×

Time-fractional diffusion of distributed order. (English) Zbl 1229.35118

Summary: The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville and the Caputo sense. For a general distribution of time orders we provide the fundamental solution, which is a probability density, in terms of an integral of Laplace type. The kernel depends on the type of the assumed fractional derivative, except for the single order case where the two approaches turn out to be equivalent. We consider in some detail two cases of order distribution: Double-order, and uniformly distributed order. Plots of the corresponding fundamental solutions and their variance are provided for these cases, pointing out the remarkable difference between the two approaches for small and large times.

MSC:

35K57 Reaction-diffusion equations
26A33 Fractional derivatives and integrals
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anh, V.V., Journal of Statistical Physics 104 (5) pp 1349– (2001) · Zbl 1034.82044
[2] Abramowitz, M., Handbook of Mathematical Functions (1965)
[3] Bagley, R.L., International Journal ofApplied Mathematics 2 pp 865– (2000)
[4] Bagley, R.L., International Journal of Applied Mathematics 2 pp 965– (2000)
[5] Caputo, M., Geophysical Journal of the Royal Astronomical Society 13 pp 529– (1967)
[6] Caputo, M., Elasticità e Dissipazione, (in Italian) (1969)
[7] Caputo, M., Scienze Mathematiche 41 pp 73– (1995)
[8] Caputo, M., Fractional Calculus and Applied Analysis 4 (4) pp 421– (2001)
[9] Caputo, M., Rivista del Nuovo Cimento (Series II) 1 pp 161– (1971)
[10] Chechkin, A.V., Physical Review E 66 pp 046129– (2002)
[11] Chechkin, A.V., Fractional Calculus and Applied Analysis 6 pp 259– (2003)
[12] Chechkin, A.V., Europhysics Letters 63 pp 326– (2003)
[13] Diethelm, K., Fractional Calculus and Applied Analysis 4 pp 531– (2001)
[14] Diethelm, K., Computational Methods of Numerical Analysis 6 pp 243– (2004)
[15] Djrbashian, M.M., Integral Transforms and Representations of Functions in the Complex Plane, (in Russian) (1966)
[16] Eidelman, S.D., Journal of Differential Equations 199 pp 211– (2004) · Zbl 1068.35037
[17] Erdélyi, A., McGraw-Hill, New York 3 pp 206– (1955)
[18] Feller, W., 1971, An Introduction to Probability Theory and its Applications , 2nd Edition, Wiley, New York, Vol. 2. · Zbl 0219.60003
[19] Gelfand, I.M., Generalized Functions (1964)
[20] Ghizzetti, A., Trasformate di Laplace e Calcolo Simbolico (1971)
[21] Gorenflo, R. and Mainardi, F., 1997, ”Fractional calculus: Integral and differential equations of fractional order,” in Fractals and Fractional Calculus in Continuum Mechanics , A. Carpinteri and F. Mainardi, eds. Springer Verlag, Vienna, pp. 223-276. http://www.fracalmo.org
[22] Gorenflo, R., Proceedings of the International Workshop on Chaotic Transport and Complexity in Fluids and Plasmas
[23] Gorenflo, R., Fractional Calculus and Applied Analysis 3 pp 75– (2000)
[24] Gorenflo, R., Fractional Calculus and Applied Analysis 2 pp 383– (1999)
[25] Gorenflo, R., Journal of Computational and Applied Mathematics 118 pp 175– (2000) · Zbl 0973.35012
[26] Hanyga, A., Proceedings of the Royal Society A 458 pp 933– (2002) · Zbl 1153.35347
[27] Hartley, T.T., Signal Processing 83 pp 2287– (2003) · Zbl 1145.93433
[28] DOI: 10.1142/3779
[29] Kilbas, A.A., Theory and Applications of Fractional Differential Equations (2006) · Zbl 1138.26300
[30] Kochubei, A.N., Differential Equations 26 pp 485– (1990)
[31] Langlands, T.A.M., Physica A 367 pp 136– (2006)
[32] Lorenzo, C.F., Nonlinear Dynamics 29 pp 57– (2002) · Zbl 1018.93007
[33] Magin, R.L., 2006, Fractional Calculus in Bioengineering, Begell House, Redding, CT, p. 684.
[34] Mainardi, F., 1993, ”On the initial value problem for the fractional diffusion-wave equation,” in Waves and Stability in Continuous Media, S. Rionero and T. Ruggeri, eds. World Scientific, Singapore, pp. 246-251.
[35] Mainardi, F., Solitons and Fractals 7 pp 1461– (1996) · Zbl 1080.26505
[36] Mainardi, F., 1997, ”Fractional calculus: Some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, eds. CISM Courses and Lecture Notes, Vol. 378, Springer Verlag , Berlin, pp. 291-348. http://www.fracalmo.org · Zbl 0917.73004
[37] Mainardi, F., Journal of Computational and Applied Mathematics 118 pp 283– (2000) · Zbl 0970.45005
[38] Mainardi, F., Applied Mathematics and Computation 141 (1) pp 51– (2003) · Zbl 1053.35008
[39] Mainardi, F., Journal of Computational andAppliedMathematics 207 (2) pp 245– (2007) · Zbl 1120.35002
[40] Mainardi, F., Fractional Calculus andAppliedAnalysis 4 pp 153– (2001)
[41] Mainardi, F., Mura, A., Pagnini, G., and Gorenflo, R., 2006a, ”Sub-diffusion equations of fractional order and their fundamental solutions,” in Mathematical Methods in Engineering, K. Tas and D. Baleanu , eds. Springer Verlag, Berlin , pp. 20-48. · Zbl 1135.35004
[42] Mainardi, F., Applied Mathematics and Computation 187 (1) pp 295– (2006) · Zbl 1122.26004
[43] Mainardi, F., Journal of Computational and Applied Mathematics 178 (1) pp 321– (2005) · Zbl 1061.33012
[44] Metzler, R., Physics Reports 339 pp 1– (2000) · Zbl 0984.82032
[45] Metzler, R., Physica A 211 pp 13– (1994)
[46] Miller, K.S., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) · Zbl 0789.26002
[47] Naber, M., Fractals 12 (1) pp 23– (2004) · Zbl 1083.60066
[48] Podlubny, I., Fractional Differential Equations (1999) · Zbl 0924.34008
[49] Saichev, A., Chaos 7 pp 753– (1997) · Zbl 0933.37029
[50] Samko, S.G., Fractional Integrals and Derivatives: Theory and Applications (1993) · Zbl 0818.26003
[51] Schneider, W.R., Journal of Mathematical Physics 30 pp 134– (1989) · Zbl 0692.45004
[52] Sokolov, I.M., Chaos 15 pp 026103– (2005) · Zbl 1080.82022
[53] Sokolov, I.M., Acta Physica Polonica 35 pp 1323– (2004)
[54] Umarov, S., Zeitschrift für Analysis und ihre Anwendungen 24 (3) pp 449– (2005)
[55] West, B.J., Physics of Fractal Operators (2003)
[56] Zaslavsky, G.M., Physics Reports 371 pp 461– (2002) · Zbl 0999.82053
[57] Zaslavsky, G.M., Hamiltonian Chaos and Fractional Dynamics (2005) · Zbl 1083.37002
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.