper develops the integral, exponential function, and Laplace transform tailored to the conformable derivative so that delay terms can be incorporated through series expansions and transform methods while preserving algebraic structure and causality. It then compares this approach to standard Caputo formulations, where convolutional memory kernels complicate both analysis and numerics. Mesh-aligned Euler and Runge-Kutta schemes applied to conformable models produce solutions that stay consistent with the closed-form expressions, whereas Caputo models require L2-σ or series-anchored predictor-corrector methods to keep discretization noise from growing over time. The work therefore positions the conformable framework as a tractable, physically interpretable route for modeling phenomena that combine fractional order with explicit time delays.

Core claim

For fractional-order delay differential equations the conformable derivative admits an associated Laplace transform that keeps the algebraic structure intact, allowing explicit solutions via series expansions of the delay terms; the resulting analytic expressions agree with basic Euler and Runge-Kutta integrations over long intervals, while the Caputo derivative's memory kernel forces the use of higher-order or series-anchored schemes to suppress accumulating discretization noise and maintain accuracy.

What carries the argument

The conformable Laplace transform together with series expansions of the delay argument, which together convert the fractional delay equation into an algebraic problem whose solutions remain causal and finite.

If this is right

  • Explicit solutions become available for a range of conformable delay models without resorting to numerical approximation from the outset.
  • Basic first-order integrators suffice to maintain long-term accuracy in conformable cases, reducing the need for specialized high-order codes.
  • Caputo-based delay models can be made reliable only by anchoring the scheme to a series expansion or by increasing the order of the discretization.
  • The conformable approach supplies a direct physical interpretation for the fractional order and the delay, unlike memory-integral formulations.

Where Pith is reading between the lines

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

  • The same conformable Laplace machinery could be applied to systems with state-dependent or distributed delays common in biology and control.
  • Switching from Caputo to conformable might simplify real-time simulation of delayed fractional systems in engineering hardware.
  • Comparative tests on experimental time-series data with known delays would reveal whether the conformable solutions better match observed transients.

Load-bearing premise

That the series expansions used to incorporate delay terms inside the conformable Laplace framework preserve causality and do not generate unphysical artifacts or lose interpretability.

What would settle it

Integrate a simple linear conformable delay equation whose closed-form solution is known, using only the basic Euler method over a long time horizon, and check whether the numerical trajectory remains within a small, non-growing error bound of the analytic solution.

Figures

Figures reproduced from arXiv: 2604.23008 by Byron Droguett, Genly Leon, Guillermo Fern\'andez-Anaya, Michel Molina del Sol, Yhon Flores, Yoelsy Leyva.

Figure 1
Figure 1. Figure 1: FIG. 1. Top panel: comparison between the analytical series solution (black solid line) and numeri view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Top panel: comparison between the analytical series solution (black solid line) and numer view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Top panel: comparison between the analytical series solution (black solid line) and numer view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Top panel: comparison between the analytical series solution (black solid line) and numer view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Top panel: comparison between the analytical series solution (black solid line) and numer view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of analytic series, Euler (L1), and L2–1 view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of analytic truncated series and series–anchored predictor–corrector algorithm view at source ↗
read the original abstract

This work presents an analytical and computational study of fractional-order delay differential equations formulated using both the conformable and Caputo derivatives. For the conformable case, we develop the associated integral, exponential function, and Laplace transform, showing how the conformable Laplace framework preserves algebraic structure and facilitates explicit solutions. Delay terms are treated through series expansions and transform-based methods, ensuring causal and finite representations. In parallel, Caputo-based formulations are examined, highlighting the challenges posed by convolutional memory kernels and the potential for long-term numerical instability. Numerical implementations are carried out using mesh-aligned algorithms: Euler and Runge--Kutta schemes for conformable dynamics, and Euler, L2--$\sigma$, and a series--anchored predictor--corrector method for Caputo dynamics. Comparative experiments demonstrate that conformable derivatives yield stable, consistent agreement between analytic and numerical solutions, whereas Caputo dynamics require higher-order or series-anchored schemes to suppress discretization noise and maintain long-term accuracy. These results underscore the advantages of the conformable formalism in modeling dynamic phenomena with delay and memory, offering a tractable and physically interpretable alternative to integral-based fractional models.

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

2 major / 2 minor

Summary. This manuscript investigates fractional delay differential equations using both conformable and Caputo fractional derivatives. Analytically, it derives the integral operator, exponential function, and Laplace transform for the conformable derivative, employing series expansions and transform methods to incorporate delay terms while maintaining causality. For the Caputo derivative, it addresses the difficulties arising from the memory kernel. Computationally, it implements Euler and Runge-Kutta methods for conformable equations and compares them to Euler, L2-σ, and a series-anchored predictor-corrector scheme for Caputo equations. The key finding is that conformable formulations achieve stable and consistent analytic-numerical agreement using standard integrators, whereas Caputo formulations exhibit discretization noise unless higher-order or specialized schemes are used, suggesting conformable derivatives as a more tractable modeling choice for delayed systems.

Significance. If the reported numerical advantages are confirmed to stem from the properties of the conformable derivative rather than differences in discretization schemes, the work would offer a valuable alternative framework for fractional delay models that avoids the computational overhead of non-local operators. The analytical tools developed for conformable derivatives with delays could enable closed-form solutions in certain cases and facilitate easier numerical implementation. This could be significant in applied mathematics and modeling of systems with memory and delay, such as in biology or control theory, provided the comparisons are made rigorous.

major comments (2)
  1. [Numerical implementations and comparative experiments] Numerical implementations (as described in the abstract): the experiments apply Euler and Runge-Kutta schemes to conformable dynamics but L2-σ and series-anchored predictor-corrector methods to Caputo dynamics. This does not hold the discretization scheme fixed, so the claimed stability and accuracy advantages of conformable derivatives cannot be unambiguously attributed to the derivative type rather than the choice of integrator; a controlled comparison using the same scheme on both is required to support the central claim that conformable is a 'tractable alternative'.
  2. [Comparative experiments] The manuscript provides no specific parameter values for the delay equations, time intervals, error metrics (e.g., L2 or maximum error), or quantitative bounds on the 'consistent agreement' and 'discretization noise' reported in the experiments. This absence makes the comparative results qualitative and prevents independent verification of the headline performance gap.
minor comments (2)
  1. [Numerical implementations] The term 'mesh-aligned algorithms' is introduced without definition or reference, which may obscure the numerical methodology for readers.
  2. [Abstract] The abstract would be strengthened by naming the specific delay differential equation examples (e.g., the form of the test problems) used to generate the reported agreement between analytic and numerical solutions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions highlight important aspects of making the numerical comparisons more rigorous, and we will revise the paper to address them directly.

read point-by-point responses

  1. Referee: Numerical implementations (as described in the abstract): the experiments apply Euler and Runge-Kutta schemes to conformable dynamics but L2-σ and series-anchored predictor-corrector methods to Caputo dynamics. This does not hold the discretization scheme fixed, so the claimed stability and accuracy advantages of conformable derivatives cannot be unambiguously attributed to the derivative type rather than the choice of integrator; a controlled comparison using the same scheme on both is required to support the central claim that conformable is a 'tractable alternative'.

    Authors: We agree that the current setup uses schemes tailored to each derivative type, which prevents isolating the effect of the derivative alone. To address this, we will add a controlled comparison in the revised manuscript by applying the same Euler method to both conformable and Caputo delay equations under identical parameters and time intervals. This will allow direct attribution of any observed differences to the derivative formulation while retaining the original results (which demonstrate that Caputo requires specialized schemes for stability) as supplementary evidence of practical tractability. revision: yes

  2. Referee: The manuscript provides no specific parameter values for the delay equations, time intervals, error metrics (e.g., L2 or maximum error), or quantitative bounds on the 'consistent agreement' and 'discretization noise' reported in the experiments. This absence makes the comparative results qualitative and prevents independent verification of the headline performance gap.

    Authors: We accept this criticism; the absence of explicit parameters and metrics renders the comparisons insufficiently quantitative. In the revision, we will specify all parameters (e.g., delay value τ, fractional order α, initial conditions, and integration interval [0,T]), report L2 and maximum absolute errors for each method, and include tables or figures with quantitative bounds on agreement and noise levels. This will enable independent verification and strengthen the central claims. revision: yes

Circularity Check

0 steps flagged

No load-bearing circularity; minor self-citations only

full rationale

The paper's derivation chain consists of explicit constructions of the conformable integral, exponential, and Laplace transform from the standard conformable derivative definition, followed by series expansions for delay terms and direct numerical comparisons using stated schemes. These steps are self-contained and do not reduce to fitted parameters, self-referential definitions, or unverified self-citations. The central comparative claim rests on the reported experiments rather than any tautological renaming or imported uniqueness theorem. Any self-citations to prior conformable work are peripheral and not load-bearing for the delay-modeling results or stability conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on established definitions of conformable and Caputo derivatives from prior literature without introducing new free parameters or invented entities; relies on standard mathematical properties for transforms and series handling of delays.

axioms (2)
  • standard math Standard algebraic properties of the conformable derivative, including existence of integral, exponential function, and Laplace transform that preserve structure
    Invoked to develop explicit solutions and facilitate analytical treatment of delays.
  • domain assumption Delay terms admit causal and finite representations via series expansions and transform methods
    Used to ensure well-posedness in both conformable and Caputo formulations.

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Reference graph

Works this paper leans on

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