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The two forms of fractional relaxation of distributed order. (English) Zbl 1165.26302
Summary: The first-order differential equation of exponential relaxation can be generalized by using either the fractional derivative in the Riemann-Liouville (R-L) sense and in the Caputo (C) sense, both of a single order less than 1. The two forms turn out to be equivalent. When, however, we use fractional derivatives of distributed order (between zero and 1), the equivalence is lost, in particular on the asymptotic behaviour of the fundamental solution at small and large times. We give an outline of the theory providing the general form of the solution in terms of an integral of Laplace type over a positive measure depending on the order-distribution. We consider in some detail two cases of fractional relaxation of distribution order: the double-order and the uniformly distributed order discussing the differences between the R-L and C approaches. For all the cases considered we give plots of the solutions for moderate and large times.

##### MSC:
 26A33 Fractional derivatives and integrals
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##### References:
  Abramowitz, M., Handbook of Mathematical Functions (1965)  Bagley, R.L., International Journal of Applied Mathematics 2 pp 865– (2000)  Caputo, M., Elasticità e Dissipazione (1969)  Caputo, M., Annali della Università di Ferrara (Sez VII, Science of Materials) 41 pp 73– (1995)  Caputo, M., Fractional Calculus and Applied Analysis 4 (4) pp 421– (2001)  Caputo, M., Rivista del Nuovo Cimento 1 pp 161– (1971) · doi:10.1007/BF02820620  Chechkin, A.V., Physical Review 66 pp 046129– (2002)  Chechkin, A.V., Fractional Calculus and Applied Analysis 6 pp 259– (2002)  Diethelm, K., Fractional Calculus and Applied Analysis 4 pp 531– (2001)  Diethelm K., Analysis 6 pp 243– (2004)  Erdélyi, A., Higher Transcendental Functions (1955) · Zbl 0064.06302  Feller, W., An Introduction to Probability Theory and its Applications, 2. ed. (1971) · Zbl 0219.60003  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, Wien, pp. 223–276. [Reprinted in http://www.fracalmo.org]  Gorenflo, R. and Mainardi, F., 2006, ”Fractional relaxation of distributed order,” Complexus Mundi: Emergent Patterns in Nature, M. Novak, ed. World Scientific, Singapore, 33–42. · Zbl 1167.26303  Hartley, T.T., Signal Processing 83 pp 2287– (2003) · Zbl 1145.93433 · doi:10.1016/S0165-1684(03)00182-8  Hilfer, R., 2000, ”Fractional time evolution”, in Applications of Fractional Calculus in Physics, R. Hilfer, ed. World Scientific, Singapore, 87–130. · Zbl 0994.34050 · doi:10.1142/9789812817747_0002  Kilbas, A.A., Theory and Applications of Fractional Differential Equations (2006) · Zbl 1138.26300  Langlands, T.A.M., Physica A 367 pp 136– (2006) · doi:10.1016/j.physa.2005.12.012  Lorenzo, C.F., Nonlinear Dynamics 29 pp 57– (2002) · Zbl 1018.93007 · doi:10.1023/A:1016586905654  Mainardi, F., Chaos, Solitons and Fractals 7 pp 1461– (1996) · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5  Mainardi, F., Sub-diffusion equations of fractional order and their fundamental solutions (2007) · Zbl 1135.35004 · doi:10.1007/978-1-4020-5678-9_3  Miller, K.S., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993) · Zbl 0789.26002  Miller, K.S., Integral Transforms and Special Functions 12 (4) pp 389– (2001) · Zbl 1035.26012 · doi:10.1080/10652460108819360  Nonnenmacher, T.F., Fractals 3 pp 557– (1995) · Zbl 0868.26004 · doi:10.1142/S0218348X95000497  Podlubny, I., Fractional Differential Equations (1999) · Zbl 0924.34008  Samko, S.G., Fractional Integrals and Derivatives: Theory and Applications (1993) · Zbl 0818.26003  Sokolov, I.M., Acta Physica Polonica 35 pp 1323– (2004)  West, B.J., Physics of Fractal Operators (2003) · doi:10.1007/978-0-387-21746-8  Zaslavsky, G.M., Hamiltonian Chaos and Fractional Dynamics (2005) · Zbl 1083.37002
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