THERMAL SCIENCE

International Scientific Journal

Xiao-Jun YangORCID

,

H M SrivastavaORCID

,

Delfim F. M. TorresORCID

,

Amar DebboucheORCID

GENERAL FRACTIONAL-ORDER ANOMALOUS DIFFUSION WITH NON-SINGULAR POWER-LAW KERNEL

ABSTRACT
In this paper, we investigate general fractional derivatives with a non-singular power-law kernel. The anomalous diffusion models with non-singular power-law kernel are discussed in detail. The results are efficient for modelling the anomalous behaviors within the frameworks of the Riemann-Liouville and Liouville-Caputo general fractional derivatives.
KEYWORDS
general fractional derivative with nonsingular power-law kernel, Riemann-Liouville general fractional derivative, Liouville-Caputo general fractional derivative, anomalous diffusion
PAPER SUBMITTED: 2017-06-10
PAPER REVISED: 2017-06-27
PAPER ACCEPTED: 2017-06-28
PUBLISHED ONLINE: 2017-09-09
DOI REFERENCE: https://doi.org/10.2298/TSCI170610193Y
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2017, VOLUME 21, ISSUE Supplement, PAGES [1 - 9]
REFERENCES
[1] Samko, S. G., et al., Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, 1993
[2] Yang, X. J., et al., Local Fractional Integral Transforms and Their Applications, Academic Press, New York, USA, 2005, 10.1016/c2014-0-04768-5
[3] Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models,World Scientific, Singapore, 2010, 10.1142/p614#t=toc
[4] Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, New York, 2006, 10.1016/s0304-0208(06)x8001-5
[5] Solomon, T. H., et al., Observation of Anomalous Diffusion and Lévy Flights in a Two-dimensional Rotating Flow, Physical Review Letters, 71 (1993), 24, pp. 3975, 10.1103/physrevlett.71.3975
[6] Klafter, J., et al., Stochastic Pathway to Anomalous Diffusion, Physical Review A, 35 (1987), 7, pp. 3081, 10.1103/physreva.35.3081
[7] Bouchaud, J. P., et al., Anomalous Diffusion in Disordered Media: Statistical Mechanisms, Models and Physical Applications, Physics Reports, 195 (1990), (4-5), pp. 127-293, 10.1016/0370-1573(90)90099-n
[8] Tsallis, C., Bukman, D. J., Anomalous Diffusion in the Presence of External Forces: Exact Time-dependent Solutions and Their Thermostatistical Basis, Physical Review E, 54 (1996), 3, R2197, 10.1103/physreve.54.r2197
[9] Metzler, R., et al., Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach, Physical review letters, 82 (1999), 18, pp. 3563, 10.1103/physrevlett.82.3563
[10] Piryatinska, A., et al., Models of Anomalous Diffusion: The Subdiffusive Case, Physica A: Statistical Mechanics and its Applications, 349 (2005), 3, pp. 375-420, 10.1016/j.physa.2004.11.003
[11] Yang, X. J., et al., Anomalous Diffusion Models with General Fractional Derivatives within the Kernels of the Extended Mittag-Leffler Type Functions, Romanian Reports in Physics, 2017, in press
[12] Yang, X. J., New General Fractional-order Rheological Models within Kernels of Mittag-Leffler Functions, Romanian Reports in Physics, 2017, in press
[13] Yang, X. J., New Rheological Problems Involving General Fractional Derivatives within Nonsingular Power-law Kernel, Proceedings of the Romanian Academy - Series A, 2017, in press
[14] Gao, F., General Fractional Calculus in Nonsingular Power-law Kernel Applied to Model Anomalous Diffusion Phenomena in Heat-Transfer Problems, Thermal Science, 21 (2017), Suppl.3, in press, 10.2298/tsci170310194g
[15] Atangana, A., et al., New Fractional Derivatives with Nonlocal and Non-singular Kernel: Theory and Application to Heat Transfer Model, Thermal Science, 20(2016), 2, 763-769., 10.2298/tsci160111018a
[16] Caputo, M., et al., A New Definition of Fractional Derivative without Singular Kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 2, pp. 73-85
[17] Lozada, J., et al., Properties of a New Fractional Derivative without Singular Kernel, Progress in Fractional Differentiation and Applications, 1 (2015), 2, pp. 87-92
[18] Yang, X. J., et al., Some New Applications for Heat and Fluid Flows Via Fractional Derivatives without Singular Kernel, Thermal Science, 20 (2016), S3, pp. 833-839, 10.2298/tsci16s3833y
[19] Gao, F., et al., Fractional Maxwell Fluid with Fractional Derivative without Singular Kernel, Thermal Science, 20 (2016), pp. 871-877, 10.2298/tsci16s3871g
[20] Yang, X. J., et al., A New Fractional Derivative without Singular Kernel: Application to the Modelling of the Steady Heat Flow, Thermal Science, 20 (2016), 2, pp. 753-756, 10.2298/tsci151224222y
[21] 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, 10.2298/tsci161216326y
[22] Yang, A. M., et al., On Steady Heat Flow Problem Involving Yang-Srivastava-Machado Fractional Derivative without Singular Kernel, Thermal Science, 20 (2016), Suppl.3, pp. 717-721, 10.2298/tsci16s3717y
PDF VERSION [DOWNLOAD]
General fractional-order anomalous diffusion with non-singular power-law kernel

© 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