THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

External Links

Yue Hu

,

Ji-Huan He

ON FRACTAL SPACE-TIME AND FRACTIONAL CALCULUS

ABSTRACT
This paper gives an explanation of fractional calculus in fractal space-time. On observable scales, continuum models can be used, however, when the scale tends to a smaller threshold, a fractional model has to be adopted to describe phenomena in micro/nano structure. A time-fractional Fornberg-Whitham equation is used as an example to elucidate the physical meaning of the fractional order, and its solution process is given by the fractional complex transform.
KEYWORDS
He's fractional derivative, Adomian's decomposition method, variational iteration method, exp-function method, fractional complex transform, fractional Fornberg-Whitham equation
PAPER SUBMITTED: 2015-09-10
PAPER REVISED: 2015-12-12
PAPER ACCEPTED: 2015-12-12
PUBLISHED ONLINE: 2016-08-13
DOI REFERENCE: https://doi.org/10.2298/TSCI1603773H
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2016, VOLUME 20, ISSUE Issue 3, PAGES [773 - 777]
REFERENCES
  1. He, J.-H., An Elementary Introduction to Recently Developed Asymptotic Methods and Nanomechanics in Textile Engineering, Int. J. Mod. Phys. B, 22 (2008), 21, pp. 3487-3578
  2. El Naschie, M. S., Elementary Prerequisites for E-Infinity (Recommended Background Readings in Nonlinear Dynamics, Geometry and Topology), Chaos, Solitones & Fractals, 30 (2006), 3, pp. 579-605
  3. El Naschie, M. S., A Review of E-Infinity Theory and the Mass Spectrum of High Energy Particle Physics, Chaos, Solitones & Fractals, 19 (2004), 1, pp. 209-236
  4. He, J.-H., A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (2014), 11, pp. 3698-3718
  5. He, J.-H., Hilbert Cube Model for Fractal Spacetime, Chaos, Solitones & Fractals, 42 (2009), 5, pp. 2754-2759
  6. Majumder, M., et al., Nanoscale Hydrodynamics: Enhanced Flow in Carbon Nanotubes, Nature, 438 (2005), Nov., p. 44
  7. Chen, R. X., et al., Silk Cocoon: “Emperor's New Clothes” for Pupa: Fractal Nano-Hydrodynamical Approach, Journal of Nano Research, 22 (2013), May, pp. 65-70
  8. Kong, H. Y., et al., The Fractal Harmonic Law and its Application to Swimming Suit, Thermal Science, 16 (2012), 5, pp. 1467-1471
  9. Kong, H. Y., et al., A Novel Friction Law, Thermal Science, 16 (2012), 5, pp. 1529-1533
  10. Fan, J., et al., Model of Moisture Diffusion in Fractal Media, Thermal Science, 19 (2015), 4, pp. 1161-1166
  11. Wu, G. C., et al., Lattice Fractional Diffusion Equation in Terms of a Riesz-Caputo Difference, Physica A, 438 (2015), Nov., pp. 335-339
  12. Wu, G. C., et al., Discrete Fractional Diffusion Equation, Nonlinear Dynamics, 80 (2015), 1, pp. 281-286
  13. Chen, F. L., A Review of Existence and Stability Results for Discrete Fractional Equations, Journal of Computational Complexity and Applications, 1 (2015), 1, pp. 22-53
  14. Zhang, M. F., et al., Efficient Homotopy Perturbation Method for Fractional Nonlinear Equations Using Sumudu Transform, Thermal Science, 19 (2015), 4, pp. 1167-1171
  15. Cui, L. X., et al., New Exact Solutions of Fractional Hirota-Satsuma Coupled Korteweg-De Vries Equation, Thermal Science, 19 (2015), 4, pp. 1173-1176
  16. Qin, Y. M., et al., Discrete Fractional Diffusion Model with Two Memory Terms, Thermal Science, 19 (2015), 4, pp. 1177-1181
  17. Ma, H. C., et al., Exact Solutions of Nonlinear Fractional Partial Differential Equations by Fractional Sub-Equation Method, Thermal Science, 19 (2015), 4, pp. 1239-1244
  18. Wu, F., Liu, J. F., Discrete Fractional Creep Model of Salt Rock, Journal of Computational Complexity and Applications, 2 (2016), 1, pp. 1-6
  19. Keshtkar, F., On Global Stability of Nonlinear Fractional Dynamical Systems, Journal of Computational Complexity and Applications, 2 (2016), 1, pp. 16-23
  20. He, J.-H., et al., Geometrical Explanation of the Fractional Complex Transform and Deriva ive Chain Rule for Frac tional Calculus, Physics Letters A, 376 (2012), 4, pp. 257-259
  21. He, J.-H., A New Fractal Derivation, Thermal Science, 15 (2011), Suppl. 1, pp. S145-S147
  22. Fan, J., He, J.-H., Fractal Derivative Model for Air Permeability in Hierarchic Porous Media, Abstract and Applied Analysis, 2012 (2012), ID354701, p. 7
  23. Fan, J., He, J.-H., Biomimic Design of Multi-Scale Fabric with Efficient Heat Transfer Property, Thermal Science, 16 (2012), 5, pp. 1349-1352
  24. Fan, J., Shang, X. M., Water Permeation in the Branching Channel Net of Wool Fiber, Heat Transfer Research, 44 (2013), 5, pp. 465-472
  25. Fan, J., Shang, X. M., Fractal Heat Transfer in Wool Fiber Hierarchy, Heat Transfer Research, 44 (2013), 5, pp. 399-407
  26. Fei, D. D., et al., Fractal Approach to Heat Transfer in Silkworm Cocoon Hierarchy, Thermal Science, 17 (2013), 5, pp. 1546-1548
  27. He, J.-H., Li, Z. B., Converting Fractional Differential Equations into Partial Differential Equations, Thermal Science, 16 (2012), 2, pp. 331-334
  28. Li, Z., He, J.-H., Fractional Complex Transform for Fractional Differential Equations, Math. Comput. Appl, 15 (2010), 5, pp. 970-973
  29. He, J.-H., et al., A New Fractional Derivative and its Application to Explanation of Polar Bear Hairs, Journal of King Saud University Science (on line first), 2015
  30. Liu, F. J., et al., He's Frac tional Derivative for Heat Conduction in a Fractal Medium Arising in Silkworm Cocoon Hierarchy, Thermal Science, 19 (2015), 4, pp. 1155-1159
  31. Liu, J. F., Modified Variational Iteration Method for Varian Boussinesq Equation, Thermal Science, 19 (2015), 4, 1195-1199
  32. Huang, L. L., Matrix Lagrange Multiplier of the VIM, Journal of Computational Complexity and Applications, 2 (2016), 3, pp. 86-88
  33. Wu, G. C., New Correction Functional and Lagrange Multipliers for Two Point Value Problems, Journal of Computational Complexity and Applications, 2 (2016), 2, pp. 46-48
  34. He, J.-H., Asymptotic Methods for Solitary Solutions and Compactons, Abstract and Applied Analysis, 2012 (2012), ID916793
  35. He, J.-H., Some Asymptotic Methods for Strongly Nonlinear Equations, Int. J. Mod. Phys. B, 20 (2006), 10, pp. 1141-1199
  36. Adomian, G., A Review of the Decomposition Method in Applied Mathematics, J. Math. Anal. Appl, 135 (1988), 2, pp. 44-501
PDF VERSION [DOWNLOAD]
On fractal space-time and fractional calculus

© 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