Weitzner, H.; Zaslavsky, G. M. Some applications of fractional equations. (English) Zbl 1041.35073 Commun. Nonlinear Sci. Numer. Simul. 8, No. 3-4, 273-281 (2003). The authors deal with the application of fractional equations in physics. They consider the kinetic equation with fractional derivatives \[ \frac {\partial P(x,t)} {\partial t'}=\frac {\partial^2P} {\partial x}^{\prime 2} +\varepsilon\frac {\partial^\alpha P} {\partial| x'|^\alpha}, \quad 1<\alpha<2,\tag{1} \] where \(\varepsilon\) is a constant, and \(\frac{\partial^\alpha P} {\partial| x'|^\alpha}\) is the Riesz derivatives. The authors study the competition between normal diffusion and diffusion induced by fractional derivatives for (1). It is shown that for large times the fractional derivative term dominates the solution and leads to power type tails. Moreover a corresponding fractional generalization of the Ginzburg-Landau and nonlinear Schrödinger equations is proposed. Reviewer: Messoud A. Efendiev (Berlin) Cited in 99 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 26A33 Fractional derivatives and integrals Keywords:application of fractional equations in physics; diffusion; fractional generalization of the Ginzburg-Landau equation and nonlinear Schrödinger equation × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Sapoval, B.; Gobron, Th.; Margolina, A., Phys. Rev. Lett., 67, 2974-2977, 1991 [2] Wyss, W., J. Math. Phys., 27, 2782-2785, 1986 · Zbl 0632.35031 [3] Zosimov, V. V.; Lyamshev, L. M., Physics-Uspekhi, 38, 347-385, 1995 [4] Montroll, E. W.; Shlesinger, M. S., 1-121 [5] Shlesinger, M. F.; Zaslavsky, G. M.; Klafter, J., Nature, 363, 31-37, 1993 [6] Nigmatullin, R. R., Phys. Stat. Solidi B, 133, 425-430, 1986 [7] Zaslavsky, G. M., 481-493 [8] Saichev, A. I.; Zaslavsky, G. M., Chaos, 7, 753-764, 1997 · Zbl 0933.37029 [9] Zaslavsky, G. M., Phys. Rep., 371, 461-580, 2002 · Zbl 0999.82053 [10] Zaslavsky, G. M.; Edelman, M., Chaos, 11, 295-305, 2001 · Zbl 1080.37584 [11] Bardos, C.; Peul, P.; Frisch, U.; Sulem, P. L., Arch. Rat. Mech. Anal., 71, 237-256, 1979 · Zbl 0421.35037 [12] Frisch, U.; Lesieur, M.; Brissand, A., J. Fluid Mech., 65, 145-152, 1974 · Zbl 0285.76021 [13] Biler, P.; Funaki, T.; Woyczynski, W. A., J. Dif. Eq., 148, 9-46, 1998 · Zbl 0911.35100 [14] Woyczynski, W. A., 241-266 · Zbl 0982.60043 [15] Lighthill, J., Waves in fluids, 1978, Cambridge University Press: Cambridge University Press Cambridge · Zbl 0375.76001 [16] Zaslavsky, G. M.; Abdullaev, S. S., Chaos, 7, 182-186, 1997 · Zbl 0938.37012 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.