THERMAL SCIENCE

International Scientific Journal

Xiao-Jun Yang

,

Abdulrahman Ali Alsolami

,

Sohail D. Nadeem

A NEW GENERAL FRACTIONAL DERIVATIVE RELAXATION PHENOMENON

ABSTRACT
This article addresses a novel anomalous relaxation model with the new general fractional derivative of the Sonine kernel. This operator is considered in the sense of general fractional derivative proposed in the work [Yang et al., General fractional derivatives with applications in viscoelasticity, Academic Press, New York, USA, 2020]. The solution of the mathematical model is obtained with the aid of Laplace transform. The comparison among the classical and anomalous relaxation models is discussed in detail. This result is proposed as a mathematical tool to model the anomalous relaxation behavior of the complex materials.
KEYWORDS
anomalous relaxation, general fractional derivative, complex materials, Sonine kernel, laplace transform
PAPER SUBMITTED: 2022-11-07
PAPER REVISED: 2022-11-21
PAPER ACCEPTED: 2022-11-29
PUBLISHED ONLINE: 2023-03-04
DOI REFERENCE: https://doi.org/10.2298/TSCI2301721Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2023, VOLUME 27, ISSUE Issue 1, PAGES [721 - 726]
REFERENCES
  1. Yang, X. J., et al., General Fractional Derivatives with Applications in Viscoelasticity, Academic Press, New York, USA, 2020
  2. Yang, X. J., General Fractional Derivatives: Theory, Methods and Applications, Chapman and Hall/CRC, London, UK, 2019
  3. Yang, X. J., Theory and Applications of Special Functions for Scientists and Engineers, Springer Singapore, New York, USA, 2021
  4. Tarasov, V. E., General Fractional Dynamics, Mathematics, 9 (2021), 13, ID1464
  5. Luchko, Y., General Fractional Integrals and Derivatives with the Sonine Kernels, Mathematics, 9 (2021), 6, 594
  6. Atangana, A., Baleanu, D., New Fractional Derivatives with Non-Local and Non-Singular Kernel: Theory and Application Heat Transfer Model, Thermal Science, 20 (2016), 2, pp. 763-769
  7. Yang, X. J., et al., A New General Fractional-Order Derivative with Rabotnov Fractional-Exponential Kernel Applied to Model the Anomalous Heat Transfer, Thermal Science, 23 (2019), 3A, pp. 1677-1681
  8. Omame, A., et al., A Fractional-Order Model for COVID-19 and Tuberculosis Co-Infection Using Atangana-Baleanu Derivative, Chaos, Solitons and Fractals, 153 (2021), Dec., ID111486
  9. Jleli, M., et al., Analytical Approach for Time Fractional Wave Equations in the Sense of Yang-Abdel-Aty-Cattani via the Homotopy Perturbation Transform Method, Alexandria Engineering Journal, 59 (2020), 5, pp. 2859-2863
  10. Yang, X. J., et al., Fundamental Solutions of Anomalous Diffusion Equations with the Decay Exponential Kernel, Mathematical Methods in the Applied Sciences, 42 (2019), 11, pp. 4054-4060
  11. Yang, X. J., et al., A New General Fractional-Order Derivative with Rabotnov Fractional-Exponential Kernel, Thermal Science, 23 (2019), 6B, pp. 3711-3718
  12. F. Mainardi, Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena, Chaos, Solitons and Fractals, 7 (1996), 9, pp. 1461-1477
  13. Metzler, R., Nonnenmacher, T. F., Fractional Relaxation Processes and Fractional Rheological Models for the Description of a Class of Viscoelastic Materials, International Journal of Plasticity, 19 (2003), 7, pp. 941-959
  14. Tofighi, A., Golestani, A., A Perturbative Study of Fractional Relaxation Phenomena, Physica A: Statistical Mechanics and its Applications, 387 (2008), 8-9, pp. 1807-1817
  15. Chen, W., et al., Investigation on Fractional and Fractal Derivative Relaxation-Oscillation Models, International Journal of Non-Linear Sciences and Numerical Simulation, 11 (2010), 1, pp. 3-10
  16. De Oliveira, et al., Fractional Models of Anomalous Relaxation Based on the Kilbas and Saigo Function, Meccanica, 49 (2014), 9, pp. 2049-2060
  17. Giona, M., et al., Fractional Diffusion Equation and Relaxation in Complex Viscoelastic Materials, Physica A: Statistical Mechanics and its Applications, 191 (1992), 1-4, pp. 449-453
  18. Yang, X. J., Fractional Derivatives of Constant and Variable Orders Applied to Anomalous Relaxation Models in Heat Transfer Problems, Thermal Science, 21 (2017), 3, pp. 1161-1171
  19. Sonine, N., Sur la Généralisation d'une Formule d'Abel (in French), Acta Mathematica, 4 (1884), 1, pp. 171-176
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A new general fractional derivative relaxation phenomenon

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