On certain applications of fractional calculus to viscoelasticity

Journal of Mathematical Sciences

Volumn 112, Issue 5, 2002, Pages 4517-4557

  Surguladze, T A ^a  

^a NONE

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EID: 30644461243     PISSN: 10723374     EISSN: 15738795     Source Type: Journal    
DOI: 10.1023/A:1020574305792     Document Type: Article

References (58)

1. G. Bateman and A. Erdelyi, Higher Transcendental Functions. Elliptic and Automorphic Functions, Lamé and Mathieu Functions [in Russian], Vol. 3, Nauka, Moscow (1967).
2. D. Bland, Linear Viscoelasticity Theory [Russian translation], Mir, Moscow (1965).
3. V. S. Vladimirov, Distributions in Mathematical Physics [in Russian], Nauka, Moscow (1979).
4. V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1988).
5. I. M. Gelfand and G. E. Shilov, Distributions and Operations on Distributions [in Russian], Fizmatgiz, Moscow (1959).
6. G. M. Gubreev, "Basis property of Mittag-Leffler's type functions, Djrbashyan's transforms and the Mackenhoupt condition," Funkts. Anal. Ego Prilozhen., 21, Issue 4, 71-72 (1987).
7. M. M. Djrbashyan, Integral Transforms and Representation of Functions in Complex Domains [in Russian], Nauka, Moscow (1966).
8. V. A. Ditkin and A. P. Prudnikov, Operational Calculus in Two Variables and Its Application [in Russian], Fizmatgiz, Moscow (1958).
9. V. A. Ditkin and A. P. Prudnikov, Operational Calculus [in Russian], Vyshaya Shkola, Moscow (1975).
10. A. A. Il'yushin and B. E. Pobedrya, Foundations of the Mathematical Theory of Viscoelasticity [in Russian], Nauka, Moscow (1970).
11. R. Christensen, An Introduction to the Theory of Viscoelasticity [Russian translation], Mir, Moscow (1974).
12. M. V. Lavrent'ev and B. V. Shabat, Methods of the Theory of Functions of a Complex Variable [in Russian], Nauka, Moscow (1987).
13. Yu. F. Luchko and S. B. Yakubovich, "Operational method for solving integrodifferential equations of certain classes," Diff. Uravn., 30, No. 2, 269-280 (1994).
14. J. Mikusinskii, Operational Calculus [Russian translation], Inostr. Lit., Moscow (1956).
15. J. Mikusinskii and R. Sikorskii, Elementary Theory of Distributions [Russian translation], Inostr. Lit., Moscow (1959).
16. R. R. Nigmatullin, "Fractional integrals and their physical treatment," Teor. Mat. Fiz., 90, No. 3, 354-368.
17. B. E. Pobedrya, "Mathematical theory of nonlinear viscoelasticity," In: Elasticity Nonelasticity, Issue 3, MGU (1973), pp. 95-173.
18. B. E. Pobedrya, Numerical Methods in the Theory of Elasticity and Plasticity [in Russian], MGU, Moscow (1995).
19. Yu. N. Rabotnov, Essentials of Hereditary Mechanics of Solids [in Russian], Nauka, Moscow (1977).
20. Yu. A. Rossikhin, M. V. Shitikova, and M. A. Kolesnikov, "Application of viscoelastic models with two fractional parameters to problems of vibrating 2-DOF mechanical systems," Deposited at VINITI, November 19, 1997, No. 3383-97.
21. Yu. A. Rossikhin, M. V. Shitikova, and M. A. Kolesnikov, "Application of the generalized model of a standard linear body with two fractional parameters to the description of the impact of a viscoelastic bar on a rigid obstacle," Deposited at VINITI, June 19, 1998, No. 1885-B98.
22. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Certain Applications of These Derivatives [in Russian], Nauka i Tekhnika, Minsk (1987).
23. T. A. Surguladze, "On certain applications of fractional calculus in viscoelasticity," Deposited at VINITI, June 7, 1999, No. 1827-99.
24. T. A. Surguladze, "On one of the applications of fractional calculus in viscoelasticity," Vestn. MGU, Mat. Mekh., 5, 62-66 (2000).
25. T. A. Surguladze, "On one of the applications of the fractional Green's function," Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, No. 1 (2001).
26. J. Feder, Fractals [Russian translation], Mir, Moscow (1991).
27. G. M. Fikhtengol'tz, A Course in Differential and Integral Calculus [in Russian], Nauka, Moscow (1970).
28. T. D. Shermergor, "On application of operators of fractional differentiation for the description of hereditary properties of materials," Prikl, Mat. Tekh. Fiz., No. 6, 118-121 (1966).
29. F. B. Adda, "Geometric interpretation of the fractional derivative," J. Fractional Calculus, 11, 21-51 (1997).
30. R. L. Bagley and P. J. Torvik, "A theoretical basis for the application of fractional calculus to viscoelasticity," J. Rheolog., 27, No. 3, 201-203 (1983).
31. R. L. Bagley and P. J. Torvik, "Fractional calculus: A different approach to the analysis of viscoelastically damped structures," AIAA J., 21, No. 5, 741-748 (1983)
32. R. L. Bagley R.L. and P. J. Torvik, "On the fractional calculus model of viscoelastic behavior," J. Rheolog., 30, No. 1, 133-155 (1986).
33. M. Dostanić, "A basis property of families of the Mittag-Leffler functions," Mat. Vesn., 48, 77-82 (1996).
34. M. Enelund, A. Fenander, and P. Olson, "Fractional integral formulation of constitutive equations of viscoelasticity," AIAA J., 35, No. 8, 1356-1362 (1997).
35. M. Enelund and B. L. Josefson, "Time-domain finite element analysis of viscoelastic structures with fractional derivatives constitutive relations," AIAA J., 35, No. 10, 1630-1637 (1997).
36. C. Friedrich, "Relaxation functions of rheological constitutive equations with fractional derivatives: thermodynamical constraints," In: J. Casas-Vazgauez and D. Jou (eds) Lecture Notes in Physics; 381, Rheological Modeling; Thermodynamical and Statistical Approaches, Springer-Verlag, Berlin-Heidelberg (1991), pp. 321-330.
37. R. Gorenflo and F. Mainardi, Signalling Problem and Dirichlet-Neumann Map for Time-Fractional Diffusion-Wave Equations, Freie Universität, Berlin; Fachbereich Mathematic and Informatic, Preprint No. A-98-07 R, 1998.
38. R. Gorenflo and F. Mainardi, Essentials of Fractional Calculus, Preprint submitted to MaPhySto Center, 28 Juanuary, 2000.
39. N. Heymans and J.-C. Bauwens, "Fractal rheological models and fractional differential equations for viscoelastic behavior," Rheologica Acta, 33, 210-219 (1994).
40. R. C. Koeller, "Applications of fractional calculus to the theory of viscoelasticity," J. Appl. Mech., 51, 299-307 (1984).
41. R. C. Koeller, "Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics," Acta Mech., 58, 251-263 (1986).
42. C. F. Lorenzo and T. Hartley, Initialization, Conceptualization, and Application in the Generalized Fractional Calculus, National Aeronautics and Space Administratrion, NASA/TR-1998-208415 (1998).
43. F. Mainardi, P. Paradisi, and R. Gorenflo, "Probability distributions generated by fractional diffusion equations," In: Ecophysics: An Emerging Science, Kluwer (1999).
44. D. Matignon, and B. d'Andrea-Novel, "Some result on controllability and observability of finite-dimensional fractional differential systems," In: Computational Engineering in Systems and Application, Multiconference, Vol. 2, Lille, France, July (1996), pp. 952-956.
45. K. S. Miller and B. Ross, "Fractional Green's functions," Indian J. Pure Appl. Math., 22, No. 9, 763-767 (1991).
46. K. S. Miller, "Properties of fractional Green's functions," Int. J. Math. Stat. Sci., 1, No. 2, 129-149 (1992).
47. K. S. Miller, "The Mittag-Leffler and related functions," Integral Transforms Special Functions, 1, 41-49 (1993).
48. A. R. Nkamnang, Discretization of Multi-Term Fractional Integral and Ordinary Differential Equations, FU Berlin, Digitale Dissertation (1999).
49. T. F. Nonnenmacher, "Fractional relaxation equations for viscoelasticity and related phenomena," Lect. Notes Phys., 381, 307-321 (1991).
50. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York-London (1974).
51. K. B. Oldham and J. Spanier, "Fractional calculus and its applications," Bul. Inst. Politehn. Iasi., Sec. 1, 24, Nos. 3-4, 29-34 (1978).
52. J. Padovan, "Computational algorithms for FE formulations involving fractional operators," Computational Mechanics, 2, No. 4, 271-287 (1987).
53. I. Podlubny, The Laplace Transform Method for Linear Differential Equations, Inst. Exp. Phys. Slovak Acad. Sci. No. UEF-02-94, Kosice (1994).
54. I. Podlubny, Fractional Differential Equations (An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications), Academic Press, San Diego-Boston-New York-London-Sydney-Tokyo-Toronto (1999).
55. B. Ross, "A brief history and exposition of the fundamental theory of fractional calculus," Lect. Notes Math., 457, 1-36 (1975).
56. Y. A. Rossikhin and M. V. Shitikova, "Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems," Acta Mech., 120, 110-125 (1997).
57. G. Scarpi, "Sulla possibilita un modello reologico intermedio di tipo evolutivo," Rend. Sc. Fis. Mat. Nat. Accad. Naz. Lincei, 52, 570-575 (1972).
58. H. Schiessel, R. Metzler, A. Blumen, and T. F. Nonnenmacher, "Generalized viscoelastic models: Their fractional equations with solutions," J. Phys. A: Math. Gen., 28, 6567-6584 (1995).