Geodesic
Stationary Solutions of the Fractional Kinetic Equation with a Symmetric Power-Law Potential
Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 1, pp. 162-176 
V. Yu. Gonchar; L. V. Tanatarov; A. V. Chechkin. Stationary Solutions of the Fractional Kinetic Equation with a Symmetric Power-Law Potential. Teoretičeskaâ i matematičeskaâ fizika, Tome 131 (2002) no. 1, pp. 162-176. http://geodesic.mathdoc.fr/item/TMF_2002_131_1_a13/
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     title = {Stationary {Solutions} of the {Fractional} {Kinetic} {Equation} with {a~Symmetric} {Power-Law} {Potential}},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The properties of stationary solutions of the one-dimensional fractional Einstein–Smoluchowski equation with a potential of the form $x^{2m+2}$, $m=1,2,\dots$, and of the Riesz spatial fractional derivative of order $\alpha$, $1\leq\alpha\leq2$ are studied analytically and numerically. We show that for $1\leq\alpha<2$, the stationary distribution functions have power-law asymptotic approximations decreasing as $x^{-(\alpha+2m+1)}$ for large values of the argument. We also show that these distributions are bimodal.

[1] R. Metzler, J. Klafter, Phys. Rep., 339 (2000), 1 | DOI | MR | Zbl

[2] G. M. Zaslavsky, M. Edelman, B. A. Niyazov, Chaos, 7 (1997), 753 | DOI | MR | Zbl

[3] F. Mainardi, Chaos, Solitons and Fractals, 7 (1996), 1461 | DOI | MR | Zbl

[4] E. Barkai, R. Metzler, J. Klafter, Phys. Rev. E, 61 (2000), 132 | DOI | MR

[5] S. Chandrasekar, Stokhasticheskie problemy v fizike i astronomii, IL, M., 1947

[6] A. V. Chechkin, V. Yu. Gonchar, ZhETF, 118 (2000), 730

[7] A. I. Saichev, G. M. Zaslavsky, Chaos, 7 (1997), 753 | DOI | MR | Zbl

[8] S. G. Samko, A. A. Kilbas, O. I. Marichev, Drobnye integraly i proizvodnye, Nauka i tekhnika, Minsk, 1987 | MR | Zbl

[9] S. Jespersen, R. Metzler, H. C. Fogedby, Phys. Rev. E, 59 (1999), 2736 | DOI | MR

[10] E. Kamke, Spravochnik po obyknovennym differentsialnym uravneniyam, Nauka, M., 1965 | MR

[11] V. I. Smirnov, Kurs vysshei matematiki, T. 3. Ch. 2, Gos. izd-vo tekhniko-teoreticheskoi l-ry, L.–M., 1949 | MR

[12] E. Titchmarsh, Vvedenie v teoriyu integralov Fure, Gostekhizdat, M., 1948

[13] G. Khardi, Raskhodyaschiesya ryady, IL, M., 1951

[14] A. N. Malakhov, Fluktuatsii v avtokolebatelnykh sistemakh, Nauka, M., 1968

[15] B. V. Gnedenko, A. N. Kolmogorov, Predelnye raspredeleniya dlya summ nezavisimykh sluchainykh velichin, Gos. izd-vo tekhniko-teoreticheskoi l-ry, M., 1949 | MR

[16] A. V. Chechkin, V. Yu. Gonchar, Chaos, Solitons and Fractals, 11 (2000), 2379 | DOI | Zbl

[17] A. V. Chechkin, V. Yu. Gonchar, Physica A, 27 (2000), 312 | DOI