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NON-DIFFERENTIABLE SOLUTIONS FOR NON-LINEAR LOCAL FRACTIONAL HEAT CONDUCTION EQUATION

ABSTRACT
Fractional calculus has many advantages. Under consideration of this paper is a (2+1)-dimensional non-linear local fractional heat conduction equation with arbitrary degree non-linearity. Backlund transformation of a reduced form of the local fractional heat conduction equation is constructed by Painleve analysis. Based on the Backlund transformation, some exact non-differentiable solutions of the local fractional heat conduction equation are obtained. To gain more insights of the obtained solutions, two solutions are constrained to a Cantor set and then two spatio-temporal fractal structures with profiles of these two solutions are shown. This paper further reveals by local fractional heat conduction equation that fractional calculus plays important role in dealing with non-differentiable problems.
KEYWORDS
PAPER SUBMITTED: 2021-06-28
PAPER REVISED: 2021-07-15
PAPER ACCEPTED: 2021-07-17
PUBLISHED ONLINE: 2021-12-18
DOI REFERENCE: https://doi.org/10.2298/TSCI21S2309Z
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE Special issue 2, PAGES [309 - 314]
REFERENCES
  1. Oldham, K. B., Spanier, J., The Fractional Calculus, Academic Press, San Diego, Cal., USA, 1974
  2. Fan, J., Liu, Y., Heat Transfer in the Fractal Channel Network of Wool Fiber, Materials Science and Technology, 26 (2010), 11, pp. 1320-1322
  3. Fan, J., et al., Influence of Hierarchic Structure on the Moisture Permeability of Biomimic Woven Fabricusing Fractal Derivative Method, Advances in Mathematical Physics, 2015 (2015), ID 817437
  4. Fan, J., Shang, X. M., Fractal Heat Transfer in Wool Fiber Hierarchy, Heat Transfer Research, 44 (2013), 5, pp. 399-407
  5. He, J, H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
  6. He, J, H., Fractal Calculus and its Geometrical Explanation, Results in Physics, 10 (2018), 1, pp. 272-276
  7. He, J. H., A New Fractal Derivation, Thermal Science, 15, (2011), Suppl. 1, pp. S145-S147
  8. Li, X., et al., A Fractal Modification of the Surface Coverage Model for an Electrochemical Arsenic Sensor, Electrochemical Acta, 296 (2019), 1, pp. 1491-493
  9. Xu, B., et al., Line Soliton Interactions for Shallow Ocean-Waves and Novel Solutions with Peakon, Ring, Conical, Columnar and Lump Structures Based on Fractional KP Equation, Advances in Mathematical Physics, 2021 (2021), ID 6664039
  10. Zhang, S., et al., Variable Separation for Time Fractional Advection-Dispersion Equation with Initial and Boundary Conditions, Thermal Science, 20 (2016), 3, pp. 789-792
  11. Zhang, S., Hong, S. Y., Variable Separation Method for a Non-linear Time Fractional Partial Differential Equation with Forcing Term, Journal of Computational and Applied Mathematics, 339 (2018), Sept., pp. 297-305
  12. Zhang, Y. F., Similarity Solutions and the Computation Formulas of a Non-linear Fractional-Order Gen-eralized Heat Equation, Modern Physics Letters B, 33 (2019), 10, ID 1950122
  13. Liu, J. G., et al., On Integrability of the Time Fractional Non-linear Heat Conduction Equation, Journal of Geometry and Physics, 144 (2019), Oct., pp. 190-198
  14. Harris, P. A., Garra, R. Non-linear Heat Conduction Equations with Memory: Physical Meaning and An-alytical Results, Journal of Mathematical Physics, 58 (2017), 6, ID 063501
  15. 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
  16. Kabir, M. M., Analytic Solutions for Generalized Forms of the Non-linear Heat Conduction Equation, Non-linear Analysis Real World Applications, 12 (2011), 5, pp. 2681-2691
  17. Djordjevic, V. D., Atanackovic, T. M. Similarity Solutions to Non-linear Heat Conduction and Burg-ers/Korteweg-de Vries Fractional Equations, Journal of Computational & Applied Mathematics, 222 (2008), 2, pp. 701-714
  18. Yang, X. J., et al., Non-linear Dynamics for Local Fractional Burgers' Equation Arising in Fractal Flow, Non-linear Dynamics, 84 (2016), 1, pp. 3-7
  19. Hristov, J., Transient Heat Diffusion with a Non-singular Fading Memory from the Cattaneo Constitutive Equation with Jeffrey's Kernel to the Caputo-Fabrizio Time-Fractional Derivative, Thermal Science, 20, (2016), 2, pp. 757-762
  20. Wen, Y. X., Zhou, X. W., Exact Solutions for the Generalized Non-linear Heat Conduction Equations Using the Exp-Function Method, Computers & Mathematics with Applications, 58 (2009), 11-12, pp. 2464-2467
  21. Yang, X. J., Advanced Local Fractional Calculus and its Applications, World Science, New York, USA, 2012
  22. Li, Z. B., He, J. H. Fractional Complex Transform for Fractional Differential Equations, Mathematical & Computational Applications, 15 (2011), 5, pp. 97-137
  23. Kruskal, M. D., et al., Analytic and Asymptotic Methods for Non-linear Singularity Analysis: a Review and Extensions of Tests for the Painleve Property, Lecture Notes in Physics, 495 (2007), Dec., pp. 171-205

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