## Langevin equation with two fractional orders.(English)Zbl 1225.82049

Summary: A new type of fractional Langevin equation of two different orders is introduced. The solutions for this equation, known as the fractional Ornstein-Uhlenbeck processes, based on Weyl and Riemann-Liouville fractional derivatives are obtained. The basic properties of these processes are studied. An example of the spectral density of ocean wind speed which has similar spectral density as that of Weyl fractional Ornstein-Uhlenbeck process is given.

### MSC:

 82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics 28A80 Fractals 26A33 Fractional derivatives and integrals 34K37 Functional-differential equations with fractional derivatives
Full Text:

### References:

 [1] () [2] Mazo, R., Brownian motion: fluctuations, dynamics and applications, (2002), Oxford Univ. Press Oxford · Zbl 1140.60001 [3] Coffey, W.T.; Kalmykov, Yu.P.; Waldron, J.T., The Langevin equation, (2004), World Scientific Singapore · Zbl 0952.82510 [4] Wang, K.G., Phys. rev. A, 45, 833, (1992) [5] Porra, J.M.; Wang, K.G.; Masoliver, J., Phys. rev. E, 53, 5872, (1996) [6] Wang, K.G.; Tokuyama, M., Physica A, 265, 341, (1999) [7] Lutz, E., Phys. rev. E, 64, 051106, (2001) [8] Fa, K.S., Phys. rev. E, 73, 061104, (2006) [9] Fa, K.S., Eur. phys. J. E, 24, 139, (2007) [10] Kobolev, V.; Romanov, E., Prog. theor. phys. suppl., 139, 470, (2000) [11] Lim, S.C.; Muniandy, S.V., Phys. rev. E, 66, 021114, (2002) [12] Picozzi, S.; West, B., Phys. rev. E, 66, 046118, (2002) [13] Lim, S.C.; Eab, C.H., Phys. lett. A, 335, 87, (2006) [14] Lim, S.C.; Li, M.; Teo, L.P., Fluc. noise lett., 7, L169, (2007) [15] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley and Sons New York · Zbl 0789.26002 [16] Samko, S.; Kilbas, A.A.; Maritchev, D.I., Integrals and derivatives of the fractional order and some of their applications, (1993), Gordon and Breach Amsterdam [17] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [18] West, B.J.; Bologna, M.; Grigolini, P., Physics of fractal operators, (2003), Springer New York [19] Metzler, R.; Klafter, J., J. phys. A, 37, R161, (2004) [20] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Amsterdam · Zbl 1092.45003 [21] Samko, S., Hypersingular integrals and their applications, (2002), Taylor & Francis London · Zbl 0998.42010 [22] Gay, R.; Heyde, C.C., Biometrica, 77, 401, (1990) [23] Leonenko, N., Limit theorems for random fields with singular spectrum, (1999), Kluwer Dordrecht · Zbl 0963.60048 [24] Barci, B.G.; Oxman, L.E.; Rocca, M., Int. J. mod. phys. A, 11, 2111, (1996) [25] Lim, S.C.; Muniandy, S.V., Phys. lett. A, 324, 396, (2004) [26] Kubo, R., Rep. prog. phys., 29, 255, (1966) [27] Grote, R.F.; Hynes, J.T., J. chem. phys., 73, 2715, (1980) [28] Hänggi, P.; Talkner, P.; Borkovec, M., Rev. mod. phys., 62, 251, (1990) [29] Lim, S.C.; Teo, L.P. [30] Gradshteyn, I.S.; Ryzhik, I.M., Tables of integrals, series and products, (1994), Academic Press San Diego · Zbl 0918.65002 [31] Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., Higher transcendental functions, vol. I, (1955), McGraw-Hill New York · Zbl 0064.06302 [32] Lim, S.C.; Sithi, V.M., Phys. lett. A, 206, 311, (1995) [33] Mandelbrot, B.B.; van Ness, J.W., SIAM rev., 10, 422, (1968) [34] () [35] Benassi, A.; Jaffard, S.; Roux, D., Rev. mat. iberoamericana, 13, 19, (1997) [36] Benassi, A.; Cohen, S.; Istas, J., C. R. acad. sci. Paris, math., 336, 267, (2003) [37] Adler, A.J., Geometry of random fields, (1981), Wiley New York · Zbl 0478.60059 [38] Chakrabarti, S.K., Offshore structure modeling, (1994), World Scientific Singapore · Zbl 0867.76003 [39] Li, G.; Li, Q., Theory of time-varying reliability for engineering structures and applications, (2001), Science Press Beijing, (in Chinese) [40] J.-W. Shen, Structure Navigability Test for 051 Ship in South China Sea, Technical Report, China Ship Scientific Research Center, 1976 (in Chinese) [41] Von Kármán, Proc. natl. acad. sci., 34, 530, (1948) [42] Davenport, A.G., Quart. J. R. meteor. soc., 87, 372, 194, (1961) [43] Kaimal, J.C.; Wyngaard, J.C.; Izumi, Y.; Coté, O.R., Quart. J. R. meteor. soc., 98, 563, (1972) [44] Antoniou, I.; Asimakopoulos, D.; Fragoulis, A.; Kotronaros, A.; Lalas, D.P.; Panourgias, I., J. wind eng. ind. aerod., 39, 343, (1992) [45] Simiu, E.; Scanlan, R.H., Wind effects on structure, (1996), John Willy and Sons New York [46] Goedecke, G.H.; Ostashev, V.E.; Wilson, D.K.; Auvermann, H.J., Boundary-layer meteorol., 112, 33, (2004) [47] J.-J. Hu, Report on Field Measurement Analysis for Navigability Test of 051B (167) Ship-Stress Measurement of Hull, Technical Report, China Ship Scientific Research Center, 2000 (in Chinese)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.