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 |
Keywords:
fractional Langevin equation; Weyl and Riemann-Liouville fractional derivatives; short range dependence; fractal dimensionReferences:
| [1] | (Wax, N., Selected Papers on Noise and Stochastic Processes (1954), Dover: Dover New York) · Zbl 0059.11903 |
| [2] | Mazo, R., Brownian Motion: Fluctuations, Dynamics and Applications (2002), Oxford Univ. Press: Oxford Univ. Press Oxford · Zbl 1140.60001 |
| [3] | Coffey, W. T.; Kalmykov, Yu. P.; Waldron, J. T., The Langevin Equation (2004), World Scientific: 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: 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: Gordon and Breach Amsterdam · Zbl 0818.26003 |
| [17] | Podlubny, I., Fractional Differential Equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010 |
| [18] | West, B. J.; Bologna, M.; Grigolini, P., Physics of Fractal Operators (2003), Springer: 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: Elsevier Amsterdam · Zbl 1092.45003 |
| [21] | Samko, S., Hypersingular Integrals and Their Applications (2002), Taylor & Francis: Taylor & Francis London · Zbl 0998.42010 |
| [22] | Gay, R.; Heyde, C. C., Biometrica, 77, 401 (1990) · Zbl 0711.62086 |
| [23] | Leonenko, N., Limit Theorems for Random Fields with Singular Spectrum (1999), Kluwer: Kluwer Dordrecht · Zbl 0963.60048 |
| [24] | Barci, B. G.; Oxman, L. E.; Rocca, M., Int. J. Mod. Phys. A, 11, 2111 (1996) · Zbl 0985.81576 |
| [25] | Lim, S. C.; Muniandy, S. V., Phys. Lett. A, 324, 396 (2004) · Zbl 1123.81376 |
| [26] | Kubo, R., Rep. Prog. Phys., 29, 255 (1966) · Zbl 0163.23102 |
| [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: 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: McGraw-Hill New York · Zbl 0064.06302 |
| [32] | Lim, S. C.; Sithi, V. M., Phys. Lett. A, 206, 311 (1995) · Zbl 1020.60501 |
| [33] | Mandelbrot, B. B.; van Ness, J. W., SIAM Rev., 10, 422 (1968) · Zbl 0179.47801 |
| [34] | (Rangarajan, G.; Ding, M., Processes with Long-Range Correlations, Theory and Applications (2003), Springer-Verlag: Springer-Verlag New York) |
| [35] | Benassi, A.; Jaffard, S.; Roux, D., Rev. Mat. Iberoamericana, 13, 19 (1997) · Zbl 0880.60053 |
| [36] | Benassi, A.; Cohen, S.; Istas, J., C. R. Acad. Sci. Paris, Math., 336, 267 (2003) · Zbl 1023.60043 |
| [37] | Adler, A. J., Geometry of Random Fields (1981), Wiley: Wiley New York · Zbl 0478.60059 |
| [38] | Chakrabarti, S. K., Offshore Structure Modeling (1994), World Scientific: World Scientific Singapore · Zbl 0867.76003 |
| [39] | Li, G.; Li, Q., Theory of Time-varying Reliability for Engineering Structures and Applications (2001), Science Press: 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); 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) · Zbl 0032.22601 |
| [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: 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); 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. 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.
