About Fractional Calculus and its Applications in Physics

C.F.L. Godinho; J.J.A. de Oliveira

REVIEW 4 major objections 5 minor 1 cited by

About Fractional Calculus and its Applications in Physics

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper's central claim is that Riemann–Liouville fractional integrals compose by adding their orders and that a fractional action yields a dissipative Euler–Lagrange equation, though the proof as printed contains misprinted identities.

desk verdict A well-intentioned review of fractional calculus undermined by a broken semigroup proof and other objective math errors; not publishable as is. read the letter →

arxiv 2507.04186 v1 pith:SDQMN66J submitted 2025-07-05 math-ph math.MP

J.J.A. de Oliveira , C.F.L. Godinho This is my paper · ORCID

classification math-phmath.MP MSC 26A3349K05

keywords Riemann-LiouvilleCaputofractionalcalculusintegralderivativesemigrouppropertyEuler-LagrangeequationFALVA

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper is a survey of fractional calculus written for physics students, moving from L'Hôpital's 1695 question about half-order derivatives through the Riemann–Liouville and Caputo operators to a variational principle called FALVA. The mathematical result it tries to establish is the semigroup law for the Riemann–Liouville fractional integral, $aI^\alpha_x[aI^\beta_x f]=aI^{\alpha+\beta}_x f$, and the physical result is a fractional Euler–Lagrange equation whose extra term plays the role of a dissipative force. If both held, fractional integral operators would compose by adding orders, and frictional systems could be derived from a fractional action. The proof of the semigroup law as printed leans on a Beta-function identity and an exponent in the change of variables; the text's versions of those two ingredients are not correct as printed, so that claim is asserted but not demonstrated in the derivation shown. A further goal is to assess whether this material fits into an undergraduate physics curriculum.

What carries the argument

The engine is the Riemann–Liouville fractional integral $aI^\alpha_x f(x)=\frac1{\Gamma(\alpha)}\int_a^x (x-t)^{\alpha-1}f(t)\,dt$, together with the claimed semigroup identity $aI^\alpha_x[aI^\beta_x f]=aI^{\alpha+\beta}_x f$. The proof is meant to reduce the double integral to a single integral via a change of variables and the Beta function identity $B(\alpha,\beta)=\Gamma(\alpha)\Gamma(\beta)/\Gamma(\alpha+\beta)$; this is the step where the text's printed formulas go astray. The variational engine is the FALVA action, which inserts the kernel $(t-\tau)^{\alpha-1}/\Gamma(\alpha)$ into the action integral and yields the dissipative correction in Eq. (41).

What would settle it

Substitute $\alpha=\beta=1/2$, $f(\xi)=1$, and $a=0$ into the paper's Eq. (18): the printed formula gives $\frac{1}{2\sqrt{\pi}}\int_0^x (x-\xi)^{-1}d\xi$, which diverges at $\xi=x$, while the claimed value of $I^1 1$ is $x$.

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Extended reading notes

Core claim

On its own terms, the paper's contribution is to present the Riemann–Liouville fractional integral as a semigroup of operators: applying $I^\beta$ and then $I^\alpha$ is the same as applying $I^{\alpha+\beta}$. It then derives, from the fractional action $S_\alpha[q]=\frac{1}{\Gamma(\alpha)}\int_a^x L(\dot q,q,\tau)(t-\tau)^{\alpha-1}d\tau$, the fractional Euler–Lagrange equation $\frac{\partial L}{\partial q_i}-\frac{d}{d\tau}\frac{\partial L}{\partial \dot q_i}-\frac{\alpha-1}{t-\tau}\frac{\partial L}{\partial \dot q_i}=0$, reading the last term as a Rayleigh-type dissipative force. It also positions the Caputo derivative as the fractional derivative whose Laplace transform takes integer-order initial conditions, which is the property that makes it attractive in physical models.

Load-bearing premise

The proof of Eq. (19) depends on the evaluation of the Beta function, Eq. (16), and on the power of $(x-\xi)$ that emerges from the change of variables in Eq. (13); as printed, the denominator in Eq. (16) is $\Gamma(\alpha)+\Gamma(\beta)$ rather than $\Gamma(\alpha+\beta)$, and the exponent is misprinted, so the semigroup law is not established by the derivation given.

Editorial extensions

If this is right

If the semigroup law holds, a fractional integral of order $\alpha+\beta$ can be computed as two successive lower-order integrals, which justifies treating fractional integration as a one-parameter semigroup of operators.
The FALVA Euler–Lagrange equation adds a term $-\frac{\alpha-1}{t-\tau}\frac{\partial L}{\partial \dot q_i}$ to the standard equation, offering a variational description of velocity-dependent dissipative forces.
The Caputo derivative's dependence on integer-order initial conditions makes fractional differential equations easier to cast in physical problems, where initial positions and velocities are usually integer-order data.
The review's scope suggests that the Riemann–Liouville and Caputo toolbox is enough to introduce fractional calculus in an undergraduate physics curriculum, with applications in anomalous diffusion, viscoelasticity, and signal processing.
Applications such as the FALVA-based dark matter model and fractional cosmological equations inherit the formalism's behavior, so the claimed semigroup property underpins the numerical and analytical treatment of those models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The semigroup statement itself is the standard theorem, and the textual errors are typographical, not conceptual.
A corrected derivation of Eq. (19) would go through with the standard Beta identity and the factor $(x-\xi)^{\alpha+\beta-1}$ from the change of variables, so the paper's conclusion is likely right even where its printed proof is not.
Applying the FALVA equation to a damped harmonic oscillator with $L=\frac12 m\dot q^2-\frac12 k q^2$ and comparing its $\alpha\to1$ limit to the Rayleigh-dissipation solution would be a direct numerical test of the variational claim.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The headline is a review, and the review's own proof of the semigroup property is wrong in at least three places. That's the first thing you should know, because the paper's only value is pedagogical. The historical account and the standard definitions are fine—they're taken from the literature, and the FALVA section is a recognizable restatement of El-Nabulsi and Torres—but the derivation in Eqs. (12)–(19) does not work. The exponent in Eq. (13) should be α+β−1, not α−β−1, and the Jacobian is (x−ξ), not (1−s)dξ. Eq. (16) gives the Beta function with a false denominator, Γ(α)+Γ(β), instead of Γ(α+β). With those errors, Eq. (18) cannot equal the right-hand side of Eq. (19). This is an internal contradiction, not a disagreement with consensus. The same kind of exponent mistake appears in Eq. (23), where the kernel of the RL integral is written with n−α+1 instead of n−α−1. For a paper that is explicitly aimed at teaching, these are load-bearing flaws: a student trying to follow the proof will be misled, and the claimed semigroup property is not established by the text. The writing is also rough—grammar, typos, and notation are inconsistent throughout—but that's secondary to the math. What the paper does well is gather the history and the main definitions into one place, and the motivation for teaching fractional calculus in a physics degree is reasonable; the FALVA section, once corrected, could be a useful worked example. But as it stands, the errors are too central and too numerous to overlook. The target audience is an undergraduate or beginning graduate student looking for an entry point. For that reader, the standard textbooks (Samko, Miller and Ross) and the original El-Nabulsi–Torres paper are better and safer options. I would not send this to peer review in its current state. The corrections are mechanical, but the paper needs to be rewritten carefully before it can serve the pedagogical purpose it sets out to fulfill.

Referee Report

4 major / 5 minor

Summary. This manuscript is a pedagogical review of fractional calculus and its applications in physics. It sketches the historical development, presents the Riemann-Liouville fractional integral and derivative and the Caputo derivative, attempts to prove the semigroup property of the RL integral, and introduces the Fractional Actionlike Variational Approach (FALVA). The paper argues that fractional calculus is a useful topic for inclusion in an undergraduate physics curriculum and points to several application areas such as anomalous diffusion, dissipative systems, and cosmology. The central content is expository: no new theorems or experimental results are presented, and the FALVA equation is a re-derivation of the known El-Nabulsi-Torres result.

Significance. If the equations were correct, this could serve as a compact, accessible introduction for physics students. The paper's value, however, rests entirely on the accuracy of the standard formulas it presents. The semigroup property and the FALVA equation are correct results in the literature, but the manuscript's derivations contain objective algebraic errors that invalidate the proofs as written. Because the stated goal is pedagogical implementation, these errors are load-bearing rather than cosmetic. The paper provides no original mathematical contribution and does not develop the cited applications in any detail, so its significance is modest and contingent on a thorough correction of the technical content.

major comments (4)

[Section 2.1, Eqs. (12)-(13)] The change of variables t = ξ + s(x − ξ) has Jacobian dt = (x − ξ) ds, not dt = (1 − s) dξ as stated. The integrand becomes (x − ξ)^{α+β−1}(1 − s)^{α−1} s^{β−1}, so Eq. (13) should carry the exponent α+β−1 rather than α−β−1. This error propagates into Eqs. (14), (17), and (18), making the subsequent derivation of the semigroup property invalid.
[Section 2.1, Eq. (16)] The Beta function identity is misstated as B(α,β) = Γ(α)Γ(β)/(Γ(α)+Γ(β)). The correct denominator is Γ(α+β). Along with the exponent error in Eq. (13), this yields Eq. (18) with kernel (x−ξ)^{α−β−1} and prefactor 1/(Γ(α)+Γ(β)), which is not the Riemann-Liouville integral of order α+β. The semigroup property (19) is true in the literature, but it is not proven by the text as written.
[Section 2.2, Eqs. (23)-(26)] The fractional integral kernel in Eq. (23) is written as (x−t)^{n−α+1} in the denominator, which is equivalent to (x−t)^{α−n−1}; the correct exponent is n−α−1 (or, equivalently, denominator exponent α−n+1). The same typo appears in Eqs. (24) and (25)-(26). Because these formulas define the Riemann-Liouville fractional derivative, the errors are not superficial and must be corrected.
[Section 3, Eqs. (39)-(40)] The integration by parts in Eq. (39) is incorrect: the right-hand side should be −∫_a^t d/dτ[∂L/∂q̇ (t−τ)^{α−1}] δq dτ, not −∫_t^a, and the boundary term vanishes only after explicitly using δq(a)=δq(t)=0. In addition, Eq. (40) introduces the symbol tI_α^a f(t) without definition and changes the integration lower limit from a to 0. While the final Euler-Lagrange equation (41) is a known result, the derivation as presented is internally inconsistent and needs to be reworked.

minor comments (5)

[Abstract and Introduction] The text contains numerous typographical and grammatical errors, including 'Leibiniz,' 'strogly believe,' and 'revisits the unfolds who followed this questions.' These should be corrected throughout.
[Section 2.1, Eq. (6)] In property P.4, the notation aI_α^t f(x)g(x)dx is ambiguous; it should be made clear over which variable the integral acts and how the fractional integral on the right is defined.
[Section 2.2, AC^n definition] The definition of the space AC^n(Ω) is imprecise: the condition 'f^{(n−1)}(x) ∈ AC^n(Ω)' should likely read f^{(n−1)} ∈ AC(Ω) or a similar standard statement.
[Section 2.3, Eq. (34)] The relation between Caputo and Riemann-Liouville derivatives appears to have incorrect index shifts and arguments (e.g., the exponent k−n−α and the argument x are suspicious). Please check against a standard reference such as Samko, Kilbas, and Marichev.
[Section 3, Eq. (38)] The notation δ(aI_α^t)f(t) is confusing; the variation should be applied to the action functional S_α[q](t), and the integrand should display the correct variable dependencies.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected; the derivation chains are self-contained despite algebraic errors.

full rationale

The paper is a pedagogical review with two derivation chains: the semigroup property of the Riemann-Liouville fractional integral (Eq. 19) and the FALVA Euler-Lagrange equation (Eq. 41). Neither reduces to its own inputs by construction. The semigroup proof attempts a direct calculation using Dirichlet's formula and the Beta function; although Eqs. (13) and (16) contain algebraic errors (wrong exponent and wrong Beta-function denominator), these are correctness defects, not circularity, because the claimed identity is not built into the definition of aI^(alpha+beta). The FALVA equation is derived by varying the fractional action (35); the extra term (alpha-1)/(t-tau) arises from differentiating the kernel, and no parameter is fitted or renamed as a prediction. The only author-overlapping citation is [12] (Godinho is a co-author), used in the conclusion as an example application of FALVA to dark matter; it is not load-bearing for Eqs. (19) or (41). Hence no circular step is present.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new entities and fits no data. Its content rests on standard definitions of fractional operators and on the FALVA variational ansatz taken from the cited literature. The axioms listed are the mathematical facts the derivations invoke; some are mis-stated in the text.

free parameters (1)

fractional order α
Introduced as the order of the fractional action integral S_α in Eq. (35) and of the fractional derivatives throughout; left unspecified rather than fitted to data. It is a model parameter in the FALVA formalism, not a fitted constant.

assumptions (5)

domain assumption Riemann-Liouville fractional integral definition (Eq. 2): aI^α_x f(x) = 1/Γ(α) ∫_a^x (x-t)^(α-1) f(t) dt.
The paper takes this as the foundational definition for the fractional integral (Section 2.1).
standard math Euler Beta function identity: B(α,β)=∫_0^1 (1-s)^(α-1)s^(β-1) ds = Γ(α)Γ(β)/Γ(α+β).
Used in the proof of P.3; the paper states a wrong version in Eq. (16), but the intended identity is standard.
standard math Dirichlet/Fubini interchanging of integration order (Eqs. 9-11).
Assumed to hold for the iterated integrals in the semigroup proof.
standard math Fundamental lemma of variational calculus (Section 3).
Used to pass from the vanishing integral Eq. (40) to the Euler-Lagrange equations Eq. (41) for independent generalized coordinates.
domain assumption The fractional action integral (35) with kernel (t-τ)^(α-1) defines a valid action principle (FALVA).
Adopted from El-Nabulsi and Torres [10]; the paper does not justify this variational principle physically.

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Cite this review

Pith. "Pith review of About Fractional Calculus and its Applications in Physics." pith.science (2026). https://pith.science/paper/SDQMN66J

@misc{pith2026250704186,
  author       = {Pith},
  title        = {Pith review of: About Fractional Calculus and its Applications in Physics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SDQMN66J}},
  note         = {Machine review of arXiv:2507.04186}
}

abstract

Historically the fractional calculus concept works an extended idea based on the question asked by Guillaume de L'H\^opital to Gottfried Wilhelm Leibniz in 1695 about the notation ${d^nf}/{dx^n}$ for the derivative operator "What if $n=\frac{1}{2}$ ?" To which Leibiniz replied : "This is an apparent paradox, from which useful consequences will be established". Our work revisits the unfolds who followed this questions with some classical definitions of fractional derivative operators and fractional integral. We still point out possible applications in areas such as Engineering, Physics, among others. Among these definitions we will focus more on the Riemann-Liouville and Caputo definitions, however other definitions are also briefly commented. In this work we begin with a historical inspection of the birth of the fractional calculus, parallels with the differential calculus and some of its developments are traced. Always focusing on the definitions of Riemann-Liouville and Caputo, more commonly found in the bibliography of the area and more frequent in scientific works. Some examples of its operability are presented, such as the direct calculation of constant function derivatives, polynomial function and exponential function. Derivative operators and fractional integrals are defined as derivatives and noninteger-order integrals. Our work revisits these two classical definitions of derivative operators and fractional integral and points out possible applications in areas such as Engineering, Physics, among others. Our main goal is to address the feasibility of implementing this content in a degree form program in Physics, we strogly believe that this theme will aggregate a lot of content mainly because its multidisciplinary character.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Gravitational Foundations and Exact Solutions in $n$-Dimensional Fractional Cosmology
gr-qc 2025-12 conditional novelty 4.0 of 10

A fractional-kernel Sáez–Ballester action in n dimensions yields exact FLRW solutions whose effective equation of state can reproduce all cosmic epochs for hand-chosen values of α and C.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper

[1]

K, and Ross

Miller, S. K, and Ross. B. An Introduction to the Fracional Calculus and Fractional Differential Equa- tions,John Wiley and Sons Inc. New York, (1993)

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G.; Kilbas, A

Samko, S. G.; Kilbas, A. A.; Marichev, O. I. Fractional Integrals and Derivatives theory and applications, Gordon and Breach Science Publishers S.A, (1993)

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S.; Oliveira, D

Teodoro, G. S.; Oliveira, D. S.; Oliveira, E. C.; Sobre derivadas fracion´ arias (on fractional derivatives). Revista Brasileira de Ensino de F ´ ısica (RBEF), no.2, vol.40, (2018)

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El-Nabulsi, R.A. and Torres D.F.M. Fractional actionlike variational problems, J. of Math. Phys. 49, 053521 (2008)

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R.A. El-Nabulsi, C.F. L. Godinho, I.V. Vancea Mod.Phys.Lett.A 39 (2024) 31n32, 2450147

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Micolta-Riascos B., Droguett B. , Marriaga G.M., Leon G. , Paliathanasis A., del Campo L. Leyva Y. Fractional Time-Delayed Differential Equations: Applications in Cosmological Studies, Fractal Fract. 9, 318, (2025) 8

2025

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

K, and Ross

Miller, S. K, and Ross. B. An Introduction to the Fracional Calculus and Fractional Differential Equa- tions,John Wiley and Sons Inc. New York, (1993)

work page 1993

[2] [2]

On differentiation with arbitrary index,Moscow Matem

Sonin N,Y. On differentiation with arbitrary index,Moscow Matem. Sbornik,6(1):1-38, (1869)

work page

[3] [3]

Letnikov A. V. Theory of differentiation with an arbitrary index (Russian), Moscow, Matem. Sbornik, 3:1- 66, (1868)

work page

[4] [4]

V., An explanation of the concepts of the theory of differentiation of arbitrary index (Russian), Moscow Matem

Letnikov A. V., An explanation of the concepts of the theory of differentiation of arbitrary index (Russian), Moscow Matem. Sbornik, 6:413-445,(1872)

work page

[5] [5]

Laurent H., Sur le calcul des derivees a indicies quelconques. Nouv. Annales de Mathematiques, 3(3):240- 252, (1884)

work page

[6] [6]

G.; Kilbas, A

Samko, S. G.; Kilbas, A. A.; Marichev, O. I. Fractional Integrals and Derivatives theory and applications, Gordon and Breach Science Publishers S.A, (1993)

work page 1993

[7] [7]

S.; Oliveira, D

Teodoro, G. S.; Oliveira, D. S.; Oliveira, E. C.; Sobre derivadas fracion´ arias (on fractional derivatives). Revista Brasileira de Ensino de F ´ ısica (RBEF), no.2, vol.40, (2018)

work page 2018

[8] [8]

Teodoro, G. S. ; Oliveira, E. C. Derivadas fracion´ arias: crit´ erios para classifica¸ c˜ ao. Revista Brasileira de Ensino de F ´ ısica (RBEF), v.37, n.3,p. 1-12,(2015)

work page 2015