THERMAL SCIENCE

International Scientific Journal

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
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 No. 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, 10.1080/713821438
[2] Jordan, R., et al., The Variational Formulation of the Fokker-Planck Equation, SIAM Journal on Mathematical Analysis, 29 (1998), 1, pp. 1-17, 10.1137/s0036141096303359
[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, 10.1016/j.cam.2003.09.028
[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, 10.1137/080714130
[6] Chen, S., et al., Finite Difference Approximations for the Fractional Fokker-Planck Equation , Applied Mathematical Modelling, 33 (2009), 1, pp. 256-273, 10.1016/j.apm.2007.11.005
[7] Kolwankar, K. M., Gangal, A. D., Local Fractional Fokker-Planck Equation, Physics Review Letter, 80 (1998), 2, pp. 214-217, 10.1103/physrevlett.80.214
[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, 10.1103/physreve.62.2213
[9] He, J. H., A Tutorial Review on Fractal Space Time and Fractional Calculus, Int. J. Theor. Phys., 53 (2014), 11, pp. 3698-718, 10.1007/s10773-014-2123-8
[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, 10.1016/j.jksus.2010.05.008
[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, 10.2298/tsci11s1145h
[14] Fan, J., He, J. H., Fractal Derivative Model for Air Permeability in Hierarchic Porous Media, Abstract and Applied Analysis, 2012 (2012), ID 354701, 10.1155/2012/354701
[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, 10.1108/hff-07-2013-0240
[16] Chen, W., et al., Anomalous Diffusion Modeling by Fractal and Fractional Derivatives, Computers and Mathematics with Applications, 59 (2010), 5, pp. 1754-1758, 10.1016/j.camwa.2009.08.020
[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, 10.1088/0022-3727/40/1/024
[18] He, J. H., Fractal Calculus and Its Geometrical Explanation, Result in physics, 10 (2018), Sept., pp. 272-276, 10.1016/j.rinp.2018.06.011
[19] Wang, Q. L., et al., Fractal Calculus and its Application to Explanation of Biomechanism of Polar Bear Hairs, Fractals, 26 (2018), 1850086, 10.1142/s0218348x1850086x
[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, 10.1016/j.jelechem.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, 10.2298/tsci170612269d
[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, 10.3390/mca15050970
[26] Ain, Q. T., He, J. H., On Two-Scale Dimension and its Applications, Thermal Science, 23 (2019), 3B, pp. 1707-1712, 10.2298/tsci190408138a
[27] He, J. H., Ji, F. Y., Two-Scale Mathematics and Fractional Calculus for Thermodynamics, Thermal Sci-ence, 23 (2019), 4, pp. 2131-2133, 10.2298/tsci1904131h
[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, 10.1142/s0218348x19501342
[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

© 2026 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