R. S. Rutman, “On physical interpretations of fractional integration and differentiation”, TMF, 105:3 (1995), 393–404; Theoret. and Math. Phys., 105:3 (1995), 1509

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Teoreticheskaya i Matematicheskaya Fizika, 1995, Volume 105, Number 3, Pages 393–404 (Mi tmf1383)

This article is cited in 67 scientific papers (total in 67 papers)

On physical interpretations of fractional integration and differentiation

R. S. Rutman

University of Massachusetts Dartmouth

Full-text PDF (1365 kB) Citations (67)

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Abstract: Is there a relation between fractional calculus and fractal geometry? Can a fractional order system be represented by a causal dynamical model? These are the questions recently debated in the scientific community. The author intends to answer to these questions. In the first part of the paper, some recently suggested models are reviewed and no convincing evidence is found for any dynamical model of a fractional order system having been built with the help of fractals. Linear filters with constant lumped parameters have a very limited use as approximations of fractional order systems. The model suggested in the paper is a state-space representation with parameters as functions of the independent variable. Regularization of fractional differentiation is considered and asymptotic error estimates, as well as simulation results, are presented.

Received: 19.12.1994

English version:
Theoretical and Mathematical Physics, 1995, Volume 105, Issue 3, Pages 1509–1519
DOI: https://doi.org/10.1007/BF02070871

Bibliographic databases:

Language: Russian

Citation: R. S. Rutman, “On physical interpretations of fractional integration and differentiation”, TMF, 105:3 (1995), 393–404; Theoret. and Math. Phys., 105:3 (1995), 1509–1519

Citation in format AMSBIB

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https://www.mathnet.ru/eng/tmf1383

https://www.mathnet.ru/eng/tmf/v105/i3/p393

This publication is cited in the following 67 articles:

Oleg Marichev, Elina Shishkina, “Overview of fractional calculus and its computer implementation in Wolfram Mathematica”, Fract Calc Appl Anal, 2024
Helen Wilson, Sarthok Sircar, Priyanka Shukla, Fluid Mechanics and Its Applications, 138, Viscoelastic Subdiffusive Flows, 2024, 125
Vasily E. Tarasov, “General Fractional Calculus in Multi-Dimensional Space: Riesz Form”, Mathematics, 11:7 (2023), 1651
S Vitali, P Paradisi, G Pagnini, “Anomalous diffusion originated by two Markovian hopping-trap mechanisms”, J. Phys. A: Math. Theor., 55:22 (2022), 224012
Jocelyn Sabatier, “Probabilistic Interpretations of Fractional Operators and Fractional Behaviours: Extensions, Applications and Tribute to Prof. José Tenreiro Machado's Ideas”, Mathematics, 10:22 (2022), 4184
Jocelyn Sabatier, Lecture Notes in Networks and Systems, 452, Proceedings of the International Conference on Fractional Differentiation and its Applications (ICFDA'21), 2022, 74
DAZHI ZHAO, YAN YU, LIANG PU, “CUT-OFF DISTRIBUTIONS AND PROBABILISTIC INTERPRETATIONS OF THE GENERAL FRACTIONAL DERIVATIVES WITH MEMORY EFFECT”, Fractals, 30:04 (2022)
Jocelyn Sabatier, Lecture Notes in Networks and Systems, 452, Proceedings of the International Conference on Fractional Differentiation and its Applications (ICFDA'21), 2022, 1
Jocelyn Sabatier, Fractional Order Systems, 2022, 551
Xing Hu, Yongkun Li, “Left Riemann–Liouville Fractional Sobolev Space on Time Scales and Its Application to a Fractional Boundary Value Problem on Time Scales”, Fractal Fract, 6:5 (2022), 268 $https://doi.org/10.3390/fractalfract6050268$
Ahmed S. Hendy, T.R. Taha, D. Suragan, Mahmoud A. Zaky, “An energy-preserving computational approach for the semilinear space fractional damped Klein–Gordon equation with a generalized scalar potential”, Applied Mathematical Modelling, 108 (2022), 512
Vasily E. Tarasov, “Entropy Interpretation of Hadamard Type Fractional Operators: Fractional Cumulative Entropy”, Entropy, 24:12 (2022), 1852
Jocelyn Sabatier, Christophe Farges, Vincent Tartaglione, Intelligent Systems, Control and Automation: Science and Engineering, 101, Fractional Behaviours Modelling, 2022, 13
Machado J.A.T., “The Bouncing Ball and the Gr Spacing Diaeresis Unwald-Letnikov Definition of Fractional Derivative”, Fract. Calc. Appl. Anal., 24:4 (2021), 1003–1014
Zhao D., Yu G., Yu T., Zhang L., “A Probabilistic Interpretation of the Dzhrbashyan Fractional Integral”, Fractals-Complex Geom. Patterns Scaling Nat. Soc., 29:08 (2021), 2150269
Jocelyn Sabatier, “Fractional Order Models Are Doubly Infinite Dimensional Models and thus of Infinite Memory: Consequences on Initialization and Some Solutions”, Symmetry, 13:6 (2021), 1099
Ehab Malkawi, Advances in Computer and Electrical Engineering, Advanced Applications of Fractional Differential Operators to Science and Technology, 2020, 198
Cresus F. de L. Godinho, Nelson Panza, José Weberszpil, J. A. Helayël-Neto, “Variational procedure for higher-derivative mechanical models in a fractional integral”, EPL, 129:6 (2020), 60001
Lu Bai, Dingyu Xue, Li Meng, 2020 Chinese Control And Decision Conference (CCDC), 2020, 3225
Jocelyn Sabatier, “Non-Singular Kernels for Modelling Power Law Type Long Memory Behaviours and Beyond”, Cybernetics and Systems, 51:4 (2020), 383

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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