zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

26A33Fractional derivatives and integrals (real functions)
Full Text: DOI
[1] Oldham, K. B.; Spanier, J.: The fractional calculus. (1974) · Zbl 0292.26011
[2] Ross, B.: Proc. int. Conf.. (1974)
[3] Mcbride, A. C.: Fractional calculus and integral transforms of generalized functions. Pitman research notes in mathematics #31 (1979) · Zbl 0423.46029
[4] Mcbride, A. C.; Roach, G. F.: Proc. int. Workshop. (1984)
[5] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Engl. transl. From russian integrals and derivatives of fractional order and some of their applications. Integrals and derivatives of fractional order and some of their applications (1987) · Zbl 0617.26004
[6] Nishimoto, K.: Proc. int. Conf.. (1989)
[7] Nishimoto, K.: An essence of nishimoto’s fractional calculus. (1991) · Zbl 0798.26007
[8] Kalia, R. N.: Recent advances in fractional calculus. (1993) · Zbl 0790.26003
[9] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations. (1993) · Zbl 0789.26002
[10] Kiryakova, V.: Generalized fractional calculus and applications. Pitman research notes in mathematics #301 (1994)
[11] Caputo, M.: Elasticità e dissipazione. (1969)
[12] Babenko, Yu.I.: Heat and mass transfer. (1986)
[13] Davis, H. T.: The theory of linear operators. (1936) · Zbl 0015.35203
[14] Erdélyi, A.: Tables of integral transforms. 2 (1954) · Zbl 0055.36401
[15] Gorenflo, R.; Vessella, S.: Abel integral equations: analysis and applications. Lecture notes in mathematics #1461 (1991) · Zbl 0717.45002
[16] Srivastava, H. M.; Owa, S.: Univalent functions, fractional calculus, and their applications. (1989) · Zbl 0683.00012
[17] Rusev, P.; Dimovski, I.; Kiryakova, V.: Proc. int. Workshop. (12--17 August 1994)
[18] Mainardi, F.: Abstract in appl. Mech. rev.. Appl. mech. Rev. 46, 549 (1993) · Zbl 0921.35087
[19] Mainardi, F.: Fractional relaxation in anelastic solids. J. alloys compds 211/212, 534-538 (1994)
[20] Mainardi, F.: Proc. VII-th WASCOM. (4--7 October 1993)
[21] Mainardi, F.: Fractional relaxation and fractional diffusion equations. Proc. 12-th IMACS world congress 1, 329-332 (1994)
[22] Mainardi, F.; Tomirotti, M.: On a special function arising in the time fractional diffusionwave equation. Proc. int. Workshop, 171-183 (1995) · Zbl 0921.33010
[23] Podlubny, I.: The Laplace transform method for linear differential equations of fractional order. Preprint UEF-02-94 (1994)
[24] Podlubny, I.: Solutions of linear fractional differential equations. Proc. int. Workshop, 227-237 (1995) · Zbl 0918.34010
[25] Gorenflo, R.; Rutman, R.: On ultraslow and on intermediate processes. Proc. int. Workshop, 61-81 (1995) · Zbl 0923.34005
[26] Gel’fand, I. M.; Shilov, G. E.: Generalized functions. 1 (1964)
[27] Doetsch, G.: Introduction to the theory and application of the Laplace transformation. (1974) · Zbl 0278.44001
[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] Erdélyi, A.: Higher transcendental functions. 3 (1955) · Zbl 0064.06302
[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)
[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)
[37] Nonnenmacher, T. F.; Glöckle, W. G.: A fractional model for mechanical stress relaxation. Phil. mag. Letters 64, No. 2, 89-93 (1991)
[38] Nigmatullin, R. R.: The physics of fractional calculus and its realization on the fractal structures. Doctorate thesis (1992) · Zbl 0795.26007
[39] Stanković, B.: On the function of E. M. wright. Publ. inst. Math. beograd (Nouv. Serie) 10, No. 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, No. 34, 91-98 (1976) · Zbl 0343.33011
[41] Mikusiński, J.: On the function whose Laplace transform is exp(-$s{\alpha}{\lambda}$). 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.: Advanced mathematical methods for scientists and engineers. (1987) · Zbl 0417.34001
[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) · Zbl 0696.34047
[47] Kochubei, A. N.: Fractional order diffusion. J. diff. Eqns 26, 485-492 (1990) · Zbl 0729.35064
[48] Mainardi, F.; Buggisch, H.: Proc. IUTAM symposium. (22--28 August 1982)
[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)
[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.: LES géometries fractáles. (1990)
[53] Oustaloup, A.: La commande CRONE. (1990)
[54] Nonnenmacher, T. F.: Fractional integral and differential equations for a class of Lévy-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. Paper presented at the 3rd European conf. On the mathematics of oil recovery (1992)
[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