Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotic and exact fundamental solutions in hereditary media with singular memory kernels

Authors: Andrzej Hanyga and M. Seredyńska
Journal: Quart. Appl. Math. 60 (2002), 213-244
MSC: Primary 35Q72; Secondary 35A08, 35B40, 74D05, 74J05
DOI: https://doi.org/10.1090/qam/1900491
MathSciNet review: MR1900491
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A method for constructing time-domain asymptotic solutions of hyperbolic partial differential equations with delay, with singular memory kernels, is presented. The asymptotic solutions are expressed in terms of basis functions that are regularizations of a sequence of distributions related by fractional integration.

References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. Stegun, Mathematical Tables, Dover, New York, 1970
  • [2] J.-F. Allard and Y. Champoux, New empirical equations for sound propagation in rigid frame fibrous materials, J. Acoust. Soc. Amer. 91, 3346-3353 (1992)
  • [3] R. L. Bagley and P. J. Torvik, On the fractional calculus model of viscoelastic behavior, J. of Rheology, 133-155 (1986)
  • [4] J. G. Berryman, L. Thigpen, and R. C. Y. Chin, Bulk elastic wave propagation in partially saturated porous solids, J. Acoust. Soc. Amer. 84, 360-373 (1988)
  • [5] M. A. Biot, Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range, J. Acoust. Soc. Amer. 28 (1956), 179–191. MR 0134057, https://doi.org/10.1121/1.1908241
  • [6] P. W. Buchen and F. Mainardi, Asymptotic expansions for transient viscoelastic waves, J. de Mécanique 14, 597-608 (1975)
  • [7] Y. Champoux and M. R. Stinson, On acoustical models for sound propagation in rigid frame porous materials and the influence of shape factors, J. Acoust. Soc. Amer. 92, 1120-1131 (1992)
  • [8] R. M. Christensen, Theory of Viscoelasticity: An Introduction, Academic Press, New York, 1971
  • [9] R. Courant, Partial Differential Equations, Wiley Interscience Publishers, New York, 1962
  • [10] C. F. Curtiss and R. B. Bird, A kinetic theory for polymer melts, I. The equation for the single-link orientational distribution function, J. Chem. Phys. 74, 2016-2025 (1981)
  • [11] Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 5, Springer-Verlag, Berlin, 1992. Evolution problems. I; With the collaboration of Michel Artola, Michel Cessenat and Hélène Lanchon; Translated from the French by Alan Craig. MR 1156075
  • [12] Gustav Doetsch, Einführung in Theorie und Anwendung der Laplace-Transformation, Birkhäuser Verlag, Basel-Stuttgart, 1976 (German). Ein Lehrbuch für Studierende der Mathematik, Physik und Ingenieurwissenschaft; Dritte Auflage; Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften-Mathematische Reihe, Band 24. MR 0454529
  • [13] Mauro Fabrizio and Angelo Morro, Mathematical problems in linear viscoelasticity, SIAM Studies in Applied Mathematics, vol. 12, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1153021
  • [14] W. Feller, On a generalization of Marcel Riesz' potentials and the semigroups generated by them, in Medd. Lund Univ. Matematiska Seminaret, 1952, pp. 73-81. Volume dedicated to M. Riesz.
  • [15] William Feller, An introduction to probability theory and its applications. Vol. II., Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971. MR 0270403
  • [16] S. Gelinsky, S. A. Shapiro, T. Müller, and B. Gurevich, Dynamic poroelasticity of thinly layered structures, Internat. J. Solids and Structures 35, 1739-4751 (1998)
  • [17] Rudolf Gorenflo and Francesco Mainardi, Random walk models for space-fractional diffusion processes, Fract. Calc. Appl. Anal. 1 (1998), no. 2, 167–191. MR 1656314
  • [18] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th edition, Academic Press, New York, 1994
  • [19] B. Gurevich and S. L. Lopatnikov, Velocity and attenuation of elastic waves in finely layered porous rocks, Geophys. J. Internat. 121, 933-947 (1995)
  • [20] A. Hanyga, Asymptotic theory of wave propagation in viscoporoelastic media, in ``Theoretical and Computational Acoustics '97", Y.-C. Teng, E.-C. Shang, Y.-H. Pao, M. H. Shultz, and A. D. Pierce, eds., World Scientific, Singapore, 1999
  • [21] A. Hanyga and V. E. Rok, An integro-differential wave equation and applications to wave propagation in micro-inhomogeneous porous media, J. Acoust. Soc. Amer. 107 (6), 2965-2972 (2000)
  • [22] Andrzej Hanyga and Małgorzata Seredyńska, Asymptotic ray theory in poro- and viscoelastic media, Wave Motion 30 (1999), no. 2, 175–195. MR 1708130, https://doi.org/10.1016/S0165-2125(98)00053-5
  • [23] A. Hanyga and M. Seredyńska, Some effects of the memory kernel singularity on wave propagation and inversion in poroelastic media, I: Forward modeling, Geophys. J. Internat. 137, 319-335 (1994)
  • [24] A. Hanyga and M. Seredyńska, Thermodynamics and asymptotic theory of wave propagation in viscoporous media, Proc. 3rd Internat. Conf. on Computational and Theoretical Acoustics, Newark, NJ, July 14-18, 1997, 1999
  • [25] M. Ya. Kelbert and I. Ya. Chaban, Relaxation and propagation of pulses in fluids, Izv. Ak. Nauk. Ser. Mechanics of Fluids and Gases 5, 153-160 (1986)
  • [26] R. C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, Trans. ASME J. Appl. Mech. 51 (1984), no. 2, 299–307. MR 747787, https://doi.org/10.1115/1.3167616
  • [27] R. C. Koeller, Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics, Acta Mech. 58 (1986), no. 3-4, 251–264. MR 844882, https://doi.org/10.1007/BF01176603
  • [28] H. Kolsky, The propagation of stress pulses in viscoelastic solids, Philos. Mag. 8, 693-710 (1956)
  • [29] Andreas Kreis and A. C. Pipkin, Viscoelastic pulse propagation and stable probability distributions, Quart. Appl. Math. 44 (1986), no. 2, 353–360. MR 856190, https://doi.org/10.1090/S0033-569X-1986-0856190-4
  • [30] A. A. Lokšin and V. E. Rok, Self-similar solutions of wave equations with time delay, Uspekhi Mat. Nauk 33 (1978), no. 6(204), 221–222 (Russian). MR 526028
  • [31] A. A. Lokshin and V. E. Rok, Fundamental solutions of the wave equation with delayed time, Doklady AN SSSR, 239, 1305-1308 (1978)
  • [32] A. A. Lokshin and Yu. V. Suvorova, \cyr Matematicheskaya teoriya rasprostraneniya voln v sredakh s pamyat′yu, Moskov. Gos. Univ., Moscow, 1982 (Russian). MR 676810
  • [33] F. Mainardi and M. Tomirotti, Seismic pulse propagation with constant Q and stable probability distributions, Annali de Geofisica 40, 1311-1328 (1997)
  • [34] Kenneth S. Miller and Bertram Ross, An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993. MR 1219954
  • [35] A. Narain and D. D. Joseph, Linearized dynamics for step jumps of velocity and displacement of shearing flows of a simple fluid, Rheol. Acta 21 (1982), no. 3, 228–250. MR 669373, https://doi.org/10.1007/BF01515712
  • [36] A. N. Norris, On the viscodynamic operator in Biot's theory, J. Wave-Material Interaction 1, 365-380 (1986)
  • [37] P. G. Nutting, Deformation in relation to time, pressure and temperature, J. Franklin Institute 242, 449-458 (1946)
  • [38] Martin Ochmann and Sergey Makarov, Representation of the absorption of nonlinear waves by fractional derivatives, J. Acoust. Soc. Amer. 94 (1993), no. 6, 3392–3399. MR 1252538, https://doi.org/10.1121/1.407192
  • [39] A. C. Pipkin, Lectures on Viscoelasticity Theory, 2nd edition, Springer-Verlag, New York, 1986
  • [40] Allen C. Pipkin, Asymptotic behaviour of viscoelastic waves, Quart. J. Mech. Appl. Math. 41 (1988), no. 1, 51–69. MR 934693, https://doi.org/10.1093/qjmam/41.1.51
  • [41] J.-D. Polack, Time domain solution of Kirchhoff's equation for sound propagation in viscothermal gases: a diffusion process, J. Acoustique 4, 17-67 (1991)
  • [42] Yu. N. Rabotnov, Creep Problems in Structural Elements, North-Holland, Amsterdam, 1969
  • [43] Ju. N. Rabotnov, Elements of hereditary solid mechanics, “Mir”, Moscow, 1980. Translated from the Russian by M. Konyaeva [M. Konjaeva]. MR 563522
  • [44] M. Renardy, Some remarks on the propagation and nonpropagation of discontinuities in linearly viscoelastic liquids, Rheol. Acta 21 (1982), no. 3, 251–254. MR 669374, https://doi.org/10.1007/BF01515713
  • [45] Michael Renardy, William J. Hrusa, and John A. Nohel, Mathematical problems in viscoelasticity, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 35, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1987. MR 919738
  • [46] Lynn Rogers, Operators and fractional derivatives for viscoelastic constitutive equations, J. Rheology 27, 351-372 (1983)
  • [47] V. E. Rok, Time-domain representation of waves in media with frequency power law of dispersion, in Extended Abstracts 58th EAGE Conference and Technical Exhibition, Amsterdam, 3-7 June 1996, p. C008, 1996
  • [48] Yu. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanisms of solids, Appl. Mech. Rev. 50, 15-67 (1997)
  • [49] A. R. Rzhanitsyn, Some problems of the mechanics of systems that are deformed in time, Gostekhizdat, Moscow, 1949
  • [50] Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
  • [51] G. L. Slonimsky, Laws of mechanical relaxation processes in polymers, J. Polymer Science C16, 1667-1672 (1967)
  • [52] W. Smit and H. de Vries, Rheological models containing fractional derivatives, Rheol. Acta, 6 525-534 (1970)
  • [53] M. R. Stinson and Y. Champoux, Propagation of sound and the assignment of shape factors in model porous materials having simple pore geometries, J. Acoust. Soc. Amer. 91, 685-695 (1992)
  • [54] P. J. Torvik and R. L. Bagley, On the appearance of the fractional derivative in the behavior of real material, J. Appl. Mechanics 51, 294-298 (1983)
  • [55] D. V. Widder, An Introduction to Transformation Theory, Academic Press, New York, 1971
  • [56] D. K. Wilson, Relaxation-matched modelling of propagation through porous media, including fractal pore structure, J. Acoust. Soc. Amer. 94, 1136-1145 (1992)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35Q72, 35A08, 35B40, 74D05, 74J05

Retrieve articles in all journals with MSC: 35Q72, 35A08, 35B40, 74D05, 74J05

Additional Information

DOI: https://doi.org/10.1090/qam/1900491
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society