Fractional relaxation-oscillation and fractional diffusion-wave phenomena

Chaos, Solitons and Fractals

Volumn 7, Issue 9, 1996, Pages 1461-1477

  Mainardi, Francesco ^a  

^a UNIVERSITY OF BOLOGNA   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

MODELLING-MATHEMATICAL; WAVE PROPAGATION;

EID: 0030464353     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/0960-0779(95)00125-5     Document Type: Article

References (69)

1. K. B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York (1974).
2. B. Ross (Ed.), Fractional Calculus and its Applications, Lecture Notes in Mathematics #457, Springer Verlag, Berlin (1975) (Proc. Int. Conf. held at Univ. of New Haven, USA, 1974).
3. B. Ross (Ed.), Fractional Calculus and its Applications, Lecture Notes in Mathematics #457, Springer Verlag, Berlin (1975) (Proc. Int. Conf. held at Univ. of New Haven, USA, 1974).
4. A. C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions, Pitman Research Notes in Mathematics #31. Pitman, London (1979).
5. A. C. McBride and G. F. Roach (Eds), Fractional Calculus, Pitman Research Notes in Mathematics #138. Pitman, London (1985) (Proc. Int. Workshop, held at Univ. of Strathclyde, UK, 1984).
6. A. C. McBride and G. F. Roach (Eds), Fractional Calculus, Pitman Research Notes in Mathematics #138. Pitman, London (1985) (Proc. Int. Workshop, held at Univ. of Strathclyde, UK, 1984).
7. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam (1993) (Engl. Transl. from Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka i Tekhnika, Minsk (1987)).
8. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Amsterdam (1993) (Engl. Transl. from Russian, Integrals and Derivatives of Fractional Order and Some of Their Applications, Nauka i Tekhnika, Minsk (1987)).
9. K. Nishimoto (Ed.), Fractional Calculus and its Applications, Nihon University, Tokyo (1990) (Proc. Int. Conf. held at Nihon Univ., Tokyo, Japan, 1989).
10. K. Nishimoto (Ed.), Fractional Calculus and its Applications, Nihon University, Tokyo (1990) (Proc. Int. Conf. held at Nihon Univ., Tokyo, Japan, 1989).
11. K. Nishimoto, An Essence of Nishimoto's Fractional Calculus. Descartes Press, Koriyama (1991).
12. R. N. Kalia (Ed.), Recent Advances in Fractional Calculus. Global, Sauk Rapids, Minnesota (1993).
13. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993).
14. V. Kiryakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Mathematics #301. Longman, Harlow (1994).
15. M. Caputo, Elasticità e Dissipazione. Zanichelli, Bologna (1969) (in Italian).
16. Yu. I. Babenko, Heat and Mass Transfer. Chimia, Leningrad (1986) (in Russian).
17. H. T. Davis, The Theory of Linear Operators. The Principia Press, Bloomington, Indiana (1936).
18. A. Erdélyi (Ed.), Tables of Integral Transforms, Vol. 2, Chap. 13. McGraw-Hill, New York (1954).
19. R. Gorenflo and S. Vessella, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics #1461, Springer, Berlin (1991).
20. H. M. Srivastava and S. Owa (Eds), Univalent Functions, Fractional Calculus, and their Applications. Ellis Horwood, Chichester (1989).
21. P. Rusev, I. Dimovski and V. Kiryakova (Eds), Transform Methods and Special Functions, Sofia 1994. Science Culture Technology, Singapore (1995) (Proc. Int. Workshop, Sofia, Bulgaria, 12-17 August 1994).
22. P. Rusev, I. Dimovski and V. Kiryakova (Eds), Transform Methods and Special Functions, Sofia 1994. Science Culture Technology, Singapore (1995) (Proc. Int. Workshop, Sofia, Bulgaria, 12-17 August 1994).
23. F. Mainardi, Fractional diffusive waves in viscoelastic solids. In IUTAM Symposium - Nonlinear Waves in Solids, edited by J. L. Wegner and F. R. Norwood, pp. 93-97. ASME/AMR, Fairfield NJ (1995) (ASME reprint No AMR137) (Proc. IUTAM Symposium, Victoria, Canada, 15-20 August 1993; Abstract in Appl. Mech. Rev., 46, 549 (1993)).
24. F. Mainardi, Fractional diffusive waves in viscoelastic solids. In IUTAM Symposium - Nonlinear Waves in Solids, edited by J. L. Wegner and F. R. Norwood, pp. 93-97. ASME/AMR, Fairfield NJ (1995) (ASME reprint No AMR137) (Proc. IUTAM Symposium, Victoria, Canada, 15-20 August 1993; Abstract in Appl. Mech. Rev., 46, 549 (1993)).
25. F. Mainardi, Fractional diffusive waves in viscoelastic solids. In IUTAM Symposium - Nonlinear Waves in Solids, edited by J. L. Wegner and F. R. Norwood, pp. 93-97. ASME/AMR, Fairfield NJ (1995) (ASME reprint No AMR137) (Proc. IUTAM Symposium, Victoria, Canada, 15-20 August 1993; Abstract in Appl. Mech. Rev., 46, 549 (1993)).
26. F. Mainardi, Fractional relaxation in anelastic solids. J. Alloys Compds 211/212, 534-538 (1994).
27. F. Mainardi, On the initial value problem for the fractional diffusion-wave equation. In Waves and Stability in Continuous Media, edited by S. Rionero and T. Ruggeri, pp. 246-251, World Scientific, Singapore (1994) (Proc. VII-th WASCOM, Bologna, Italy, 4-7 October 1993).
28. F. Mainardi, On the initial value problem for the fractional diffusion-wave equation. In Waves and Stability in Continuous Media, edited by S. Rionero and T. Ruggeri, pp. 246-251, World Scientific, Singapore (1994) (Proc. VII-th WASCOM, Bologna, Italy, 4-7 October 1993).
29. F. Mainardi, Fractional relaxation and fractional diffusion equations, in Proc. 12-th IMACS World Congress, edited by W. F. Ames, Vol. 1, pp. 329-332. GeorgiaTech, Atlanta (1994).
30. F. Mainardi and M. Tomirotti, On a special function arising in the time fractional diffusion-wave equation, in ref. 17 (1995), pp. 171-183.
31. I. Podlubny, The Laplace transform method for linear differential equations of fractional order, Preprint UEF-02-94, Inst. Exp. Phys., Slovak Acad. Sci., Kosice (1994).
32. I. Podlubny, Solutions of linear fractional differential equations, in ref. 17 (1995), pp. 227-237.
33. R. Gorenflo and R. Rutman, On ultraslow and on intermediate processes, in ref. 17 (1995), pp. 61-81.
34. I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. 1. Academic Press, New York (1964).
35. G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation. Springer, Berlin (1974).
36. M. Caputo, Vibrations of an infinite viscoelastic layer with a dissipative memory. J. Acoust. Soc. Am. 56, 897-904 (1974).
37. A. Erdélyi (Ed.), Higher Transcendental Functions, Vol. 3, Chap. 18. McGraw-Hill, New York (1955).
38. B. Gross, On creep and relaxation. J. Appl. Phys. 18, 212-221 (1947).
39. M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, 134-147 (1971).
40. M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids. Riv. Nuovo Cimenta (Ser. II) 1, 161-198 (1971).
41. R. R. Nigmatullin, On the theory of relaxation with "remnant" memory. Phys. Stat. Sol. B 124, 389-393 (1984) (English transl. from Russian).
42. P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivatives in the behavior of real materials, J. Appl. Mech. (Trans. ASME) 51, 294-298 (1984).
43. R. C. Koeller, Applications of fractional calculus to the theory of viscoelasticity. J. Appl. Mech. (Trans. ASME) 51, 299-307 (1984).
44. F. Mainardi and E. Bonetti, The application of real-order derivatives in linear viscoelasticity. Rheol. Acta 26 Suppl., 64-67 (1988).
45. T. F. Nonnenmacher and W. G. Glöckle, A fractional model for mechanical stress relaxation. Phil. Mag. Letters, 64(2), 89-93 (1991).
46. R. R. Nigmatullin, The physics of fractional calculus and its realization on the fractal structures, Doctorate Thesis, Kazan University (1992) (in Russian).
47. B. Stankovic, On the function of E. M. Wright, Publ. Inst. Math. Beograd (Nouv. serie) 10(24), 113-124 (1970).
48. Lj. Gajić and B. Stanković, Some properties of Wright's function, Publ. Inst. Math. Beograd (Nouv. serie) 20(34), 91-98 (1976).
49. αγ). Studio Math. 18, 191-198 (1959).
50. P. W. Buchen and F. Mainardi, Asymptotic expansions for transient viscoelastic waves, J. Mécaniq. 14, 597-608 (1975).
51. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Chap. 3. McGraw-Hill, Singapore (1987).
52. W. Wyss, Fractional diffusion equation, J. Math. Phys. 27, 2782-2785 (1986).
53. W. R. Schneider and W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30, 134-144 (1989).
54. A. N. Kochubei, A Cauchy problem for evolution equations of fractional order, J. Diff. Eqns 25, 967-974 (1989) (English transl. from Russian).
55. A. N. Kochubei, Fractional order diffusion, J. Diff. Eqns 26, 485-492 (1990) (English transl. from Russian).
56. F. Mainardi and H. Buggisch, On nonlinear waves in liquid-filled elastic tubes, in Nonlinear Deformation Waves, edited by U. Nigul and J. Engelbrecht, pp. 87-100, Springer, Berlin (1983) (Proc. IUTAM Symposium, Tallinn, Estonia, 22-28 August 1982).
57. F. Mainardi and H. Buggisch, On nonlinear waves in liquid-filled elastic tubes, in Nonlinear Deformation Waves, edited by U. Nigul and J. Engelbrecht, pp. 87-100, Springer, Berlin (1983) (Proc. IUTAM Symposium, Tallinn, Estonia, 22-28 August 1982).
58. R. R. Nigmatullin, 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).
59. W. R. Young, A. Pumir and Y. Pomeau. Anomalous diffusion of tracer in convection rolls. Phys. Fluids A 1, 462-469 (1989).
60. U. J. Choi and R. C. MacCamy, Fractional order Volterra equations with applications to elasticity. J. Math. Anal. Applics 139, 448-464 (1989).
61. A. LeMéhauté, Les Géometries Fractáles. Hermes, Paris (1990).
62. A. Oustaloup, La Commande CRONE. Hermes, Paris (1990).
63. T. F. Nonnenmacher, Fractional integral and differential equations for a class of Levy-type probability densities. J. Phys A: Math. Gen. 23, L697-L700 (1990).
64. N. Sugimoto, Burgers equation with a fractional derivative; hereditary effects of nonlinear acoustic waves, J. Fluid Mech. 225, 631-653 (1991).
65. M. Giona and H. E. Roman, Fractional diffusion equation for transport phenomena in random media. Physica A 185, 82-97 (1992).
66. R. Lenormand, Use of fractional derivatives for fluid flow in heterogeneous media (paper presented at the 3rd European Conf. on the Mathematics of Oil Recovery, 17-19 June 1992, Delft, The Netherlands) (1992).
67. M. Ochmann and S. Makarov, Representation of the absorption of nonlinear waves by fractional derivatives. J. Acoust. Soc. Am. 94, 3392-3399 (1993).
68. M. Caputo, The splitting of the seismic rays due to dispersion in the Earth's interior. Rend. Fis. Acc. Lincei (Ser. IX) 4, 279-286 (1993).
69. G. M. Zaslavasky, Fractional kinetic equation for Hamiltonian chaos. Physica D 76, 110-122 (1994).