Delayed Langevin type equations with two fractional derivatives

N. I. Mahmudov

arxiv: 1907.01746 · v1 · pith:5K35ACBUnew · submitted 2019-07-03 · 🧮 math.DS · math.CA

Delayed Langevin type equations with two fractional derivatives

N. I. Mahmudov This is my paper

Pith reviewed 2026-05-25 10:14 UTC · model grok-4.3

classification 🧮 math.DS math.CA

keywords delayed Mittag-Leffler functionRiemann-Liouville derivativetime-delay Langevin equationfractional differential equationexistence and uniquenessUlam-Hyers stabilitynonhomogeneous linear system

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The pith

The delayed Mittag-Leffler type function provides explicit solution formulas for fractional time-delay Langevin equations involving two Riemann-Liouville derivatives.

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

The paper defines a delayed version of the Mittag-Leffler function to incorporate time-delay effects into fractional differential equations. It then derives an explicit formula expressing the solutions of linear nonhomogeneous fractional time-delay Langevin equations that contain two Riemann-Liouville derivatives. Existence and uniqueness of solutions are established by bounding the new function with an exponential and applying fixed-point theorems in a weighted norm. Ulam-Hyers stability is obtained as a further consequence of the same estimates. A reader would care because the explicit formula turns abstract existence statements into concrete expressions that can be evaluated or differentiated directly.

Core claim

By introducing the delayed Mittag-Leffler type function, the paper obtains an explicit formula of solutions to linear nonhomogeneous fractional time-delay Langevin equations involving two Riemann-Liouville fractional derivatives. The existence and uniqueness of solutions are obtained by using an estimation of delayed Mittag-Leffler type functions in terms of exponential functions and a weighted norm via fixed point theorems. Further, Ulam-Hyers stability results are presented.

What carries the argument

The delayed Mittag-Leffler type function, which extends the classical Mittag-Leffler function to delayed arguments and supplies the variation-of-constants formula for the two-derivative delayed system.

If this is right

The explicit formula supplies the unique solution once the forcing term and initial history are given.
Existence and uniqueness hold globally in time because the exponential bound prevents finite-time blow-up in the contraction estimate.
Ulam-Hyers stability follows directly from the same bound, controlling how much an approximate solution can deviate from the exact one.
The homogeneous equation is recovered immediately by setting the nonhomogeneous term to zero in the formula.

Where Pith is reading between the lines

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

The same delayed special function may produce closed-form solutions for other linear fractional delay equations that are not of Langevin type.
Direct numerical evaluation of the explicit formula could serve as a benchmark for discretization schemes that approximate the two fractional derivatives.
The stability constants derived from the exponential bound might be improved by sharper estimates on the delayed Mittag-Leffler function.
Models in anomalous transport or viscoelasticity that already contain fractional derivatives could incorporate explicit delay terms without losing solvability.

Load-bearing premise

The delayed Mittag-Leffler type function admits an a priori bound by an exponential function that is sufficient to run the fixed-point argument in the weighted space.

What would settle it

A concrete choice of delay, fractional orders, initial history, and forcing term for which the proposed explicit formula fails to satisfy the original equation or for which two distinct solutions exist.

read the original abstract

In this paper, we introduce a delayed Mittag-Leffler type function. With the help of the delayed Mittag-Leffler type functions, we give an explicit formula of solutions to linear nonhomogeneous fractional time-delay Langevin equations involving two Riemann-Liouville fractional derivatives. The existence and uniqueness of solutions are obtained by using an estimation of delayed Mittag-Leffler type functions in terms of exponential functions and a weighted norm via fixed point theorems. Further, we present Ulam--Hyers stability results.

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 introduces a delayed Mittag-Leffler type function to write explicit solutions for linear nonhomogeneous fractional delay equations with two RL derivatives; existence, uniqueness and Ulam-Hyers stability then depend on an exponential bound whose derivation is not shown in the abstract.

read the letter

The paper gives explicit solution formulas for a narrow class of linear fractional delay equations by introducing a delayed Mittag-Leffler type function. The existence, uniqueness, and stability results then rest on an exponential bound for that function. They define this new function and use it to express the solutions to the nonhomogeneous linear equations involving two Riemann-Liouville derivatives and a delay. That explicit representation is the main new element. The proofs of well-posedness rely on showing that the new function is bounded by an exponential, which allows them to set up a contraction in a weighted norm and apply a fixed-point theorem. Ulam-Hyers stability follows from the same estimates. This is standard technique once the special function is available. The soft spot is the bound itself. The abstract mentions an estimation in terms of exponential functions, but it is not clear from the given information whether this bound is derived rigorously for the case of two different fractional orders or whether it remains valid independently of the delay size. If the bound has any dependence on the parameters that was overlooked, the whole argument collapses. This work is for specialists in fractional differential equations who are looking for explicit formulas in delay settings. Most readers outside that niche will not find much to use. It is worth sending to peer review because the new function could be of interest if the supporting estimates hold up, and a referee can check the details that are missing from the abstract.

Referee Report

2 major / 2 minor

Summary. The paper introduces a delayed Mittag-Leffler type function and employs it to derive an explicit solution formula for linear nonhomogeneous fractional time-delay Langevin equations involving two Riemann-Liouville fractional derivatives. Existence and uniqueness of solutions are obtained via an exponential bound on the new function combined with a weighted-norm fixed-point argument; Ulam-Hyers stability results are also presented.

Significance. If the exponential bound on the delayed Mittag-Leffler function is rigorously established and uniform in the delay and fractional orders, the explicit formulas would constitute a concrete advance for this class of equations, enabling direct verification of stability without relying solely on abstract estimates. The construction of the special function itself is a technical contribution that could be reusable.

major comments (2)

[Abstract] Abstract (existence/uniqueness paragraph): the central fixed-point argument for existence/uniqueness rests on an a-priori estimate that bounds the newly introduced delayed Mittag-Leffler type function by an exponential; the manuscript must supply the explicit derivation of this bound (including uniformity with respect to the delay parameter and the pair of fractional orders) because any gap directly affects both the well-posedness claim and the subsequent Ulam-Hyers stability result.
[Definition and solution formula sections] The section defining the delayed Mittag-Leffler type function and the explicit solution formula: verify that the formula for the nonhomogeneous linear equation is obtained without circular appeal to the same bound used later for the contraction mapping; the construction must be shown to be consistent with the two-derivative structure.

minor comments (2)

Notation for the two fractional orders and the delay should be introduced once and used consistently; currently the abstract and later sections appear to switch between symbols without a central table of notation.
The weighted norm used in the fixed-point argument should be written explicitly (including the precise weight function) rather than described only qualitatively.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments. We address each point below and will revise the manuscript to strengthen the presentation of the bound and the logical flow of the solution formula.

read point-by-point responses

Referee: [Abstract] Abstract (existence/uniqueness paragraph): the central fixed-point argument for existence/uniqueness rests on an a-priori estimate that bounds the newly introduced delayed Mittag-Leffler type function by an exponential; the manuscript must supply the explicit derivation of this bound (including uniformity with respect to the delay parameter and the pair of fractional orders) because any gap directly affects both the well-posedness claim and the subsequent Ulam-Hyers stability result.

Authors: We agree that the exponential bound requires an explicit, self-contained derivation that also establishes uniformity in the delay and the two fractional orders. In the revised manuscript we will insert a dedicated lemma (with full proof) immediately after the definition of the delayed Mittag-Leffler function; the lemma will derive the bound from the series representation and the integral kernel properties, and will track the dependence on the delay and orders to confirm uniformity. This addition will also be referenced in the existence/uniqueness and stability sections. revision: yes
Referee: [Definition and solution formula sections] The section defining the delayed Mittag-Leffler type function and the explicit solution formula: verify that the formula for the nonhomogeneous linear equation is obtained without circular appeal to the same bound used later for the contraction mapping; the construction must be shown to be consistent with the two-derivative structure.

Authors: The solution formula is obtained by direct application of the Laplace transform (or equivalently by variation of constants) to the linear non-homogeneous equation, using only the series definition of the delayed Mittag-Leffler function and the two distinct Riemann-Liouville kernels; the exponential bound is not invoked at this stage. The two-derivative structure is respected by employing separate convolution terms for each fractional order. In the revision we will add an explicit remark after the formula stating this logical order and confirming that the kernels match the two-derivative operator, thereby removing any possible perception of circularity. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit solution formula and fixed-point argument rest on independently introduced special function and its derived bound.

full rationale

The paper introduces the delayed Mittag-Leffler type function as a new object, then constructs an explicit solution formula for the linear nonhomogeneous equations directly in terms of this function. Existence/uniqueness follows from a separate a-priori exponential bound on the same function, inserted into a weighted norm to obtain a contraction mapping. No step reduces by definition or by self-citation to a fitted parameter or to the target result itself; the bound is invoked as an independent estimate rather than being tautological with the solution expression. The derivation chain therefore remains self-contained against external benchmarks and contains no load-bearing self-referential reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on the definition of the new delayed Mittag-Leffler function, standard properties of Riemann-Liouville operators, and an exponential bound on that function. No free parameters or invented physical entities are introduced.

axioms (2)

standard math Riemann-Liouville fractional derivatives satisfy the usual semigroup and integration-by-parts properties.
Invoked implicitly when writing the solution formula for the linear nonhomogeneous equation.
domain assumption The delayed Mittag-Leffler type function admits an exponential upper bound.
This bound is the key estimate used to close the fixed-point argument for existence and uniqueness.

invented entities (1)

delayed Mittag-Leffler type function no independent evidence
purpose: To express explicit solutions of the linear fractional delay Langevin equation.
Newly defined object whose properties are derived in the paper; no independent physical evidence is claimed.

pith-pipeline@v0.9.0 · 5598 in / 1383 out tokens · 21469 ms · 2026-05-25T10:14:26.483704+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Theory and applications of fr actional diﬀerential equations

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work page 2006

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M. Li, J. Wang, Exploring delayed Mittag-Leﬄer type matrix funct ions to study ﬁnite time stability of fractional delay diﬀerential equations, Appl. Math. Comput. 32 4 (2018) 254–265

work page 2018

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From dynamical systems to the langevin eq uation

Beck C , Roepstorﬀ G . From dynamical systems to the langevin eq uation. Phys A 1987;145:1–14

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The langevin equation

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Anomalous transport: fundatio ns and applications

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Ulam–Hyers stability of fractional langevin equatio ns

Wang JR , Li X . Ulam–Hyers stability of fractional langevin equatio ns. Appl Math Comput 2015;258:72–83

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Fractional diﬀusion equations by the kansa m ethod

Chen W , Ye L , Sun H . Fractional diﬀusion equations by the kansa m ethod. Comput Math Appl 2010;59:1614–20

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Method of approximate particular solutions for constant–and variable–order fractional diﬀusion models

Fu ZJ , Chen W , Ling L . Method of approximate particular solutions for constant–and variable–order fractional diﬀusion models. Eng Anal Bound Elem 2015;57:37–46

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Boundary particle method for laplace tra nsformed time fractional diﬀusion equations

Fu ZJ , Chen W , Yang HT . Boundary particle method for laplace tra nsformed time fractional diﬀusion equations. J Comput Phys 2013;235:52–66 .,,

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Yu T , Deng K , Luo M . Existence and uniqueness of solutions of init ial value problems for non- linear langevin equation involving two fractional orders. Commun Non linear Sci Numer Simulat 2014;19:1661–8

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Langevin equation with two fractional orde rs

Lim SC , Li M , Teo LP . Langevin equation with two fractional orde rs. Phys Lett A 2008;372:6309–20

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The fractional oscillator process with two indices

Lim SC , Teo LP . The fractional oscillator process with two indices . J Phys A 2009;42:065208

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Ahmad B , Nieto JJ . Solvability of nonlinear langevin equation involvin g two fractional orders with dirichlet boundary conditions. Int J Diﬀer Equ 2010:164 94 86

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A study of nonlinear la ngevin equation involving two fractional orders in diﬀerent intervals

Ahmad B , Nieto JJ , Alsaedi A , El-Shahed M . A study of nonlinear la ngevin equation involving two fractional orders in diﬀerent intervals. Nonlinear Anal R W A 201 2;13:599–606

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Wei Z , Li Q , Chea J . Initial value problems for fractional diﬀere ntial equations involving Riemann– Liouville sequential fractional derivative. J Math Anal Appl 2010;36 7:260–72

work page 2010