THERMAL SCIENCE

International Scientific Journal

Authors of this Paper

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Ji-Huan He

,

Qura-Tul Ain

NEW PROMISES AND FUTURE CHALLENGES OF FRACTAL CALCULUS: FROM TWO-SCALE THERMODYNAMICS TO FRACTAL VARIATIONAL PRINCIPLE

ABSTRACT
Any physical laws are scale-dependent, the same phenomenon might lead to debating theories if observed using different scales. The two-scale thermodynamics observes the same phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption, and fractal calculus has to be adopted to establish governing equations. Here basic properties of fractal calculus are elucidated, and the relationship between the fractal calculus and traditional calculus is revealed using the two-scale transform, fractal variational principles are discussed for 1-D fluid mechanics. Additionally planet distribution in the fractal solar system, dark energy in the fractal space, and a fractal ageing model are also discussed.
KEYWORDS
two-scale fractal dimension, two scale mathematics, fractal space, fractal variational theory, local property
PAPER SUBMITTED: 2020-01-27
PAPER REVISED: 2020-01-27
PAPER ACCEPTED: 2020-01-28
PUBLISHED ONLINE: 2020-02-08
DOI REFERENCE: https://doi.org/10.2298/TSCI200127065H
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE Issue 2, PAGES [659 - 681]
REFERENCES
  1. Kozłowski, J., & Konarzewski, M. Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant? Functional Ecology, 18(2004), 2, pp. 283-289.
  2. Riisgard, H. U. No foundation of a "3/4 power scaling law" for respiration in biology, Ecol. Lett. 1(1998),2, pp.71-73.
  3. Li, X. X., He, J.H. Along the evolution process Kleiber's 3/4 law makes way for Rubner's surface law: a fractal approach, Fractals, 272019), 2, 1950015
  4. Ain, Q.T., He, J.H. On two-scale dimension and its applications, Therm. Sci., 23(2019), 3B, pp.1707-1712
  5. He, J.H,, Ji, F.Y. Two-scale mathematics and fractional calculus for thermodynamics, Therm. Sci., 23(2019),4, pp.2131-2133
  6. Zhou, C. J., Tian, D., He,J.H. Highly selective penetration of red ink in a saline water, Thermal Science, 23(2019),4,pp. 2265-2270
  7. He, J.H. Fractal calculus and its geometrical explanation, Results in Physics,10(2018), 272-276
  8. El Naschie, M. S. The theory of Cantorian space time and high energy particle physics (An informal review). Chaos, Solitons & Fractals 41(2009), pp. 2635- 2646.
  9. El Naschie, M. S. A review of E-Infinity theory and the mass spectrum of high energy particle physics. Chaos, Solitons & Fractals. 19(2004), pp. 209-236.
  10. Marek-Crnjac, L. A short history of fractal-Cantorian space-time. Chaos, Solitons & Fractals 41(2009), pp. 2697-2705.
  11. He, J.H. Hilbert cube model for fractal spacetime, Chaos, Solitons & Fractals, 42(2009), 5, pp.2754-2759
  12. El Naschie, M.S. World Formula Interpretation of E = mc2, International Journal of Applied Science and Mathematics, 5(2018), 6, pp.67-75
  13. Marek-Crnjac, L. On El Naschie's Fractal-Cantorian Space-Time and Dark Energy—A Tutorial Review, Natural Science, 7(2015),13, Article ID:61825
  14. Majumder, M., et al. Nanoscale hydrodynamics - Enhanced flow in carbon nanotubes, Nature, 438 (2005), 7064, p. 44
  15. Yang, X.J. Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012.
  16. Mohebbi, A. and Saffarian, M. Implicit RBF Meshless Method for the Solution of Two-dimensional Variable Order Fractional Cable Equation, Journal of Applied and Computational Mechanics, 6(2020), 2, pp. 235-247
  17. Jena, R.M. and Chakraverty, S. Residual Power Series Method for Solving Timefractional Model of Vibration Equation of Large Membranes, Journal of Applied and Computational Mechanics, 5(2019), 4, pp.603-615
  18. Ahmed, N., et al., Transient MHD Convective Flow of Fractional Nanofluid between Vertical Plates, Journal of Applied and Computational Mechanics, 5(2019), 4, pp.592-602
  19. Rauf, A. and Mahsud, Y. Electro-magneto-hydrodynamics Flows of Burgers' Fluids in Cylindrical Domains with Time Exponential Memory, Journal of Applied and Computational Mechanics, 5(2019), 4, pp. 577-591
  20. Ali, N., et al., Transient Electro-osmotic Slip Flow of an Oldroyd-B Fluid with Time-fractional Caputo-Fabrizio Derivative, Journal of Applied and Computational Mechanics, 5(2019), 4, pp. 779-790
  21. Jassim, H.K. and Baleanu, D. A Novel Approach for Korteweg-de Vries Equation of Fractional Order, Journal of Applied and Computational Mechanics, 5(2019), 2, pp.192-198
  22. Yang, Y.J. and Wang, S.Q. A local fractional homotopy perturbation method for solving the local fractional Korteweg-de Vries equations with non-homogeneous term, Thermal Science, 23(2019), 3A, pp. 1495-1501
  23. Zenkour, A. and Abouelregal, A. Fractional Thermoelasticity Model of a 2D Problem of Mode-I Crack in a Fibre-Reinforced Thermal Environment, Journal of Applied and Computational Mechanics, 5(2019), 2, pp.269-280
  24. Shah, N.A., et al., Magnetohydrodynamic Free Convection Flows with Thermal Memory over a Moving Vertical Plate in Porous Medium, Journal of Applied and Computational Mechanics, 5(2019), 1, pp. 150-161
  25. Zhang, S., et al. Bilinearization and fractional soliton dynamics of fractional Kadomtsev-Petviashvili equation, Thermal Science, 23(2019), 3A, pp. 1425-1431
  26. He, J.H. Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Meth. Appl. Mech. Eng., 167(1998), pp. 57- 68
  27. He, J.H., et al., Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376(2012), pp.257-259
  28. He JH, et al. A new fractional derivative and its application to explanation of polar bear hairs, Journal of King Saud University Science, 28(2016), 2, pp.190-192
  29. Liu, H.Y., et al. Fractional calculus for nanoscale flow and heat transfer, International Journal of Numerical Methods for Heat & Fluid Flow, 24(2014), 6, pp.1227-1250
  30. He, J.H. A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53(2014), 11, pp. 3698-3718
  31. Wang, K.L., Yao, S.W., Numerical method for fractional Zakharov-Kuznetsov equations with He's fractional derivative, Thermal Science, 23(2019), 4, pp. 2163- 2170
  32. Guner, O. Exp-Function Method and Fractional Complex Transform for Space- Time Fractional KP-BBM Equation, Communications in Theoretical Physics, 68(2017), 2, pp. 149-154
  33. Wang, Y., et al., A short review on analytical methods for fractional equations with He's fractional derivative, Thermal Science, 21(2017), 4, pp. 1567-1574
  34. Wang, Y., An, J.Y., Amplitude-frequency relationship to a fractional Duffing oscillator arising in microphysics and tsunami motion, Journal of Low Frequency Noise Vibration and Active Control, 38(2019), 3-4, pp. 1008-1012
  35. Wang, Y., et al., Shallow water waves in porous medium for coast protection, Thermal Science, 21(2017), S1, S145-S151
  36. Wang, Y., et al., A fractional Whitham-Broer-Kaup equation and its possible application to tsunami prevention, Thermal Science, 21(2017), 4, pp.1847-1855
  37. Sayevand, K., et al. A reliable implicit difference scheme for treatments of fourthorder fractional sub-diffusion equation, Scientia Iranica, 24(2018), 3, pp.1100- 1107
  38. Sayevand, K., Rostami, M. Fractional optimal control problems: optimality conditions and numerical solution, IMA Journal of Mathematical Control and Information, 35(2018), 1, pp. 123-148
  39. Lin, L., et al., Experimental verification of the fractional model for silver ion release from hollow fibers, Journal of Low Frequency Noise Vibration and Active Control, 38(2019),Nos.3-4, pp.1041-1044
  40. Lin, L. and Yao, S.W. Release oscillation in a hollow fiber - Part 1: Mathematical model and fast estimation of its frequency, Journal of Low Frequency Noise Vibration and Active Control, 38(2019), Nos.3-4, pp.1703-1707
  41. Liu, H.Y., et al., A fractional nonlinear system for release oscillation of silver ions from hollow fibers, Journal of Low Frequency Noise Vibration and Active Control, 38(2019), 1, 88-92
  42. Liu, F.J., et al., Silkworm(Bombyx Mori) cocoon vs wild cocoon: Multi-Layer Structure and Performance Characterization, Thermal Science, 23(2019), 4, pp. 2135-2142
  43. He, J.-H. Frontier of Modern Textile Engineering and Short Remarks on Some Topics in Physics, International Journal of Nonlinear Sciences and Numerical Simulation, 11 (2010) 555-563
  44. Mandelbrot, B.B. The Fractal Geometry of Nature, W. H. Freeman and Company, 1982
  45. Shen, J.F., et al. Fractal study in soil spatial variability and thermal conductivity, Thermal Science, 23(2019), 5, pp. 2849-2856
  46. Cao, A.Y., et al., An analytical solution for solving a new general fractional order model for wave in mining rock, Thermal Science, 23(2019), Supplement 3, pp. S983-S987
  47. Chen, J., et al., A non-linear creep constitutive model for salt rock based on fractional derivatives, Thermal Science, 23(2019), Supplement 3, pp. S773-S779
  48. Chen Z.Q., et al., A new fractional derivative model for the anomalous diffusion problem, Thermal Science, 23(2019), Supplement 3, pp. S1005-S1011
  49. Deng S.X., Approximate analytical solutions of non-linear local fractional heat equations, Thermal Science, 23(2019), Supplement 3, pp. S837-S841
  50. Dou L.M., et al., A new general fractional-order wave model involving Miller- Ross kernel, Thermal Science, 23(2019), Supplement 3, pp. S953-S957
  51. Gao Y.N., et al., The mechanical properties and fractal characteristics of the coal under temperature-gas-confining pressure, Thermal Science, 23(2019), Supplement 3, pp.S789-S798
  52. Yao, S.W., Wang, K.L. A new approximate analytical method for a system of fractional differential equations, Thermal Science, 23(2019) Supplement 3, pp. S853-S858
  53. Wei, C.F. Application of the homotopy perturbation method for solving fractional Lane-Emden type equation, Thermal Science, 23(2019), 4, pp.: 2237-2244
  54. Qiu, Y.Y., Numerical approach to the time-fractional reaction-diffusion equation, Thermal Science, 23(2019), 4, pp. 2245-2251
  55. Ma, H.C., et al., Exact solutions of the space-time fractional equal width equation, Thermal Science, 23(2019), 4, pp. 2307-2313
  56. Yang, X.H., et al., A fractional-order genetic algorithm for parameter optimization of the moisture movement in a bio-retention system, Thermal Science, 23(2019), 4, pp. 2343-2350
  57. Tian, A.H., et al., Land surface temperature vs soil spectral reflectance land surface temperature, Thermal Science, 23(2019), 4, pp. 2389-2395
  58. Tian, A.H., et al., Lathe tool chatter vibration diagnostic using general regression neural network based on Chua's circuit and fractional-order Lorenz master/slave chaotic system, Journal of Low Frequency Noise Vibration and Active Control 38(2019), Nos.3-4, pp.953-966
  59. Deng S.X., et al., Global well-posedness of a class of dissipative thermoelastic fluids based on fractal theory and thermal science analysis, Thermal Science, 23(2019), 4, pp. 2461-2469
  60. Sun, J.C., et al., Analytical study of (3+1)-dimensional fractional ultralowfrequency dust acoustic waves in a dual-temperature plasma, Journal of Low Frequency Noise Vibration and Active Control 38(2019), Nos.3-4, pp. 928-952
  61. He, J.H. A new fractal derivation, Thermal Science, 15(2011),1,pp. S145-S147
  62. He, J. H. The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, J. Low Freq. Noise V. A., 38(2019),Nos.3-4, pp.1252-1260.
  63. Wang, Q. L., et al., Fractal calculus and its application to explanation of biomechanism of polar bear hairs, Fractals, 27(2019) , 5, 1992001.
  64. Wang, Q. L., et al., Fractal calculus and its application to explanation of biomechanism of polar bear hairs, Fractals, 26(2018), 6, 1850086.
  65. Fan, J., et al., Fractal calculus for analysis of wool fiber: Mathematical insight of its biomechanism, J. Eng. Fiber. Fabr., 14 (2019). DOI: 10.1177/1558925019872200
  66. Li, X. X., et al., A fractal modification of the surface coverage model for an electrochemical arsenic sensor, Electrochm. Acta, 296 (2019), 491-493.
  67. Wang, Y., Deng, Q.G. Fractal derivative model for tsunami travelling, Fractals, 27 (2019), 1, 1950017.
  68. Wang, Y., et al., A fractal derivative model for snow's thermal insulation property, Therm. Sci., 23 (2019), 4, pp. 2351-2354.
  69. Liu, H. Y., et al., A fractal rate model for adsorption kinetics at solid/solution interface, Therm. Sci., 23(2019), 4, pp. 2477-2480.
  70. Fan, J., He J.H. Biomimic design of multi-scale fabric with efficient heat transfer property, Thermal Science, 16(2012), 5, pp. 1349-1352
  71. Fan, J., He, J.H. Fractal Derivative Model for Air Permeability in Hierarchic Porous Media, Abstract and Applied Analysis, 2012, 354701
  72. Lin, L., Yao, S.W., Fractal diffusion of silver ions in hollow cylinders with unsmooth inner surface, Journal of Engineered Fibers and Fabrics, 14(2019), Dec., Article Number: 1558925019895643
  73. Yang, Z.P. Filtration efficiency of a cigarette filter with X-or Y-shaped fibers, Thermal Science, 23(2019), 4, pp. 2517-2522
  74. Liu, Y.Q., et al., Air permeability of nanofiber membrane with hierarchical structure, Thermal Science, 22(2018), 4,pp.1637-1643
  75. Yu, D.Y., et al., Wetting and supercontraction properties of spider-based nanofibers, Thermal Science, 23(2019), 4, pp. 2189-2193
  76. Tian, D., et al. Sea-silk based nanofibers and their diameter prediction, Thermal Science, 23(2019), 4, pp. 2253-2256
  77. Zhou, C.J., et al., Highly selective penetration of red ink in a saline water, Thermal Science, 23(2019), 4, pp. 2265-2270
  78. Li, X.X., et al., Thermal property of rock powder-based nanofibers for high temperature filtration and adsorption, Thermal Science, 23(2019), 4, pp. 2501- 2507
  79. Calvini, P. The influence of levelling-off degree of polymerisation on the kinetics of cellulose degradation. Cellulose 12(2005), pp.445-447
  80. Calvini, P. Comments on the article ‘‘On the degradation evolution equations of cellulose'' by Hongzhi Ding and Zhongdong Wang, Cellulose 15(2008), pp.225- 228
  81. Calvini, P. The role of the Ekenstam equation on the kinetics of cellulose hydrolytic degradation, Cellulose 19(2012), pp. 313-318
  82. Calvini, P. On the meaning of the Emsley, Ding & Wang and Calvini equations applied to the degradation of cellulose, Cellulose 21(2014), pp.1127-1134
  83. Calvini, P., et al., On the kinetics of cellulose degradation: looking beyond the pseudo zero order rate equation. Cellulose 15(2008), pp.193-203
  84. Ding, H-Z, Wang, Z.D. Time-temperature superposition method for predicting the permanence of paper by extrapolating accelerated ageing data to ambient conditions. Cellulose 14(2007), pp.171-181
  85. Ding H-Z, Wang Z.D. On the degradation evolution equations of cellulose. Cellulose 15 (2008), pp.205-224
  86. Fan, W., et al. Random process model of mechanical property degradation in carbon fiber-reinforced plastics under thermo-oxidative aging, Journal of Composite Materials 51(2017) , 9, pp. 1253-1264
  87. He, C.H., et al. Taylor series solution for fractal Bratu-type equation arising in electrospinning process, Fractals, (2019) DOI: 10.1142/S0218348X20500115
  88. He, J.H. The simplest approach to nonlinear oscillators, Results in Physics, 15(2019), 102546
  89. He, J.H. A simple approach to one-dimensional convection-diffusion equation and its fractional modification for E reaction arising in rotating disk electrodes, Journal of Electroanalytical Chemistry, 854(2019), 113565
  90. He, J. H., Ji, F.Y. Taylor series solution for Lane-Emden equation, J. Math. Chem., 57 (2019),8, pp.1932-1934.
  91. He, J.H. Generalized Variational Principles for Buckling Analysis of Circular Cylinders, Acta Mechanica, 2019, DOI: 10.1007/s00707-019-02569-7
  92. He, J.-H. Variational Principle for the Generalized KdV-Burgers Equation with Fractal Derivatives for Shallow Water Waves, J. Appl. Comput. Mech., 6 (2020), 4. DOI: 10.22055/JACM.2019.14813
  93. He, J.H. A modified Li-He's variational principle for plasma, International Journal of Numerical Methods for Heat and Fluid Flow, (2019) DOI: 10.1108/HFF-06-2019-0523
  94. He, J.H. Lagrange Crisis and Generalized Variational Principle for 3D unsteady flow, International Journal of Numerical Methods for Heat and Fluid Flow, 2019, DOI: 10.1108/HFF-07-2019-0577
  95. He, J.H., Sun, C. A variational principle for a thin film equation, Journal of Mathematical Chemistry. 57(2019), 9, pp 2075-2081
  96. He, J.H. A fractal variational theory for one-dimensional compressible flow in a microgravity space, Fractals doi.org/10.1142/S0218348X20500243
  97. Wang, Y., et al., A variational formulation for anisotropic wave traveling in a porous medium, Fractals, 27(2019), 4, 19500476
  98. He, J.-H., Latifizadeh, H. A general numerical algorithm for nonlinear differential equations by the variational iteration method, International Journal of Numerical Methods for Heat and Fluid Flow, 2020, DOI :10.1108/HFF-01-2020- 0029
  99. He JH, et al. A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar, Applied Mathematical Modelling, 2020, doi.org/10.1016/j.apm.2020.01.027
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New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

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