THERMAL SCIENCE

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Authors of this Paper

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Xiao-Jun Yang

,

Minvydas Ragulskis

,

Thiab Taha

A NEW GENERAL FRACTIONAL-ORDER DERIVATIVE WITH RABOTNOV FRACTIONAL-EXPONENTIAL KERNEL

ABSTRACT
In this article, a general fractional-order derivative of the Riemann-Liouville type with the non-singular kernel involving the Rabotnov fractional-exponential function is addressed for the first time. A new general fractional-order derivative model for the anomalous diffusion is discussed in detail. The general fractional-order derivative operator formula is as a novel and mathematical approch proposed to give the generalized presentation of the physical models in complex phenomena with power law.
KEYWORDS
anomalous diffusion, general fractional-order derivataive, Rabotnov fractional-exponential function, non-singular kernel, power law
PAPER SUBMITTED: 2018-08-25
PAPER REVISED: 2018-10-11
PAPER ACCEPTED: 2018-12-22
PUBLISHED ONLINE: 2019-06-08
DOI REFERENCE: https://doi.org/10.2298/TSCI180825254Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2019, VOLUME 23, ISSUE Issue 6, PAGES [3711 - 3718]
REFERENCES
  1. Yang, X. J., General Fractional Derivatives: Theory, Methods and Applications, New York, CRC Press, 2019
  2. Kochubei, A. N., General Fractional Calculus, Evolution Equations, and Renewal Processes, Integral Equations and Operator Theory, 71(2011), 4, pp.583-600
  3. Luchko, Y., et al., General Time-Fractional Diffusion Equation: some Uniqueness and Existence Results for the Initial-Boundary-Value Problems, Fractional Calculus and Applied Analysis, 19(2016), 3, pp.676-695
  4. Yang, X. J., et al., Anomalous Diffusion Models with General Fractional Derivatives within the Kernels of the Extended Mittag-Leffler Type Functions, Romanian Reports in Physics, 69 (2017), 4, Article ID 115
  5. Liouville, J., Memoire sur le calcul des different idles a indices quelconques, Journal de EcolePolytechnique, 13(1832), 21, pp.71-162
  6. Riemann, B., Versucheinerallgemeinen auffassung der integration und differentiation, Bernhard RiemannsGesammelteMathematischeWerke, 1847, Janvier, pp. 353-362
  7. Weyl, H., Bemerkungenzum begriff des differential quotienten gebrochener ordnung, Vierteljal&Rechrift tier NtdrforchentlenGeellchaft in Zirich, 62(1917), 1-2, pp.296-302
  8. Sonine, N., Sur la differentiation a indice quelconque, MatematicheskiiSbornik, 6(1872), 1, pp.1-38
  9. Caputo, M., Linear Models of Dissipation whose Q is almost Frequency Independent II, Geophysical Journal International, 13(1967), 5, pp.529-539
  10. Caputo, M., et al., A New Definition of Fractional Derivative without Singular Kernel, Progress in Fractional Differentiation and Applications, 1(2015), 2, pp.1-13
  11. Miller, K. S., et al., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons. New York, 1993
  12. Lorenzo, C. F., et al., The Fractional Trigonometry: With Applications to Fractional Differential Equations and Science, John Wiley & Sons, 2016
  13. Goreno R., et al., Fractional Oscillations and Mittag-Leffler Functions, International Workshop on the Recent Advances in Applied Mathematics (RAAM96), State of Kuwait, Proceedings, Kuwait University, 193-208, 1996
  14. Samko, S. G., et al., Fractional Integrals and Derivatives: Theory and Applications, Switzerland, Gordon and Breach Science, 1993
  15. Hille, E., et al., On the Theory of Linear Integral Equations, Annals of Mathematics, 31(1930), 3, pp.479-528
  16. Yang, X. J., Theoretical Studies on General Fractional-Order Viscoelasticity, Ph.D Thesis, China University of Mining and Technology, Xuzhou, China, December, 2017
  17. Tomovski, Ž., et al., Fractional and Operational Calculus with Generalized Fractional Derivative Operators and Mittag-Leffler Type Functions, Integral Transforms and Special Functions, 21(2010), 11, pp.797-814
  18. Yang, X. J., et al., A New Fractional Derivative Involving the Normalized Sinc Function without Singular Kernel, The European Physical Journal Special Topics, 226(2017), 16-18, pp.3567-3575
  19. Yang, X. J., et al., Local Fractional Integral Transforms and their Applications, Academic Press, 2015
  20. Rabotnov, Y., Equilibrium of an Elastic Medium with after Effect (in Russian), Prikladnaya Matematikai Mekhanika, 12(1948), 1, pp.53-62
  21. Meshkov, S. I., et al., Internal Friction Described with the aid of Fractionally-Exponential Kernels, Journal of Applied Mechanics and Technical Physics, 7(1969), 3, pp.63-65
  22. Yang, X. J., et al., A New General Fractional-Order Derivatiive with Rabotnov Fractional-Exponential Kernel Applied to Model the Anomalous Heat Transfer, Thermal Science, 23(2019), 3, DOI: 10.2298/TSCI180320239Y
  23. Kolmogorov, A. N., et al., Fundamentals of the Theory of Functions and Functional Analysis, Moscow, Nauka, 1968
  24. Prabhakar, T. R., A Singular Integral Equation with a Generalized MittagLeffler Function in the Kernel, Yokohama Mathematical Journal, 19(1971),1, pp.7-15
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A new general fractional-order derivative with Rabotnov fractional-exponential kernel

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