THERMAL SCIENCE

International Scientific Journal

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Geng Li

,

Kang-Jia Wang

DYNAMIC BEHAVIORS OF THE NON-LINEAR LOCAL FRACTIONAL HEAT CONDUCTION EQUATION ON THE CANTOR SETS

ABSTRACT
Based on the local fractional derivative, a fractal non-linear heat conduction equation, which can model the behavior of the heat transfer in the fractal medium, is extracted in this work. On defining the Mittag-Leffler function on the Cantor sets, two special functions namely the THυ(μυ) function and CHυ(μυ) function are constructed, and then are employed along with Yang's non-differentiable transfor­mation seek for the non-differentiable exact solutions. The obtained results confirm that the proposed method iseffective and powerful, and can provide a promising way to find the exact solutions of the fractal PDE.
KEYWORDS
local fractional derivative, Mittag-Leffler function, Cantor sets, Yang’s non-differentiable transformation, special functions
PAPER SUBMITTED: 2024-02-22
PAPER REVISED: 2024-03-29
PAPER ACCEPTED: 2024-05-09
PUBLISHED ONLINE: 2024-09-28
DOI REFERENCE: https://doi.org/10.2298/TSCI2404391L
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2024, VOLUME 28, ISSUE Issue 4, PAGES [3391 - 3396]
REFERENCES
  1. Sohail, M., et al., Computational Exploration for Radiative Flow of Sutterby Nanofluid with Variable Temperature-Dependent Thermal Conductivity and Diffusion Coefficient, Open Physics, 18 (2020), 1, pp. 1073-1083
  2. Wang, K. J., et al., Resonant Y-Type Soliton, Interaction Wave and Other Diverse Wave Solutions to the (3+1)-Dimensional Shallow Water Wave Equation, Journal of Mathematical Analysis and Applications, 542 (2025), 128792
  3. Wang, K. J., et al., Complexiton, Complex Multiple Kink Soliton and the Rational Wave Solutions to the Generalized (3+1)-Dimensional Kadomtsev-Petviashvili Equation, Physica Scripta, 99 (2024), 7, 075214
  4. Wang, K. J., et al., Novel Complexiton Solutions to the New Extended (3+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation for Incompressible Fluid, EPL, 146 (2024), 6, 62003
  5. Yildirim, Y., et al., Optical Soliton Perturbation and Conservation Law with Kudryashov's Refractive Index Having Quadrupled Power-Law and Dual Form of Generalized Non-Local Non-Linearity, Optik, 240 (2021), 166966
  6. Wang, K., Si, J., Dynamic Properties of the Attachment Oscillator Arising in the Nanophysics, Open Physics, 21 (2023), 1, 20220214
  7. He, J. H., et al., A simple Frequency Formulation for the Tangent Oscillator, Axioms, 10 (2021), 4, 320
  8. Ahmad, I., et al., Solution of Multi-Term Time-Fractional PDE Models Arising in Mathematical Biology and Physics by Local Meshless Method, Symmetry, 12 (2020), 7, 1195
  9. Centenera, M. M., et al., A Patient‐Derived Explant (PDE) Model of Hormone‐Dependent Cancer, Molecular Oncology, 12 (2018), 9, pp. 1608-1622
  10. Herisanu, N., Marinca, V., Explicit Analytical Approximation Large-Amplitude Non-Linear Oscillations of a Uniform Cantilever Beam Carrying an Intermediate Lumped Mass and Rotary Inertia, Meccanica, 45 (2010), 6, pp. 847-855
  11. Akbarzade, M., Khan, Y., Dynamic Model of Large Amplitude Non-Linear Oscillations Arising in the Structural Engineering: Analytical Solutions, Mathematical and Computer Modelling, 55 (2012), 3, pp. 480-489
  12. Guo, A., et al., Predicting the Future Chinese Population Using Shared Socioeconomic Pathways, The Sixth National Population Census, and a PDE Model, Sustainability, 11 (2019), 3, 3686
  13. Wazwaz, A. M., The Tanh Method for Generalized Forms of Non-Linear Heat Conduction and Burgers-Fisher Equations, Applied Mathematics and Computation, 169 (2005), 1, pp. 321-338
  14. He, J. H., Solitary Waves Travelling along an Unsmooth Boundary, Res. Phys., 24 (2021), 104104
  15. Xu, P., et al., Semi-Domain Solutions to the Fractal (3+1)-Dimensional Jimbo-Miwa Equation, Fractals, 32 (2024), 2440042
  16. He, J. H., Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves, Journal of Applied and Computational Mechanics, 6 (2020), 4, pp. 735-740
  17. Xu, P., et al., The Fractal Modification of the Rosenau-Burgers Equation and its Fractal Variational Principle, Fractals, 32 (2024), 5, 2450121
  18. He, J. H., A fractal Variational Theory for 1-D Compressible Flow in a Microgravity Space, Fractals, 28 (2020), 2, 2050024
  19. Wang, K. L., A Study of the Fractal Foam Drainage Model in a Microgravity Space, Mathematical Methods in the Applied Sciences, 44 (2021), 13, pp. 10530-10540
  20. Wang, K. J., Variational Principle and Approximate Solution for the Fractal Vibration Equation in a Microgravity Space, Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 46 (2022), 1, pp.161-165
  21. Liu, F. J., et al., He's fractional Derivative for Heat Conduction in a Fractal Medium Arising in Silkworm Cocoon Hierarchy, Thermal Science, 19 (2015), 4, pp. 1155-1159
  22. Wang, K. J., Research on the Non-Linear Vibration of Carbon Nanotube Embedded in Fractal Medium, Fractals, 30 (2022), 1, 2250016
  23. Yang, X. J., et al., Local Fractional Integral Transforms and Their Applications, Academic Press, Elsevier, Cambridge, Mass. USA, 2015
  24. Yang, X. J., et al., Non-Differentiable Exact Solutions for the Non-Linear ODE Defined on Fractal Sets, Fractals, 25 (2017), 4, 1740002
  25. Wang, K. J., et al., Study on the Local Fractional (3+1)-Dimensional Modified Zakharov-Kuznetsov Equation by a Simple Approach, Fractals, 32 (2024), 5, 2450091
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Dynamic behaviors of the non-linear local fractional heat conduction equation on the cantor sets

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence