THERMAL SCIENCE

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Shuxian Deng

,

Xinxin Ge

FRACTIONAL FOKKER-PLANCK EQUATION IN A FRACTAL MEDIUM

ABSTRACT
This paper studies a fractal modification of Fokker-Planck equation for a heat conduction in a fractal medium. Fourier transform and Darboux transformation are used to solve the equation, some new results are obtained.
KEYWORDS
fractional Fokker-Planck equation, fractal Fourier transform, fractal media, Darboux transformation
PAPER SUBMITTED: 2018-03-22
PAPER REVISED: 2018-10-30
PAPER ACCEPTED: 2018-10-30
PUBLISHED ONLINE: 2020-06-21
DOI REFERENCE: https://doi.org/10.2298/TSCI2004589D
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE Issue 4, PAGES [2589 - 2595]
REFERENCES
  1. Risken, H., Caugheyz, T. K., The Fokker-Planck Equation: Methods of Solution and Applications, Optica Acta, 31 (1996), 11, pp. 1206-1207
  2. Jordan, R., et al., The Variational Formulation of the Fokker-Planck Equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1, pp. 1-17
  3. Liu, F., et al., Numerical Solution of the Space Fractional Fokker-Planck Equation, Journal of Computational & Applied Mathematics, 166 (2004), 1, pp. 209-219
  4. Tarasov, V. E., Fractional Fokker-Planck Equation for Fractal Media, Chaos, 15 (2005), 2, pp. 461-478
  5. Deng, W. H., Finite Element Method for the Space and Time Fractional Fokker-Planck Equation, SIAM Journal on Numerical Analysis, 47 (2008), 1, pp. 204-226
  6. Chen, S., et al., Finite Difference Approximations for the Fractional Fokker-Planck Equation , Applied Mathematical Modelling, 33 (2009), 1, pp. 256-273
  7. Kolwankar, K. M., Gangal, A. D., Local Fractional Fokker-Planck Equation, Physics Review Letter, 80 (1998), 2, pp. 214-217
  8. Bologna, M., et al., Anomalous Diffusion Associated with Nonlinear Fractional Derivative Fokker-Planck-like equation: Exact time-dependent solutions, Phys. Rev. E, 62 (2000), 2, pp. 2213-2218
  9. He, J. H., A Tutorial Review on Fractal Space Time and Fractional Calculus, Int. J. Theor. Phys., 53 (2014), 11, pp. 3698-718
  10. Yıldırım, A., Analytical Approach to Fokker-Planck Equation with Space- and Time-Fractional Derivatives by Means of the Homotopy Perturbation Method, Journal of King Saud University - Science, 22 (2010), 4, pp. 257-264
  11. İbiş, B., Application of Fractional Variational Iteration Method for Solving Fractional Fokker-Planck equation, Romanian Journal of Physics, 60 (2015), 7-8, pp. 971-979
  12. Hahn, M. G., et al., Fokker-Planck-Kolmogorov Equations Associated with Time-Changed Fractional Brownian Motion, Proceedings of the American Mathematical Society, 139 (2011), 2, pp. 691-705
  13. He, J. H., A New Fractal Derivation, Thermal Science, 15 (2011), Suppl. 1, pp. S145-S147
  14. Fan, J., He, J. H., Fractal Derivative Model for Air Permeability in Hierarchic Porous Media, Abstract and Applied Analysis, 2012 (2012), ID 354701
  15. Liu, H. Y., et al., Fractional Calculus for Nanoscale Flow and Heat Transfer, International Journal of Numerical Methods for Heat and Fluid Flow, 24 (2014), 6, pp. 1227-1250
  16. Chen, W., et al., Anomalous Diffusion Modeling by Fractal and Fractional Derivatives, Computers and Mathematics with Applications, 59 (2010), 5, pp. 1754-1758
  17. Wang, M. R., et al., Three-Dimensional Effect on the Effective Thermal Conductivity of Porous Media, Journal of Physics D, 40 (2007), 1, pp. 260-265
  18. He, J. H., Fractal Calculus and Its Geometrical Explanation, Result in physics, 10 (2018), Sept., pp. 272-276
  19. Wang, Q. L., et al., Fractal Calculus and its Application to Explanation of Biomechanism of Polar Bear Hairs, Fractals, 26 (2018), 1850086
  20. Wang Y., Deng, Q. G., Fractal Derivative Model for Tsunami Travelling, Fractals, 27 (2019), 1, 1950017
  21. 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
  22. Baleanu, D., et al., A Modified Fractional Variational Iteration Method for Solving Nonlinear Gas Dynamic and Coupled KdV Equations Involving Local Fractional Operator, Thermal Science, 22 (2018), Suppl. 1, pp. S165-S175
  23. Durgun, D. D., Konuralp, A., Fractional Variational Iteration Method for Time-Fractional Nonlinear Functional Partial Differential Equation Having Proportional Delays, Thermal Science, 22 (2018), Suppl. 1, pp. S33-S46
  24. Yang, X. J., Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, 2012
  25. Li, Z. B., He, J. H., Fractional Complex Transform for Fractional Differential Equations, Math. Comput. Appl, 15 (2010), 5, pp. 970-973
  26. Ain, Q. T., He, J. H., On Two-Scale Dimension and its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712
  27. He, J. H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Sci-ence, 23 (2019), 4, pp. 2131-2133
  28. Bruckner, A. M., Bruckner, J. B., Darboux Transformations, Transactions of the American Mathematical Society, 128 (1967), 1, pp. 103-111
  29. Wang, Y., et al., A Variational Formulation for Anisotropic Wave Travelling in a Porous Medium, Fractals, 27 (2019), 4, 1950047
  30. Wang, K. L., He, C. H., A Remark on Wang's Fractal Variational Principle, Fractals, 27 (2019), 8, ID 1950134
  31. Wang, Q. L., et al., Fractal Calculus and its Application to Explanation of Biomechanism of Polar Bear Hairs Fractals, 26 (2018), 6, ID 1850086
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Fractional Fokker-Planck equation in a fractal medium

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