Fractional behaviors analysis and modelling using DRT

Jocelyn Sabatier; Hamida Hallil; Jocelyn Sabatier; Hamida Hallil

doi:10.3934/math.20261063

AIMS Mathematics

2026, Volume 11, Issue 8: 26503-26537. doi: 10.3934/math.20261063

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Fractional behaviors analysis and modelling using DRT

Jocelyn Sabatier ^,,
Hamida Hallil

IMS laboratory, UMR 5218, University of Bordeaux, CNRS, Bordeaux INP, Talence 33400, France

Received: 17 May 2026 Revised: 10 July 2026 Accepted: 05 August 2026 Published: 25 August 2026
MSC : 93B30, 93A30, 26A33

In this paper, we revisit the analysis and modelling of dynamical systems exhibiting apparent fractional behaviors using the framework of Distribution of Relaxation Times (DRT). We establish a formal mathematical equivalence between the DRT framework and diffusive representation. Moreover, we introduce a direct time-domain identification algorithm that extracts physically interpretable log-normal continuous distributions directly from raw input-output measurements, bypassing noisy frequency transforms. Through multiple engineering case studies (solar panels, epidemiological data, and tight sandstones), it is demonstrated, using the DRT concepts, that apparent fractional behaviors can often be accurately captured by compact, physically sound integer-order distributions, providing a robust structural diagnostic tool for fractional-order modelling.
- fractional behaviors,
- distribution of relaxation time,
- diffusive representation,
- fractional models,
- fractional calculus
Citation: Jocelyn Sabatier, Hamida Hallil. Fractional behaviors analysis and modelling using DRT[J]. AIMS Mathematics, 2026, 11(8): 26503-26537. doi: 10.3934/math.20261063

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Abstract

In this paper, we revisit the analysis and modelling of dynamical systems exhibiting apparent fractional behaviors using the framework of Distribution of Relaxation Times (DRT). We establish a formal mathematical equivalence between the DRT framework and diffusive representation. Moreover, we introduce a direct time-domain identification algorithm that extracts physically interpretable log-normal continuous distributions directly from raw input-output measurements, bypassing noisy frequency transforms. Through multiple engineering case studies (solar panels, epidemiological data, and tight sandstones), it is demonstrated, using the DRT concepts, that apparent fractional behaviors can often be accurately captured by compact, physically sound integer-order distributions, providing a robust structural diagnostic tool for fractional-order modelling.

References

[1]	M. Wiśniewski, J. Łuczka, J. Spiechowicz, Memory corrections to Markovian Langevin dynamics, Entropy, 26 (2024), 425. https://doi.org/10.3390/e26050425 doi: 10.3390/e26050425
[2]	S. Brandani, S. Hwang, J. Kärger, E. Mangano, Diffusion anomaly in nanopores as a rich field for theorists and a challenge for experimentalists, Nature Commun. , 15 (2024), 5721. https://doi.org/10.1038/s41467-024-49821-w doi: 10.1038/s41467-024-49821-w
[3]	A. Klimek, B. A. Dalton, R. R. Netz, Subdiffusion from competition between multi-exponential friction memory and energy barriers, Eu. r Phys. J. E, 48 (2025). https://doi.org/10.1140/epje/s10189-025-00518-y
[4]	I. Podlubny, Geometric and physical Interpretation of fractional Integration and fractional differentiation, Fract. Calc. Appl. Anal. , 5 (2002), 367-386. https://doi.org/10.48550/arXiv.math/0110241 doi: 10.48550/arXiv.math/0110241
[5]	V. Tarasov, Geometric interpretation of fractional-order derivative, Fract. Calc. Appl. Anal. , 19 (2016), 1200-1221. https://doi.org/10.1515/fca-2016-0062 doi: 10.1515/fca-2016-0062
[6]	J. Sabatier, C. Farges, V. Tartaglione, Fractional behaviours modeling: Analysis and application of several unusual tools, Intelligent Systems, Control and Automation: Science and Engineering series, Cham, Switzerland: Springer, 101 (2022). https://doi.org/10.1007/978-3-030-96749-9
[7]	A. M. Balint, S. Balint, Mathematical description of the groundwater flow and that of the impurity spread, which use temporal Caputo or Riemann-Liouville fractional partial derivatives, is non-objective, Frac. Fract. , 4 (2020), 36. https://doi.org/10.3390/fractalfract4030036 doi: 10.3390/fractalfract4030036
[8]	A. Dokoumetzidis, R. Magin, P. A. Macheras, Commentary on fractionalization of multi-compartmental models, J. Pharmaconet. Phar. , 37 (2010), 203-207. https://doi.org/10.1007/s10928-010-9153-5 doi: 10.1007/s10928-010-9153-5
[9]	Pantokratoras, Comment on the paper "Fractional order model of thermo-solutal and magnetic nanoparticles transport for drug delivery applications, Subrata Maiti, Sachin Shaw, G. C. Shit, Colloid. Surface. B, 222 (2023), 113074. https://doi.org/10.1016/j.colsurfb.2022.113074 doi: 10.1016/j.colsurfb.2022.113074
[10]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Different. Appl. , 1 (2015), 73-85. https://doi.org/10.12785/pfda/010201 doi: 10.12785/pfda/010201
[11]	A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Therm. Sci. , 20 (2016), 763-769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
[12]	J. Hristov, Derivatives with non-singular Kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models, In: Frontiers in Fractional Calculus, Chapter 10, edited by Sachin Bhalekar, Bentham Science Publishers, 2017. https://doi.org/10.2174/9781681085999118010013
[13]	J. Sabatier, M. Merveillaut, R. Malti, A. Oustaloup, How to impose physically coherent initial conditions to a fractional system? Commun. Nonlinear Sci. Numer. Simul., 15 (2010), 1318-1326. https://doi.org/10.1016/j.cnsns.2009.05.070
[14]	M. D. Ortigueira, F. Coito, System initial conditions vs derivative initial conditions, Comput. Math. Appl. , 59 (2010), 1782-1789. https://doi.org/10.1016/j.camwa.2009.08.036 doi: 10.1016/j.camwa.2009.08.036
[15]	J. Sabatier, Fractional order models are doubly infinite dimensional models and thus of infinite memory: Consequences on initialization and some solutions, Symmetry, 13 (2021), 1099. https://doi.org/10.3390/sym13061099 doi: 10.3390/sym13061099
[16]	A. V. Milovanov, K. Rypdal, J. J. Rasmussen, Stretched exponential relaxation and ac universality in disordered dielectrics, Phys Rev B, 76 (2007), 104201. https://doi.org/10.1103/PhysRevB.76.104201 doi: 10.1103/PhysRevB.76.104201
[17]	E. C. de Oliveira, J. A. T. Machado, A review of definitions for fractional derivatives and integral, Math. Probl. Eng. , 2014 (2014), 238459. https://doi.org/10.1155/2014/238459 doi: 10.1155/2014/238459
[18]	C. Plank, T. Rüther, L. Jahn, M. Schamel, J. P. Schmidt, F. Ciucci, et al., A review on the distribution of relaxation times analysis: A powerful tool for process identification of electrochemical systems, J. Power Sources, 594 (2024), 233845. https://doi.org/10.1016/j.jpowsour.2023.233845 doi: 10.1016/j.jpowsour.2023.233845
[19]	G. Montseny, Diffusive representation of pseudo-differential time-operators, ESAIM Pro. Surveys, 5 (1998), 159-175. https://doi.org/10.1051/proc:1998005 doi: 10.1051/proc:1998005
[20]	G. Montseny, Representation diffusive, Hermès Science, 2005.
[21]	M. Saccoccio, T. H. Wan, C. Chen, F. Ciucci, Optimal regularization in distribution of relaxation times applied to electrochemical impedance spectroscopy: Ridge and Lasso regression methods-A theoretical and experimental study, Electrochimica Acta, 147 (2014), 470-482. https://doi.org/10.1016/j.electacta.2014.09.058 doi: 10.1016/j.electacta.2014.09.058
[22]	T. H. Wan, M. Saccoccio, C. Chen, F. Ciucci, Influence of the discretization as well as the linear and nonlinear regularization on the distribution of relaxation times, Electrochimica Acta, 184 (2015), 483-499. https://doi.org/10.1016/J.ELECTACTA.2015.09.097 doi: 10.1016/J.ELECTACTA.2015.09.097
[23]	E. H. Balaguera, A full capacitive picture of the constant phase element: Estimations from impedance spectroscopy, distribution of relaxation times and fractional calculus, Electrochem. Commun. , 180 (2025), 108053, https://doi.org/10.1016/j.elecom.2025.108053 doi: 10.1016/j.elecom.2025.108053
[24]	A. Allagui, A. S. Elwakil, Procedure for obtaining the analytical distribution function of relaxation times for the analysis of impedance spectra using the fox H-function, J. Phys. Chem. C, 128 (2024), 2788-2795. https://doi.org/10.1021/acs.jpcc.3c08411 doi: 10.1021/acs.jpcc.3c08411
[25]	M. D. Ortigueira, V. Martynyuk, V. Kosenkov, A. G. Batista, A new look at the capacitor theory, Fractal Fract. , 7 (2023), 86. https://doi.org/10.3390/fractalfract7010086 doi: 10.3390/fractalfract7010086
[26]	M. D. Ortigueira, Fractional calculus for fractional signals and systems, Springer Science and Business Media, 2011. https://doi.org/10.1007/978-94-007-0747-4
[27]	W. A. Yager, The distribution of relaxation times in typical dielectrics, J. Appl. Phys. , 7 (1936), 434-450. https://doi.org/10.1063/1.1745355 doi: 10.1063/1.1745355
[28]	K. S. Cole, R. H. Cole, Dispersion and absorption in dielectrics I. Alternating current characteristics, J. Chem. Phys. , 9 (1941), 341-351. https://doi.org/10.1063/1.1750906 doi: 10.1063/1.1750906
[29]	R. M. Fuoss, J. G. Kirkwood, Electrical properties of solids VIII. Dipole moments in polyvinyl Chloride-Diphenyl systems, J. Am. Chem. Soc. , 63 (1941), 385-394. https://doi.org/10.1021/ja01847a013 doi: 10.1021/ja01847a013
[30]	P. Debye, Polar molecules, New York: The Chemical Catalogue Company, 1929. https://doi.org/10.1002/jctb.5000484320
[31]	K. T. Malkow, Analytical inversion of complex-valued DRT in impedance spectroscopy by integral transforms, Electrochimica Acta, 535 (2025), 146500. https://doi.org/10.1016/j.electacta.2025.146500 doi: 10.1016/j.electacta.2025.146500
[32]	J. R. Macdonald, Impedance spectroscopy: Old problems and new developments, Electrochimica Acta, 35 (1990), 1483-1492. https://doi.org/10.1016/0013-4686(90)80002-6 doi: 10.1016/0013-4686(90)80002-6
[33]	E. Tuncer, J. R. Macdonald, Comparison of methods for estimating continuous distributions of relaxation times, J. Appl. Phys. , 99 (2006), 074106. https://doi.org/10.1063/1.2188053 doi: 10.1063/1.2188053
[34]	T. H. Wan, M. Saccoccio, C. Chen, F. Ciucci, Influence of the discretization methods on the distribution of relaxation times deconvolution: Implementing radial basis functions with Tikhonov regularization, Electrochimica Acta, 184 (2015), 483-499. https://doi.org/10.1016/j.electacta.2015.09.097 doi: 10.1016/j.electacta.2015.09.097
[35]	R. Ebead, B. Abo-Zalam, E. Nabil, System identification of photovoltaic system based on fractional-order model, J. Comput. Electron. , 22 (2023), 471-484. https://doi.org/10.1007/s10825-022-01996-5 doi: 10.1007/s10825-022-01996-5
[36]	Y. Chen, F. Liu, Q. Yu, T. Li, Review of fractional epidemic models, Appl. Math. Model. , 97 (2021), 281-307. https://doi.org/10.1016/j.apm.2021.03.044 doi: 10.1016/j.apm.2021.03.044
[37]	P. A. Naik, M. Farman, K. Jamil, K. S. Nisar, M. A. Hashmi, Z. Huang, Modeling and analysis using piecewise hybrid fractional operator in time scale measure for ebola virus epidemics under Mittag-Leffler kernel, Sci. Rep. , 14 (2024), 24963. https://doi.org/10.1038/s41598-024-75644-2 doi: 10.1038/s41598-024-75644-2
[38]	A. Oustaloup, F. Levron, S. Victor, L. Dugard, Non-integer (or fractional) power model to represent the complexity of a viral spreading: Application to the COVID-19, Annu. Rev. Control, 52 (2021), 523-542. https://doi.org/10.1016/j.arcontrol.2021.09.003 doi: 10.1016/j.arcontrol.2021.09.003
[39]	A. Oustaloup, F. Levron, S. Victor, L. Dugard, From the FPM model in viral spreading to predictions of CO₂ concentration in the atmosphere, Nonlinear Dynam. , 114 (2026), 17. https://doi.org/10.1007/s11071-025-12033-y doi: 10.1007/s11071-025-12033-y
[40]	A. Oustaloup, F. Levron, S. Victor, L. Dugard, Addendum: Predictive form of the FPM model, Annu. Rev. Control, 55 (2023), 291-296. https://doi.org/10.1016/j.arcontrol.2023.02.001 doi: 10.1016/j.arcontrol.2023.02.001
[41]	X. Ding, G. Zhang, B. Zhao, Y. Wang, Unexpected viscoelastic deformation of tight sandstone: Insights and predictions from the fractional Maxwell model, Sci. Rep. , 7 (2017), 11336. https://doi.org/10.1038/s41598-017-11618-x doi: 10.1038/s41598-017-11618-x
[42]	A. Allagui, A. Elwakil, Generalized distribution function of relaxation times with the Davidson-Cole model as a Kernel, Adv. Theor. Simul. , 8 (2025), e00792. https://doi.org/10.1002/adts.202500792 doi: 10.1002/adts.202500792
[43]	I. Nikolaou, H. Hallil, V. Conédéra, B. Plano, O. Tamarin, J. L. Lachaud, et al., Electro-mechanical properties of inkjet-printed graphene oxide nanosheets, Phys. Status Solidi A, 214 (2017), 1600492. https://doi.org/10.1002/pssa.201600492 doi: 10.1002/pssa.201600492
[44]	I. Nikolaou, H. Hallil, V. Conédéra, G. Deligeorgis, C. Dejous, D. Rebiere, Inkjet-printed graphene oxide thin layers on love wave devices for humidity and vapor detection, IEEE Sens. J. , 16 (2016), 7620-7627. https://doi.org/10.1109/JSEN.2016.2600269 doi: 10.1109/JSEN.2016.2600269
[45]	J. Sabatier, Probabilistic interpretations of fractional operators and fractional behaviours: Extensions, applications and tribute to Prof. José Tenreiro Machado's ideas, Mathematics, 10 (2022), 4184. https://doi.org/10.3390/math10224184 doi: 10.3390/math10224184
[46]	E. Limpert, W. A. Stahel, M. Abbt, Log-normal Distributions across the sciences: Keys and clues: On the charms of statistics, and how mechanical models resembling gambling machines offer a link to a handy way to characterize log-normal distributions, which can provide deeper insight into variability and probability—normal or log-normal: That is the question, BioScience, 51 (2001), 341-352. https://doi.org/10.1641/0006-3568(2001)051[0341: LNDATS]2.0.CO; 2
[47]	R. Leonhardt, J. K. von Nidda, D. Andrae, A. Schmidt, J. Kowal, T. Tichter, Reconstructing the distribution of relaxation times with analytical basis functions, J. Power Sources, 652 (2025), 237403. https://doi.org/10.1016/j.jpowsour.2025.237403 doi: 10.1016/j.jpowsour.2025.237403
[48]	J. Sabatier, C. Farges, J. C. Trigeassou, Fractional systems state space description: Some wrong ideas and proposed solutions, J. Vib. Control, 20 (2014), 1076-1084. https://doi.org/10.1177/1077546313481839 doi: 10.1177/1077546313481839
[49]	D. Zwillinger, V. Moll, Table of integrals, series, and products, Academic Press, 2014, https://doi.org/10.1016/c2013-0-10754-4
[50]	A. Erdelyi, Tables of integral transforms, McGraw-Hill, 1 (1954).
[51]	M. Abramowitz, I. Stegun, Handbook of mathematical functions, graphs, and mathematical tables, National Bureau of Standards, 1964.

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