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

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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.

**[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)

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DOI:
https://doi.org/10.1090/qam/1900491

Article copyright:
© Copyright 2002
American Mathematical Society