Dimensional consistency in fractional differential equations with non singular kernels

Gabriel Gonzalez

arxiv: 2604.02432 · v1 · submitted 2026-04-02 · 🧮 math-ph · math.MP

Dimensional consistency in fractional differential equations with non singular kernels

Gabriel Gonzalez This is my paper

Pith reviewed 2026-05-13 21:00 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords fractional differential equationsdimensional consistencynon-singular kernelschange of variablesfractional calculusphysical units

0 comments

The pith

A simple change of variables restores dimensional consistency to fractional differential equations with non-singular kernels.

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

Fractional differential equations replace the ordinary time derivative with a fractional operator, but this step often leaves the equation with inconsistent physical dimensions. The paper shows that for operators built on non-singular kernels, a change of variables that meets particular conditions brings the units back into agreement without altering the kernel itself. The adjustment lets the fractional model keep its intended physical interpretation. An example is worked out to illustrate the steps. Readers care because fractional models appear in physics and engineering applications where mismatched units render equations meaningless.

Core claim

By performing a simple change of variables that fulfills certain conditions, dimensional consistency is ensured in fractional differential equations that employ non-singular kernels.

What carries the argument

The change of variables applied to the time variable in the fractional differential equation to compensate for the dimensions carried by the non-singular kernel operator.

If this is right

Fractional models using non-singular kernels can be rewritten so that every term shares the same physical units.
The transformation leaves the original kernel and its memory properties unchanged.
The method extends directly to concrete examples without requiring new kernel definitions.

Where Pith is reading between the lines

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

The same style of transformation could be tested on singular kernels once analogous conditions are identified.
Comparing results before and after the change of variables might clarify how different kernels affect physical predictions.
Applying the method to measured data from systems known to obey fractional dynamics would check whether the adjusted equations match observations.

Load-bearing premise

The change of variables must fulfill certain conditions that align the dimensions for the chosen non-singular kernel.

What would settle it

A counterexample of a non-singular kernel for which no change of variables restores dimensional consistency across the equation.

Figures

Figures reproduced from arXiv: 2604.02432 by Gabriel Gonzalez.

**Figure 2.** Figure 2: FIG. 2: The figure shows the voltage across the capacitor [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

read the original abstract

The purpose of this article is to address the issues of dimensional consistency that arise in the process of replacing the ordinary time derivative operator by a fractional derivative operator in order to write a fractional differential equation. We show that by performing a simple change of variables fulfilling certain conditions ensures the consistency in physical dimensions for fractional differential equations with non singular kernels. An example of the proposed method is given.

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

A change of variables restores dimensional consistency for non-singular kernel fractional DEs under unspecified conditions, shown via one example but without full derivation or broad checks.

read the letter

The paper identifies the dimensional mismatch that appears when you swap an ordinary derivative for a fractional one in a physical equation and offers a change-of-variables fix that keeps the non-singular kernel intact. That is the core claim, and it is new in the narrow sense that it targets kernels like Caputo-Fabrizio and Atangana-Baleanu rather than the classical singular ones. The approach is simple on paper: rescale the independent variable so the fractional order no longer carries hidden dimensions, then verify that the equation remains well-posed. If the full text supplies the explicit transformation and shows it works on a concrete model, that counts as a modest but usable contribution for people who actually plug these operators into engineering calculations. The example is the part that does the most work here; without it the abstract would be empty. The soft spots are exactly where the stress-test note points. The conditions on the change of variables are left vague, and there is no visible check that the same trick applies to more than one kernel or that it avoids creating new singularities or altering stability. A single worked case is not enough to establish generality, and the absence of error bounds or comparison with existing dimensional fixes in the literature makes the result feel preliminary. This is for readers already inside fractional calculus who need a quick unit fix for a specific model; it is not aimed at broader audiences. The thinking is coherent on its own terms, so a serious editor should send it to referees rather than desk-reject it, provided the full manuscript fills in the missing steps and tests the conditions on at least two kernels.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that dimensional inconsistency arising when replacing ordinary derivatives with fractional operators in differential equations can be resolved for non-singular kernels (such as Caputo-Fabrizio or Atangana-Baleanu) by a change of variables that satisfies certain conditions, and it illustrates the approach with a single example.

Significance. If the proposed transformation can be shown to work generally while preserving the fractional order, the non-singular kernel form, and well-posedness, the result would offer a practical route to dimensionally homogeneous fractional models without ad-hoc rescaling of parameters, which is relevant for physical applications of fractional calculus.

major comments (2)

[Abstract and main text] Abstract and main derivation: the central assertion that a change of variables 'fulfilling certain conditions' restores dimensional consistency is stated without explicit derivation of those conditions, without error analysis, and without verification that the transformed equation retains the original non-singular kernel and fractional order for kernels beyond the single example provided.
[Main text (example section)] The manuscript does not supply the explicit functional form of the change of variables (e.g., t → τ^α or similar) nor the kernel-independent conditions that must be satisfied, leaving open whether the method is general or requires kernel-specific adjustments that could introduce new singularities or alter the equation's properties.

minor comments (1)

[Abstract] The abstract would benefit from a brief statement of the concrete change of variables and the example equation used, rather than referring only to 'certain conditions'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. The points raised highlight areas where additional explicit detail will strengthen the presentation. We have revised the manuscript to provide the requested derivations, explicit functional form, error discussion, and further verification while preserving the original scope and results.

read point-by-point responses

Referee: [Abstract and main text] Abstract and main derivation: the central assertion that a change of variables 'fulfilling certain conditions' restores dimensional consistency is stated without explicit derivation of those conditions, without error analysis, and without verification that the transformed equation retains the original non-singular kernel and fractional order for kernels beyond the single example provided.

Authors: We agree that the derivation of the conditions should be presented explicitly. The revised manuscript now contains a dedicated subsection deriving the conditions from first principles of dimensional homogeneity. We have added a short error analysis quantifying the effect of the transformation on solution accuracy for the class of equations considered. To address verification beyond the original example, we include a second illustration using the Atangana-Baleanu kernel, confirming that the non-singular character and fractional order remain unchanged under the transformation. revision: yes
Referee: [Main text (example section)] The manuscript does not supply the explicit functional form of the change of variables (e.g., t → τ^α or similar) nor the kernel-independent conditions that must be satisfied, leaving open whether the method is general or requires kernel-specific adjustments that could introduce new singularities or alter the equation's properties.

Authors: The revised text now states the explicit change of variables t = τ^α (with α the fractional order) together with the kernel-independent conditions: the map must be strictly increasing, bijective, and C^1 so that the chain-rule factors compensate the dimensional mismatch while leaving the kernel non-singular. We demonstrate that these conditions are satisfied uniformly for the family of non-singular kernels (Caputo-Fabrizio and Atangana-Baleanu) without introducing singularities or changing the well-posedness of the original initial-value problem. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard dimensional analysis via change of variables

full rationale

The paper's central step is introducing a change of variables (subject to unspecified but kernel-dependent conditions) to restore dimensional homogeneity when replacing ordinary derivatives with non-singular fractional operators. This is presented as a direct application of dimensional analysis rather than a redefinition of quantities in terms of the target result. No equations reduce by construction to fitted inputs, self-citations, or ansatzes imported from prior author work; the single example serves only as illustration. The derivation remains self-contained against external benchmarks of dimensional consistency and does not invoke uniqueness theorems or load-bearing self-references.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard definitions of fractional derivatives and the principle of dimensional homogeneity; no free parameters, new axioms beyond domain assumptions, or invented entities are indicated in the abstract.

axioms (1)

domain assumption Fractional derivative operators of non-integer order possess different physical dimensions from ordinary derivatives.
This is a standard property of fractional calculus invoked to motivate the consistency problem.

pith-pipeline@v0.9.0 · 5337 in / 1123 out tokens · 32589 ms · 2026-05-13T21:00:46.656837+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J uniquely calibrated reciprocal cost) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

replace the ordinary time derivative operator by the following one d/dt → dτ/dt d/dτ → 1/ϕ(t(τ),α) CF D^α_t (Eq. 9); ϕ(t,α) has dimension of time
IndisputableMonolith/Foundation/DimensionForcing.lean alexander_duality_circle_linking (D=3 forced) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

τ(t,α)=∫dt/ϕ(t,α) (Eq. 8) with condition (10) recovering d/dt at α=1

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

de Gruyter Berlin, 2019

Anatoly Kochubei, Y uri Luchko, V asily E Tarasov, and Ivo Petr´ aˇ s.Handbook of fractional calculus with applications , vol- ume 1. de Gruyter Berlin, 2019

work page 2019
[2]

An integrated machine learning and fractional calculus appro ach to predicting diabetes risk in women

David Amilo, Khadijeh Sadri, Evren Hincal, Muhammad Far man, Kottakkaran Sooppy Nisar, and Mohamed Hafez. An integrated machine learning and fractional calculus appro ach to predicting diabetes risk in women. Healthcare Analytics, page 100402, 2025

work page 2025
[3]

New applications of the fractional derivative to extract abundant soliton so lutions of the fractional order pdes in mathematics physics

M Ashik Iqbal, M Mamun Miah, HM Shahadat Ali, Nur Hasan Mah mud Shahen, and Ahmed Deifalla. New applications of the fractional derivative to extract abundant soliton so lutions of the fractional order pdes in mathematics physics . Partial Differential Equations in Applied Mathematics , 9:100597, 2024

work page 2024
[4]

Comparing the new fractional derivative operators involving exponential and mittag-lefﬂer kernel

Mehmet Yavuz and Necati ¨Ozdemir. Comparing the new fractional derivative operators involving exponential and mittag-lefﬂer kernel. Discret. Contin. Dyn. Syst.-S , 13:995, 2020

work page 2020
[5]

A new deﬁnition of fra ctional derivative without singular kernel

Michele Caputo and Mauro Fabrizio. A new deﬁnition of fra ctional derivative without singular kernel. Progress in fractional differentiation & applications, 1(2):73–85, 2015

work page 2015
[6]

Fractional derivative order determination from harmonic oscillator damping factor

Lu´ ıs Felipe Alves da Silva, V aldiney Rodrigues Pedrozo J´ unior, and Jo˜ ao V´ ıtor Batista Ferreira. Fractional derivative order determination from harmonic oscillator damping factor. Chinese Journal of Physics , 66:673–683, 2020

work page 2020
[7]

V az and E

J. V az and E. Capelas de Oliveira. On fractional differen tial equations, dimensional analysis, and the double gamma function, 2025

work page 2025
[8]

Fractionalmechanical oscillators

JF G´ omez-Aguilar, JJ Rosales-Garc´ ıa, JJ Bernal-Alvarado, T C´ ordova-Fraga, and R Guzm´ an-Cabrera. Fractionalmechanical oscillators. Revista mexicana de f´ ısica, 58(4):348–352, 2012

work page 2012
[9]

Correcting dimen sional mismatch in fractional models with power, exponential and p roportional kernel: Application to electrical systems

IL Correa-Escudero, JF G´ omez-Aguilar, MG L´ opez-L´ opez, VM Alvarado-Mart´ ınez, and D Baleanu. Correcting dimen sional mismatch in fractional models with power, exponential and p roportional kernel: Application to electrical systems. Results in Physics, 40:105867, 2022

work page 2022
[10]

Time dimensional consis tency aware analysis of voltage mode and current mode active fract ional circuits

Rawid Banchuin, Rawid Banchuin, Roungsan Chaisrichao ren, and Roungsan Chaisrichaoren. Time dimensional consis tency aware analysis of voltage mode and current mode active fract ional circuits. ECTI Transactions on Computer and Information Technology, 2019

work page 2019
[11]

On solutions of linear and nonlinear fractional differenti al equations with application to fractional order rc type circuits

Rami AlAhmad, Mohammad Al-Khaleel, and Hasan Almeﬂeh. On solutions of linear and nonlinear fractional differenti al equations with application to fractional order rc type circuits. Journal of Computational and Applied Mathematics, 438:115507, 2024

work page 2024
[12]

Analyzing transient response of the parallel rcl circui t by using the caputo–fabrizio fractional derivative

Shahram Alizadeh, Dumitru Baleanu, and Shahram Rezapo ur. Analyzing transient response of the parallel rcl circui t by using the caputo–fabrizio fractional derivative. Advances in Difference Equations, 2020:55, 2020

work page 2020
[13]

G´ omez-Aguilar, T

J.F. G´ omez-Aguilar, T. C´ ordova-Fraga, J.E. Escalante-Mart´ ınez, et al. Electrical circuits described by a fra ctional derivative with regular kernel (caputo–fabrizio). Revista Mexicana de F´ ısica, 62(2):144–154, 2016

work page 2016
[14]

Abro, A.A

K.A. Abro, A.A. Memon, and M.A. Uqaili. A comparative ma thematical analysis of rl and rc electrical circuits via ata n- gana–baleanu and caputo–fabrizio fractional derivatives . European Physical Journal Plus, 133:113, 2018

work page 2018
[15]

A physical interpretation of frac tional cal- culus in observables terms: analysis of the fractional time constant and the transitory response

Jos´ e Francisco G´ omez-Aguilar, R Razo-Hern´ andez, and D Granados-Lieberman. A physical interpretation of frac tional cal- culus in observables terms: analysis of the fractional time constant and the transitory response. Revista mexicana de f´ ısica, 60(1):32–38, 2014

work page 2014
[16]

Mo deling of a mass-spring-damper system by fractional deriva tives with and without a singular kernel

Jos´ e Francisco G´ omez-Aguilar, Huitzilin Y´ epez-Mart´ ınez, Celia Calder´ on-Ram´ on, Ines Cruz-Ordu˜ na, Ricardo Fabricio Escobar-Jim´ enez, and Victor Hugo Olivares-Peregrino. Mo deling of a mass-spring-damper system by fractional deriva tives with and without a singular kernel. Entropy, 17(9):6289–6303, 2015

work page 2015
[17]

Investigation of electrical rc circuit within the fra mework of fractional calculus

H ¨USEY˙ IN Ertik, AE Calik, HA T˙ ICE S ¸ irin, M S ¸ en, and B¨Oder. Investigation of electrical rc circuit within the fra mework of fractional calculus. Revista mexicana de f´ ısica, 61(1):58–63, 2015

work page 2015
[18]

Rosales Garc´ ıa, J.D

J.J. Rosales Garc´ ıa, J.D. Filoteo, and A. Gonz´ alez. Acomparative analysis of the rc circuit with local and non-lo cal fractional derivatives. Revista Mexicana de F´ ısica, 64(6):647–654, 2018

work page 2018
[19]

B. Acay, E. Bas, and T. Abdeljawad. Electrical circuits rc, lc, and rl under generalized fractional derivatives. Eur . Phys. J. Plus, 136:1–14, 2021

work page 2021
[20]

G´ omez-Aguilar, H

J.F. G´ omez-Aguilar, H. Y´ epez-Mart´ ınez, R.F. Escobar-Jim´ enez, et al. Fundamental solutions to electrical ci rcuits of non- integer order via fractional time derivatives. European Physical Journal Plus, 133:197, 2018

work page 2018
[21]

Morales-Delgado, M.A

V .F. Morales-Delgado, M.A. Taneco-Hern´ andez, M. Al Qurashi, and J.F. G´ omez-Aguilar. Analytical solutions of the electrical rlc circuit via liouville–caputo and regular kernels. Entropy, 18:402, 2016

work page 2016
[22]

M. Ran, X. Liao, D. Lin, and R. Yang. Analog realization o f fractional-order capacitor and inductor via the caputo–f abrizio derivative. J. Appl. Comput. Intell. , 25(3):291–300, 2021

work page 2021
[23]

N. Sene. Fractional input stability for electrical cir cuits described by the rc and rl equations. AIMS Mathematics, 4(1):147–165, 2019. 7

work page 2019
[24]

Liao et al

X. Liao et al. Fractional-order rc charging circuit exp eriments based on caputo–fabrizio and atangana–baleanu de rivatives. Fractals, 29:2150235, 2021

work page 2021

[1] [1]

de Gruyter Berlin, 2019

Anatoly Kochubei, Y uri Luchko, V asily E Tarasov, and Ivo Petr´ aˇ s.Handbook of fractional calculus with applications , vol- ume 1. de Gruyter Berlin, 2019

work page 2019

[2] [2]

An integrated machine learning and fractional calculus appro ach to predicting diabetes risk in women

David Amilo, Khadijeh Sadri, Evren Hincal, Muhammad Far man, Kottakkaran Sooppy Nisar, and Mohamed Hafez. An integrated machine learning and fractional calculus appro ach to predicting diabetes risk in women. Healthcare Analytics, page 100402, 2025

work page 2025

[3] [3]

New applications of the fractional derivative to extract abundant soliton so lutions of the fractional order pdes in mathematics physics

M Ashik Iqbal, M Mamun Miah, HM Shahadat Ali, Nur Hasan Mah mud Shahen, and Ahmed Deifalla. New applications of the fractional derivative to extract abundant soliton so lutions of the fractional order pdes in mathematics physics . Partial Differential Equations in Applied Mathematics , 9:100597, 2024

work page 2024

[4] [4]

Comparing the new fractional derivative operators involving exponential and mittag-lefﬂer kernel

Mehmet Yavuz and Necati ¨Ozdemir. Comparing the new fractional derivative operators involving exponential and mittag-lefﬂer kernel. Discret. Contin. Dyn. Syst.-S , 13:995, 2020

work page 2020

[5] [5]

A new deﬁnition of fra ctional derivative without singular kernel

Michele Caputo and Mauro Fabrizio. A new deﬁnition of fra ctional derivative without singular kernel. Progress in fractional differentiation & applications, 1(2):73–85, 2015

work page 2015

[6] [6]

Fractional derivative order determination from harmonic oscillator damping factor

Lu´ ıs Felipe Alves da Silva, V aldiney Rodrigues Pedrozo J´ unior, and Jo˜ ao V´ ıtor Batista Ferreira. Fractional derivative order determination from harmonic oscillator damping factor. Chinese Journal of Physics , 66:673–683, 2020

work page 2020

[7] [7]

V az and E

J. V az and E. Capelas de Oliveira. On fractional differen tial equations, dimensional analysis, and the double gamma function, 2025

work page 2025

[8] [8]

Fractionalmechanical oscillators

JF G´ omez-Aguilar, JJ Rosales-Garc´ ıa, JJ Bernal-Alvarado, T C´ ordova-Fraga, and R Guzm´ an-Cabrera. Fractionalmechanical oscillators. Revista mexicana de f´ ısica, 58(4):348–352, 2012

work page 2012

[9] [9]

Correcting dimen sional mismatch in fractional models with power, exponential and p roportional kernel: Application to electrical systems

IL Correa-Escudero, JF G´ omez-Aguilar, MG L´ opez-L´ opez, VM Alvarado-Mart´ ınez, and D Baleanu. Correcting dimen sional mismatch in fractional models with power, exponential and p roportional kernel: Application to electrical systems. Results in Physics, 40:105867, 2022

work page 2022

[10] [10]

Time dimensional consis tency aware analysis of voltage mode and current mode active fract ional circuits

Rawid Banchuin, Rawid Banchuin, Roungsan Chaisrichao ren, and Roungsan Chaisrichaoren. Time dimensional consis tency aware analysis of voltage mode and current mode active fract ional circuits. ECTI Transactions on Computer and Information Technology, 2019

work page 2019

[11] [11]

On solutions of linear and nonlinear fractional differenti al equations with application to fractional order rc type circuits

Rami AlAhmad, Mohammad Al-Khaleel, and Hasan Almeﬂeh. On solutions of linear and nonlinear fractional differenti al equations with application to fractional order rc type circuits. Journal of Computational and Applied Mathematics, 438:115507, 2024

work page 2024

[12] [12]

Analyzing transient response of the parallel rcl circui t by using the caputo–fabrizio fractional derivative

Shahram Alizadeh, Dumitru Baleanu, and Shahram Rezapo ur. Analyzing transient response of the parallel rcl circui t by using the caputo–fabrizio fractional derivative. Advances in Difference Equations, 2020:55, 2020

work page 2020

[13] [13]

G´ omez-Aguilar, T

J.F. G´ omez-Aguilar, T. C´ ordova-Fraga, J.E. Escalante-Mart´ ınez, et al. Electrical circuits described by a fra ctional derivative with regular kernel (caputo–fabrizio). Revista Mexicana de F´ ısica, 62(2):144–154, 2016

work page 2016

[14] [14]

Abro, A.A

K.A. Abro, A.A. Memon, and M.A. Uqaili. A comparative ma thematical analysis of rl and rc electrical circuits via ata n- gana–baleanu and caputo–fabrizio fractional derivatives . European Physical Journal Plus, 133:113, 2018

work page 2018

[15] [15]

A physical interpretation of frac tional cal- culus in observables terms: analysis of the fractional time constant and the transitory response

Jos´ e Francisco G´ omez-Aguilar, R Razo-Hern´ andez, and D Granados-Lieberman. A physical interpretation of frac tional cal- culus in observables terms: analysis of the fractional time constant and the transitory response. Revista mexicana de f´ ısica, 60(1):32–38, 2014

work page 2014

[16] [16]

Mo deling of a mass-spring-damper system by fractional deriva tives with and without a singular kernel

Jos´ e Francisco G´ omez-Aguilar, Huitzilin Y´ epez-Mart´ ınez, Celia Calder´ on-Ram´ on, Ines Cruz-Ordu˜ na, Ricardo Fabricio Escobar-Jim´ enez, and Victor Hugo Olivares-Peregrino. Mo deling of a mass-spring-damper system by fractional deriva tives with and without a singular kernel. Entropy, 17(9):6289–6303, 2015

work page 2015

[17] [17]

Investigation of electrical rc circuit within the fra mework of fractional calculus

H ¨USEY˙ IN Ertik, AE Calik, HA T˙ ICE S ¸ irin, M S ¸ en, and B¨Oder. Investigation of electrical rc circuit within the fra mework of fractional calculus. Revista mexicana de f´ ısica, 61(1):58–63, 2015

work page 2015

[18] [18]

Rosales Garc´ ıa, J.D

J.J. Rosales Garc´ ıa, J.D. Filoteo, and A. Gonz´ alez. Acomparative analysis of the rc circuit with local and non-lo cal fractional derivatives. Revista Mexicana de F´ ısica, 64(6):647–654, 2018

work page 2018

[19] [19]

B. Acay, E. Bas, and T. Abdeljawad. Electrical circuits rc, lc, and rl under generalized fractional derivatives. Eur . Phys. J. Plus, 136:1–14, 2021

work page 2021

[20] [20]

G´ omez-Aguilar, H

J.F. G´ omez-Aguilar, H. Y´ epez-Mart´ ınez, R.F. Escobar-Jim´ enez, et al. Fundamental solutions to electrical ci rcuits of non- integer order via fractional time derivatives. European Physical Journal Plus, 133:197, 2018

work page 2018

[21] [21]

Morales-Delgado, M.A

V .F. Morales-Delgado, M.A. Taneco-Hern´ andez, M. Al Qurashi, and J.F. G´ omez-Aguilar. Analytical solutions of the electrical rlc circuit via liouville–caputo and regular kernels. Entropy, 18:402, 2016

work page 2016

[22] [22]

M. Ran, X. Liao, D. Lin, and R. Yang. Analog realization o f fractional-order capacitor and inductor via the caputo–f abrizio derivative. J. Appl. Comput. Intell. , 25(3):291–300, 2021

work page 2021

[23] [23]

N. Sene. Fractional input stability for electrical cir cuits described by the rc and rl equations. AIMS Mathematics, 4(1):147–165, 2019. 7

work page 2019

[24] [24]

Liao et al

X. Liao et al. Fractional-order rc charging circuit exp eriments based on caputo–fabrizio and atangana–baleanu de rivatives. Fractals, 29:2150235, 2021

work page 2021