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A SPATIAL STRUCTURAL DERIVATIVE MODEL FOR ULTRASLOW DIFFUSION

ABSTRACT
This study investigates the ultraslow diffusion by a spatial structural derivative, in which the exponential function ex is selected as the structural function to construct the local structural derivative diffusion equation model. The analytical solution of the diffusion equation is a form of Biexponential distribution. Its corresponding mean squared displacement is numerically calculated, and increases more slowly than the logarithmic function of time. The local structural derivative diffusion equation with the structural function ex in space is an alternative physical and mathematical modeling model to characterize a kind of ultraslow diffusion.
KEYWORDS
PAPER SUBMITTED: 2017-03-10
PAPER REVISED: 2017-05-01
PAPER ACCEPTED: 2017-05-20
PUBLISHED ONLINE: 2017-12-02
DOI REFERENCE: https://doi.org/10.2298/TSCI17S1121X
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2017, VOLUME 21, ISSUE Supplement 1, PAGES [S121 - S127]
REFERENCES
  1. Gorenflo, R., et al., Discrete Random Walk Models for Space-Time Fractional Diffusion, Chemical Physics, 284 (2002), 1-2, pp. 521-541
  2. Metzler, R., Klafter, J., The Random Walk's Guide to Anomalous Diffusion: A Fractional Dynamics Approach, Physics Reports, 339 (2000), 1, pp. 1-77
  3. Yong, Z., et al., On Using Random Walks to Solve the Space-Fractional Advection-Dispersion Equations, Journal of Statistical Physics, 123 (2006), 1, pp. 89-110
  4. Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, World Scientific, Singapore, 2010
  5. Kopf, M., et al., Anomalous Diffusion of Water in Biological Tissues, Biophysical Journal, 70 (1996), 6, pp. 2950-2958
  6. Shlesinger, M. F., Asymptotic Solutions of Continuous-Time Random Walks, Journal of Statistical Physics, 10 (1974), 5, pp. 421-434
  7. Weiss, G. H., Rubin, R. J., Random Walks: Theory and Selected Applications, Advances in Chemical Physics. 52 (2007), Mar., pp. 363-505
  8. Metzler, R., et al., Anomalous Diffusion Models and Their Properties: Non-Stationarity, Non-Ergodicity, and Ageing at the Centenary of Single Particle Tracking, Physical Chemistry Chemical Physics, 16 (2014), 44, pp. 24128-24164
  9. Yang, X. J., Machado, J. T., A New Fractional Operator of Variable Order: Application in the Description of Anomalous Diffusion, Physical A: Statistical Mechanics and its Applications, 481 (2017), Sept., pp. 276-283
  10. 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.
  11. Alsaedi, A., et al., On Coupled Systems of Time-Fractional Differential Problems by Using a New Fractional Derivative, Journal of Function Spaces, 2016 (2016), ID 4626940
  12. Caputo, M., Fabrizio, M., Applications of New Time and Spatial Fractional Derivatives with Exponential Kernels, Progress in Fractional Differentiation and Applications, 2 (2016), 2, pp. 1-11
  13. Hou, S., Stochastic Model for Ultraslow Diffusion, Stochastic Processes & Their Applications, 116 (2006), 9, pp. 1215-1235
  14. Sinai, Y. G., The Limiting Behavior of a One-Dimensional Random Walk in a Random Medium, Theory of Probability & Its Applications, 27 (2006), 2, pp. 256-268
  15. Lomholt, M. A., et al., Microscopic Origin of the Logarithmic Time Evolution of Aging Processes in Complex Systems, Physical Review Letters, 110 (2013), 20, 208301
  16. Chen, W., et al., Local Structure Derivative and its Application, Journal of Solid Mechanics, 37 (2016), 5, pp. 456-460
  17. Chen, W., Time-Space Fabric Underlying Anomalous Diffusion, Chaos, Solitons & Fractals, 28 (2006), 4, pp. 923-929
  18. Chen, W., Liang, Y. J., New Methodologies in Fractional and Fractal Derivatives Modeling, Chaos Solitons & Fractals, 3 (2017), 6, pp. 1-6
  19. Chen, W., et al., A Fractional Structural Derivative Model for Ultra-slow Diffusion, Applied Mathematics and Mechanics, 37 (2016), 6, pp. 599-608
  20. Podlubny, I., Fractional Differential Equations, Academic Press, New York, USA, 1999
  21. Chen, W., et al., Structural Derivative Based on Inverse Mittag-Leffler Function for Modeling Ultraslow Diffusion, Fractional Calculus & Applied Analysis, 19 (2016), 5, pp. 1316-1346
  22. Bazi, Y., et al., Image Thresholding Based on the EM Algorithm and the Generalized Gaussian Distribution, Pattern Recognition, 40 (2007), 2, pp. 619-634
  23. Chaurasia, B. L. V., Pandey, C. S., On the Fractional Calculus of Generalized Mittag-Leffler Function, Series A: Mathematical Sciences, 20 (2010), pp. 113-122
  24. Oliveira, E. C. D., et al., Models Based on Mittag-Leffler Functions for Anomalous Relaxation in Dielectrics, The European Physical Journal Special Topics, 193 (2011), 1, pp. 161-171
  25. Zeng, C., Chen, Y. Q., Global Pad'e Approximations of the Generalized Mittag-Leffler Function and its Inverse, Fractional Calculus & Applied Analysis, 18 (2013), 8, pp. 1492-1506

© 2017 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, 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