# zbMATH — the first resource for mathematics

Fractional relaxation-oscillation and fractional diffusion-wave phenomena. (English) Zbl 1080.26505
Summary: The processes involving the basic phenomena of relaxation, diffusion, oscillations and wave propagation are of great relevance in physics; from a mathematical point of view they are known to be governed by simple differential equations of order 1 and 2 in time. The introduction of fractional derivatives of order $$a$$ in time, with $$0 < a < 1$$ or $$1 < a < 2$$, leads to processes that, in mathematical physics, we may refer to as fractional phenomena. The objective of this paper is to provide a general description of such phenomena adopting a mathematical approach to the fractional calculus that is as simple as possible. The analysis carried out by the Laplace transform leads to certain special functions in one variable, which generalize in a straightforward way the characteristic functions of the basic phenomena, namely the exponential and the Gaussian.

##### MSC:
 26A33 Fractional derivatives and integrals
CRONE
Full Text:
##### References:
 [1] Oldham, K.B.; Spanier, J., () [2] () [3] McBride, A.C., Fractional calculus and integral transforms of generalized functions, () · Zbl 0423.46029 [4] () [5] Samko, S.G.; Kilbas, A.A.; Marichev, O.I.; Samko, S.G.; Kilbas, A.A.; Marichev, O.I., (), Engl. Transl. from Russian · Zbl 0818.26003 [6] () [7] Nishimoto, K., () [8] () [9] Miller, K.S.; Ross, B., () [10] Kiryakova, V., Generalized fractional calculus and applications, () · Zbl 1189.33034 [11] Caputo, M., (), (in Italian) [12] Babenko, Yu.I., (), (in Russian) [13] Davis, H.T., () [14] (), Chap. 13 [15] Gorenflo, R.; Vessella, S., Abel integral equations: analysis and applications, () · Zbl 0717.45002 [16] () [17] (), Sofia 1994 [18] Mainardi, F.; Mainardi, F.; Mainardi, F., Fractional diffusive waves in viscoelastic solids, (), Appl. mech. rev., 46, 549-97, (1993), Abstract in · Zbl 0879.35036 [19] Mainardi, F., Fractional relaxation in anelastic solids, J. alloys compds, 211/212, 534-538, (1994) [20] Mainardi, F.; Mainardi, F., On the initial value problem for the fractional diffusion-wave equation, (), 246-251 · Zbl 0948.60006 [21] Mainardi, F., Fractional relaxation and fractional diffusion equations, (), 329-332 [22] Mainardi, F.; Tomirotti, M., On a special function arising in the time fractional diffusionwave equation, (), 171-183 · Zbl 0921.33010 [23] Podlubny, I., The Laplace transform method for linear differential equations of fractional order, () · Zbl 0893.65051 [24] Podlubny, I., Solutions of linear fractional differential equations, (), 227-237 · Zbl 0918.34010 [25] Gorenflo, R.; Rutman, R., On ultraslow and on intermediate processes, (), 61-81 · Zbl 0923.34005 [26] Gel’fand, I.M.; Shilov, G.E., () [27] Doetsch, G., () [28] Caputo, M., Vibrations of an infinite viscoelastic layer with a dissipative memory, J. acoust. soc. am., 56, 897-904, (1974) · Zbl 0285.73031 [29] (), Chap. 18 [30] Gross, B., On creep and relaxation, J. appl. phys., 18, 212-221, (1947) [31] Caputo, M.; Mainardi, F., A new dissipation model based on memory mechanism, Pure appl. geophys., 91, 134-147, (1971) [32] Caputo, M.; Mainardi, F., Linear models of dissipation in anelastic solids, Riv. nuovo cimento (ser. II), 1, 161-198, (1971) [33] Nigmatullin, R.R., On the theory of relaxation with “remnant” memory, Phys. stat. sol. B, 124, 389-393, (1984), (English transl. from Russian) [34] Torvik, P.J.; Bagley, R.L., On the appearance of the fractional derivatives in the behavior of real materials, J. appl. mech. (trans. ASME), 51, 294-298, (1984) · Zbl 1203.74022 [35] Koeller, R.C., Applications of fractional calculus to the theory of viscoelasticity, J. appl. mech. (trans. ASME), 51, 299-307, (1984) · Zbl 0544.73052 [36] Mainardi, F.; Bonetti, E., The application of real-order derivatives in linear viscoelasticity, Rheol. acta, 26, 64-67, (1988), Suppl. [37] Nonnenmacher, T.F.; Glöckle, W.G., A fractional model for mechanical stress relaxation, Phil. mag. letters, 64, 2, 89-93, (1991) [38] Nigmatullin, R.R., The physics of fractional calculus and its realization on the fractal structures, (), (in Russian) · Zbl 0795.26007 [39] Stanković, B., On the function of E. M. wright, Publ. inst. math. beograd (nouv. serie), 10, 24, 113-124, (1970) · Zbl 0204.08404 [40] Gajić, Lj.; Stanković, B., Some properties of Wright’s function, Publ. inst. math. beograd (nouv. serie), 20, 34, 91-98, (1976) · Zbl 0343.33011 [41] Mikusiński, J., On the function whose Laplace transform is exp(−sαλ), Studia math., 18, 191-198, (1959) · Zbl 0087.10501 [42] Buchen, P.W.; Mainardi, F., Asymptotic expansions for transient viscoelastic waves, J. Mécaniq., 14, 597-608, (1975) · Zbl 0351.73033 [43] Bender, C.M.; Orszag, S.A., (), Chap. 3 [44] Wyss, W., Fractional diffusion equation, J. math. phys., 27, 2782-2785, (1986) · Zbl 0632.35031 [45] Schneider, W.R.; Wyss, W., Fractional diffusion and wave equations, J. math. phys., 30, 134-144, (1989) · Zbl 0692.45004 [46] Kochubei, A.N., A Cauchy problem for evolution equations of fractional order, J. diff. eqns, 25, 967-974, (1989), (English transl. from Russian) · Zbl 0696.34047 [47] Kochubei, A.N., Fractional order diffusion, J. diff. eqns, 26, 485-492, (1990), (English transl. from Russian) · Zbl 0729.35064 [48] Mainardi, F.; Buggisch, H.; Mainardi, F.; Buggisch, H., On nonlinear waves in liquid-filled elastic tubes, (), 87-100 · Zbl 0505.76140 [49] Nigmatullin, R.R., The realization of the generalized transfer equation in a medium with fractal geometry, Phys. stat. sol. B, 133, 425-430, (1986), (English transl. from Russian) [50] Young, W.R.; Pumir, A.; Pomeau, Y., Anomalous diffusion of tracer in convection rolls, Phys. fluids A, 1, 462-469, (1989) · Zbl 0659.76097 [51] Choi, U.J.; MacCamy, R.C., Fractional order Volterra equations with applications to elasticity, J. math. anal. applics, 139, 448-464, (1989) · Zbl 0674.45007 [52] LeMéhauté, A., () [53] Oustaloup, A., () [54] Nonnenmacher, T.F., Fractional integral and differential equations for a class of levy-type probability densities, J. phys A: math. gen., 23, L697-L700, (1990) [55] Sugimoto, N., Burgers equation with a fractional derivative; hereditary effects of nonlinear acoustic waves, J. fluid mech., 225, 631-653, (1991) · Zbl 0721.76011 [56] Giona, M.; Roman, H.E., Fractional diffusion equation for transport phenomena in random media, Physica A, 185, 82-97, (1992) [57] Lenormand, R., Use of fractional derivatives for fluid flow in heterogeneous media, () · Zbl 0738.76074 [58] Ochmann, M.; Makarov, S., Representation of the absorption of nonlinear waves by fractional derivatives, J. acoust. soc. am., 94, 3392-3399, (1993) [59] Caputo, M., The splitting of the seismic rays due to dispersion in the Earth’s interior, Rend. fis. acc. lincei (ser. IX), 4, 279-286, (1993) [60] Zaslavasky, G.M., Fractional kinetic equation for Hamiltonian chaos, Physica D, 76, 110-122, (1994) · Zbl 1194.37163
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.