Abstract
A relationship is established between Cantor's fractal set (Cantor's bars) and a fractional integral. The fractal dimension of the Cantor set is equal to the fractional exponent of the integral. It follows from analysis of the results that equations in fractional derivatives describe the evolution of physical systems with loss, the fractional exponent of the derivative being a measure of the fraction of the states of the system that are preserved during evolution timet. Such systems can be classified as systems with “residual” memory, and they occupy an intermediate position between systems with complete memory, on the one hand, and Markov systems, on the other. The use of such equations to describe transport and relaxation processes is discussed. Some generalizations that extent the domain of applicability of the fractional derivative concept are obtained.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Mandelbrot,Fractal Geometry of Nature, Freeman, San-Francisco (1983).
Fractals in Physics. Proceedings of Sixth International Symposium on Fractals in Physics [Russian translation], Mir, Moscow (1988).
T. Pajkossy and L. Nyikos,Electrokhim. Acta,34, 171 (1989).
T. Kaplan, L. J. Gray, and S. H. Lin,Phys. Rev. B,35, 5379 (1987).
B. Sapoval,Silid State Ionics,23, 253 (1987).
A. Le Mehaute, A. Gnibert, M. Delaye, and C. Filippi,C. R. Acad. Sci. Ser. II,294, 835 (1982).
A. Le Mehaute and G. Crepy,Solid State Ionics,9/10, 359 (1983).
A. Le Mehaute and A. Dugast,J. Power Sources,9, 359 (1983).
R. R. Nigmatullin,Phys. Status Solidi B,123, 739 (1984).
R. R. Nigmatullin,Phys. Status Solidi B,124, 389 (1984).
K. Oldham and J. Spanier,Fractional Calculus, Academic Press, New York (1973).
S. G. Samko, A. A. Kilbas, and O. I. Marichev,Integrals and Derivatives of Fractional Order and Some Applications of Them [in Russian], Nauka i Tekhnika, Minsk (1987).
Yu. I. Babenko,Heat and Mass Transfer. A Method of Calculating and Diffusion Fluxes [in Russian]., Khimiya, Leningrad (1986).
R. Sh. Nigmatullin and B. A. Belavin,Tr. KAI,82, 58 (1964).
R. R. Nigmatullin,Phys. Status Solidi B,133, 425 (1986).
L. A. Dissado, R. R. Nigmatullin, and R. M. Hill, in:Dynamical Processes in Condensed Matter (ed. M. Evans),63, 253 (1985).
R. R. Nigmatullin,Fiz Tverd. Tela (Leningrad),27, 1583 (1985).
A. K. Jonscher,Dielectric Relaxation in Solids, Chelsea Dielectric Press, London (1983).
W. F. Brown (Jr.),Dielectrics, in:Handbuch der Physik, Vol. 17, Springer Verlag, Berlin (1956).
D. N. Zubarev,Nonequilibrium Statistical Thermodynamics Plenum, New York (1974).
Additional information
Kazan State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 90, No. 3, pp. 354–368, March, 1992.
Rights and permissions
About this article
Cite this article
Nigmatullin, R.R. Fractional integral and its physical interpretation. Theor Math Phys 90, 242–251 (1992). https://doi.org/10.1007/BF01036529
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01036529