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Fractional Calculus and Applied Analysis

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On a fractional Zener elastic wave equation

Sven Näsholm / Sverre Holm
Published Online: 2012-12-27 | DOI: https://doi.org/10.2478/s13540-013-0003-1

Abstract

This survey concerns a causal elastic wave equation which implies frequency power-law attenuation. The wave equation can be derived from a fractional Zener stress-strain relation plus linearized conservation of mass and momentum. A connection between this four-parameter fractional wave equation and a physically well established multiple relaxation acoustical wave equation is reviewed. The fractional Zener wave equation implies three distinct attenuation power-law regimes and a continuous distribution of compressibility contributions which also has power-law regimes. Furthermore it is underlined that these wave equation considerations are tightly connected to the representation of the fractional Zener stress-strain relation, which includes the spring-pot viscoelastic element, and by a Maxwell-Wiechert model of conventional springs and dashpots. A purpose of the paper is to make available recently published results on fractional calculus modeling in the field of acoustics and elastography, with special focus on medical applications.

MSC: Primary 26A33; Secondary 33E12, 34A08, 34K37, 35L05, 92C50, 92C55, 35R11, 74J10

Keywords: fractional calculus; acoustical wave equations; elastic wave equations; fractional wave equations; fractional viscoelasticity; fractional ordinary and partial differential equations

  • [1] N. H. Abel, Auflösung einer mechanischen Aufgabe (Resolution of a mechanical problem). J. Reine. Angew. Math. 1 (1826), 153–157. http://dx.doi.org/10.1515/crll.1826.1.153CrossrefGoogle Scholar

  • [2] K. Adolfsson, M. Enelund, and P. Olsson, On the fractional order model of viscoelasticity. Mech. Time-Dep. Mater. 9, No 1 (2005), 15–34. http://dx.doi.org/10.1007/s11043-005-3442-1CrossrefGoogle Scholar

  • [3] M. Ainslie and J. G. McColm, A simplified formula for viscous and chemical absorption in sea water. J. Acoust. Soc. Am. 103, No 3 (1998), 1671–1672. http://dx.doi.org/10.1121/1.421258CrossrefGoogle Scholar

  • [4] T. M. Atanackovic, A modified Zener model of a viscoelastic body. Continuum Mech. Therm. 14, No 2 (2002), 137–148. http://dx.doi.org/10.1007/s001610100056CrossrefGoogle Scholar

  • [5] T. M. Atanackovic, S. Konjik, L. Oparnica, and D. Zorica, Thermodynamical restrictions and wave propagation for a class of fractional order viscoelastic rods. Abstr. Appl. Anal. 2011 (2011), Article ID 975694. CrossrefGoogle Scholar

  • [6] R. L. Bagley. The thermorheologically complex material. Int. J. Eng. Sci. 29, No 7 (1991), 797–806. http://dx.doi.org/10.1016/0020-7225(91)90002-KCrossrefGoogle Scholar

  • [7] R. L. Bagley and P. J. Torvik, Fractional calculus — A different approach to the analysis of viscoelastically damped structures. AIAA J. 21, No 5 (1983), 741–748. http://dx.doi.org/10.2514/3.8142CrossrefGoogle Scholar

  • [8] R. L. Bagley and P. J. Torvik, On the fractional calculus model of viscoelastic behavior. J. Rheol. 30, No 1 (1986), 133–155. http://dx.doi.org/10.1122/1.549887CrossrefGoogle Scholar

  • [9] J. C. Bamber, Attenuation and Absorption, Ch. 4, pp. 93–166. John Wiley & Sons, Chichester, 2005. Google Scholar

  • [10] C. T. Barry, B. Mills, Z. Hah, R. A. Mooney, C. K. Ryan, D. J. Rubens, and K. J. Parker, Shear wave dispersion measures liver steatosis. Ultrasound Med. Biol. 38, No 2 (2012), 175–182. http://dx.doi.org/10.1016/j.ultrasmedbio.2011.10.019CrossrefGoogle Scholar

  • [11] H. Bass, L. Sutherland, A. Zuckerwar, D. Blackstock, and D. Hester, Atmospheric absorption of sound: Further developments. J. Acoust. Soc. Am. 97 (1995), 680–683. http://dx.doi.org/10.1121/1.412989CrossrefGoogle Scholar

  • [12] P. Beard, Biomedical photoacoustic imaging. Interface Focus 1, No 4 (2011), 602–631. http://dx.doi.org/10.1098/rsfs.2011.0028Google Scholar

  • [13] M.N. Berberan-Santos, E.N. Bodunov, B. Valeur, Luminescence decays with underlying distributions of rate constants: General properties and selected cases. In: M.N. Berberan-Santos, M. Hof, Eds., Fluorescence of Supermolecules, Polymers, and Nanosystems, Vol. 4. Springer Ser. on Fluorescence, pp. 67–103, Springer, Berlin-Heidelberg, 2008. http://dx.doi.org/10.1007/4243_2007_001CrossrefGoogle Scholar

  • [14] J. Bercoff, M. Tanter, and M. Fink, Supersonic shear imaging: a new technique for soft tissue elasticity mapping. IEEE Trans. Ultrason. Ferroelectr., Freq. Control 51, No 4 (2004), 396–409. http://dx.doi.org/10.1109/TUFFC.2004.1295425CrossrefGoogle Scholar

  • [15] S. Bhalekar, V. Daftardar-Gejji, D. Baleanu, and R. Magin, Generalized fractional order bloch equation with extended delay. Int. J. Bifurcat. Chaos 22, No 04 (2012), 1250071-1–1250071-15. CrossrefGoogle Scholar

  • [16] M. Caputo, Linear models of dissipation whose Q is almost frequency independent — II. Geophys. J. Roy. Astr. S. 13, No 5 (1967), 529–539; Reprinted in: Fract. Calc. Appl. Anal. 11, No 1 (2008), 3–14; http://www.blackwell-synergy.com/toc/gji/13/5. http://dx.doi.org/10.1111/j.1365-246X.1967.tb02303.xCrossrefGoogle Scholar

  • [17] M. Caputo, J. M. Carcione, and F. Cavallini, Wave simulation in biologic media based on the Kelvin-Voigt fractional-derivative stress-strain relation. Ultrasound Med. Biol. 37, No 6 (2011), 996–1004. http://dx.doi.org/10.1016/j.ultrasmedbio.2011.03.009CrossrefGoogle Scholar

  • [18] M. Caputo and F. Mainardi. A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, No 1 (1971), 134–147; Reprinted in: Fract. Calc. Appl. Anal. 10, No 3 (2007), 309–324; at http://www.math.bas.bg/~fcaa. http://dx.doi.org/10.1007/BF00879562CrossrefGoogle Scholar

  • [19] J.M. Carcione, A generalization of the Fourier pseudospectral method. Geophysics 75, No 6 (2010), A53–A56. http://dx.doi.org/10.1190/1.3509472CrossrefGoogle Scholar

  • [20] S. Chatelin, S. A. Lambert, L. Jugé, X. Cai, S. P. Näsholm, V. Vilgrain, B. E. Van Beers, L. E. Maitre, X. Bilston, B. Guzina, S. Holm, and R. Sinkus. Measured elasticity and its frequency dependence are sensitive to tissue microarchitecture in mr elastography. In: Proc. 20th Annual Meeting of ISMRM, May 2012. Google Scholar

  • [21] A. Chatterjee, Statistical origins of fractional derivatives in viscoelasticity. J. Sound. Vib. 284, No 3–5 (2005), 1239–1245. http://dx.doi.org/10.1016/j.jsv.2004.09.019CrossrefGoogle Scholar

  • [22] S. Chen, M. Fatemi, and J. F. Greenleaf, Quantifying elasticity and viscosity from measurement of shear wave speed dispersion. J. Acoust. Soc. Am. 115, No 6 (2004), 2781–2785. http://dx.doi.org/10.1121/1.1739480CrossrefGoogle Scholar

  • [23] W. Chen and S. Holm, Modified Szabo’s wave equation models for lossy media obeying frequency power law. J. Acoust. Soc. Am. 114, No 5 (2003), 2570–2574. http://dx.doi.org/10.1121/1.1621392CrossrefGoogle Scholar

  • [24] K. S. Cole and R. H. Cole, Dispersion and absorption in dielectrics I. Alternating current characteristics. J. Chem. Phys. 9, No 4 (1941), 341–351. http://dx.doi.org/10.1063/1.1750906CrossrefGoogle Scholar

  • [25] C. Coussot, S. Kalyanam, R. Yapp, and M. Insana, Fractional derivative models for ultrasonic characterization of polymer and breast tissue viscoelasticity. IEEE Trans. Ultrason. Ferroelectr., Freq. Control 56, No 4 (2009), 715–725. http://dx.doi.org/10.1109/TUFFC.2009.1094CrossrefGoogle Scholar

  • [26] D. O. Craiem, F. J. Rojo, J. M. Atienza, G. V. Guinea, and R. L. Armentano, Fractional calculus applied to model arterial viscoelasticity. Latin. Am. Appl. Res. 38, No 2 (2008), 141–145. Google Scholar

  • [27] G. B. Davis, M. Kohandel, S. Sivaloganathan, and G. Tenti, The constitutive properties of the brain paraenchyma. Part 2. Fractional derivative approach. Med. Eng. Phys. 28, No 5 (2006), 455–459. http://dx.doi.org/10.1016/j.medengphy.2005.07.023CrossrefGoogle Scholar

  • [28] E. C. de Oliveira, F. Mainardi, and J. Vaz, Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur. J. Phys. 193 (2011), 161–171. Google Scholar

  • [29] F. Dinzart and P. Lipinski, Improved five-parameter fractional derivative model for elastomers. Arch. Mech. 61, No 6 (2009), 459–474. Google Scholar

  • [30] V. D. Djordjević, J. Jarić, B. Fabry, J. J. Fredberg, and D. Stamenović, Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng. 31 (2003), 692–699. http://dx.doi.org/10.1114/1.1574026CrossrefGoogle Scholar

  • [31] M. M. Djrbashian, Integral Transforms and Representations of Functions in the Complex Domain, Chs. 3–4. Nauka, Moscow, 1966 (In Russian). Google Scholar

  • [32] M. M. Djrbashian, Harmonic Analysis and Boundary Value Problems in the Complex Domain, Ch. 1. Birkhäuser, Basel, 1993. http://dx.doi.org/10.1007/978-3-0348-8549-2CrossrefGoogle Scholar

  • [33] F. A. Duck, Physical Properties of Tissue. Academic Press, 1990. Google Scholar

  • [34] R. L. Ehman, K. J. Glaser, and A. Manduca, Review of MR elastography applications and recent developments. J. Magn. Reson. 36, No 4 (2012), 757–774. http://dx.doi.org/10.1002/jmri.23597CrossrefGoogle Scholar

  • [35] Y. Feldman, Y. A. Gusev, and M. A. Vasilyeva, Dielectric Relaxation Phenomena in Complex Systems. Tutorial, Kazan Federal University, Institute of Physics, 2012. Google Scholar

  • [36] C. Friedrich and H. Braun, Generalized cole-cole behavior and its rheological relevance. Rheol. Acta 31 (1992), 309–322. http://dx.doi.org/10.1007/BF00418328CrossrefGoogle Scholar

  • [37] J. Garnier and K. Sølna, Effective fractional acoustic wave equations in one-dimensional random multiscale media. J. Acoust. Soc. Am. 127, No 1 (2010), 62–72. http://dx.doi.org/10.1121/1.3263608CrossrefGoogle Scholar

  • [38] W. G. Glöckle and T. F. Nonnenmacher, Fractional integral operators and Fox functions in the theory of viscoelasticity. Macromolecules 24, No 24 (1991), 6426–6434 http://dx.doi.org/10.1021/ma00024a009CrossrefGoogle Scholar

  • [39] N. M. Grahovac and M. Zigic, Modelling of the hamstring muscle group by use of fractional derivatives. Comput. Math. Appl. 59, No 5 (2010), 1695–1700. http://dx.doi.org/10.1016/j.camwa.2009.08.011CrossrefGoogle Scholar

  • [40] H. J. Haubold, A. M. Mathai, and R. K. Saxena, Mittag-Leffler functions and their applications. J. of Appl. Math. 2011 (2011), 1–51. http://dx.doi.org/10.1155/2011/298628CrossrefGoogle Scholar

  • [41] S. Holm and S. P. Näsholm, A causal and fractional all-frequency wave equation for lossy media. J. Acoust. Soc. Am. 130, No 4 (2011), 2195–2202. http://dx.doi.org/10.1121/1.3631626CrossrefGoogle Scholar

  • [42] S. Holm and R. Sinkus, A unifying fractional wave equation for compressional and shear waves. J. Acoust. Soc. Am. 127 (2010), 542–548. http://dx.doi.org/10.1121/1.3268508CrossrefGoogle Scholar

  • [43] L. Jugé, S. A. Lambert, S. Chatelin, L. ter Beek, V. Vilgrain, B. E. Van Beers, L. E. Bilston, B. Guzina, S. Holm, and R. Sinkus, Sub-voxel micro-architecture assessment by diffusion of mechanical shear waves. In: Proc. 20th Annual Meeting of ISMRM, May 2012. Google Scholar

  • [44] J. F. Kelly and R. J. McGough, Fractal ladder models and power law wave equations. J. Acoust. Soc. Am. 126, No 4 (2009), 2072–2081. http://dx.doi.org/10.1121/1.3204304CrossrefGoogle Scholar

  • [45] D. Klatt, U. Hamhaber, P. Asbach, J. Braun, and I. Sack, Noninvasive assessment of the rheological behavior of human organs using multifrequency MR elastography: A study of brain and liver viscoelasticity. Phys. Med. Biol. 52, No 24 (2007), 7281–7294. http://dx.doi.org/10.1088/0031-9155/52/24/006CrossrefGoogle Scholar

  • [46] M. Kohandel, S. Sivaloganathan, G. Tenti, and K. Darvish, Frequency dependence of complex moduli of brain tissue using a fractional Zener model. Phys. Med. Biol. 50, No 12 (2005), 2799–2805. http://dx.doi.org/10.1088/0031-9155/50/12/005CrossrefGoogle Scholar

  • [47] S. Konjik, L. Oparnica, and D. Zorica, Waves in fractional Zener type viscoelastic media. J. Math. Anal. Appl. 365, No 1 (2010), 259–268. http://dx.doi.org/10.1016/j.jmaa.2009.10.043CrossrefGoogle Scholar

  • [48] R. Kowar and O. Scherzer, Attenuation models in photoacoustics. In: H. Ammari, Ed., Mathematical Modeling in Biomedical Imaging II, Vol. 2035 of L.N.M., pp. 85–130, Springer, Berlin-Heidelberg, 2012. http://dx.doi.org/10.1007/978-3-642-22990-9_4Google Scholar

  • [49] M. Liebler, S. Ginter, T. Dreyer, and R. E. Riedlinger, Full wave modeling of therapeutic ultrasound: Efficient time-domain implementation of the frequency power-law attenuation. J. Acoust. Soc. Am. 116 (2004), 2742–2750. http://dx.doi.org/10.1121/1.1798355CrossrefGoogle Scholar

  • [50] J. G. Liu and M. Y. Xu, Higher-order fractional constitutive equations of viscoelastic materials involving three different parameters and their relaxation and creep functions. Mech. Time-Depend. Mat. 10 (2006), 263–279. http://dx.doi.org/10.1007/s11043-007-9022-9CrossrefGoogle Scholar

  • [51] Y. Luchko, Fractional wave equation and damped waves. ArXiv e-prints, May 2012. Google Scholar

  • [52] J. A. T. Machado and A. Galhano, Fractional dynamics: A statistical perspective. J. Comput. Nonlin. Dynam. 3, No 2 (2008), 021201-1–021201-4. CrossrefGoogle Scholar

  • [53] F. Mainardi, Fractional relaxation in anelastic solids. Journal of Alloys and Compounds 211 (1994), 534–538. http://dx.doi.org/10.1016/0925-8388(94)90560-6CrossrefGoogle Scholar

  • [54] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelesticity: An Introduction to Mathematical Models, pp. 1–347. Imperial College Press, London, 2010. http://dx.doi.org/10.1142/p614CrossrefGoogle Scholar

  • [55] F. Mainardi, An historical perspective on fractional calculus in linear viscoelasticity. Fract. Calc. Appl. Anal. 15, No 4 (2012), 712–717; DOI:10.2478/s13540-012-0048-6; at http://link.springer.com/article/10.2478/s13540-012-0048-6. CrossrefGoogle Scholar

  • [56] F. Mainardi and G. Spada, Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur. J. Phys. 193 (2011), 133–160. Google Scholar

  • [57] T. Meidav, Viscoelastic properties of the standard linear solid. Geophys. Prospect. 12, No 1 (1964), 1365–2478. http://dx.doi.org/10.1111/j.1365-2478.1964.tb01891.xCrossrefGoogle Scholar

  • [58] S. I. Meshkov, G. N. Pachevskaya, V. S. Postnikov, and U. A. Rossikhin, Integral representations of ∋γ-functions and their application to problems in linear viscoelasticity. Int. J. Eng. Sci. 9, No 4 (1971), 387–398. http://dx.doi.org/10.1016/0020-7225(71)90059-0CrossrefGoogle Scholar

  • [59] R. Metzler and T. F. Nonnenmacher, Fractional relaxation processes and fractional rheological models for the description of a class of viscoelastic materials. Int. J. Plasticity 19, No 7 (2003), 941–959. http://dx.doi.org/10.1016/S0749-6419(02)00087-6CrossrefGoogle Scholar

  • [60] M. G. Mittag-Leffer, Sur la nouvelle fonction E α(x) (On the new function E α(x)). C. R. Acad. Sci. Paris 137 (1903), 554–558. Google Scholar

  • [61] R. Muthupillai, D. J. Lomas, P. J. Rossman, J. F. Greenleaf, A. Manduca, and R. L. Ehman, Magnetic resonance elastography by direct visualization of propagating acoustic strain waves. Science 269, No 5232 (1995), 1854–1857. http://dx.doi.org/10.1126/science.7569924CrossrefGoogle Scholar

  • [62] A. I. Nachman, J. F. Smith III, and R. C. Waag, An equation for acoustic propagation in inhomogeneous media with relaxation losses. J. Acoust. Soc. Am. 88 (1990), 1584–1595. http://dx.doi.org/10.1121/1.400317CrossrefGoogle Scholar

  • [63] S. P. Näsholm and S. Holm, Linking multiple relaxation, power-law attenuation, and fractional wave equations. J. Acoust. Soc. Am. 130, No 5 (2011), 3038–3045. http://dx.doi.org/10.1121/1.3641457CrossrefGoogle Scholar

  • [64] S. P. Näsholm and S. Holm, A fractional acoustic wave equation from multiple relaxation loss and conservation laws. Proc. 5th Int. Workshop on Fractional Differentiation and its Applications’ 2012, China. Google Scholar

  • [65] R. R. Nigmatullin, Theory of dielectric relaxation in non-crystalline solids: From a set of micromotions to the averaged collective motion in the mesoscale region. Physica B 358, No 1–4 (2005), 201–215. http://dx.doi.org/10.1016/j.physb.2005.01.173CrossrefGoogle Scholar

  • [66] R. F. O’Doherty and N. A. Anstey, Reflections on amplitudes. Geophys. Prosp. 19 (1971), 430–458. http://dx.doi.org/10.1111/j.1365-2478.1971.tb00610.xCrossrefGoogle Scholar

  • [67] M. L. Palmeri and K. R. Nightingale, Acoustic radiation force-based elasticity imaging methods. Interface Focus 1, No 4 (2011), 553–564. http://dx.doi.org/10.1098/rsfs.2011.0023CrossrefGoogle Scholar

  • [68] S. Papazoglou, S. Hirsch, J. Braun, and I. Sack, Multifrequency inversion in magnetic resonance elastography. Phys. Med. Biol. 57, No 8 (2012), 2329–2346. http://dx.doi.org/10.1088/0031-9155/57/8/2329CrossrefGoogle Scholar

  • [69] K. Papoulia, V. Panoskaltsis, N. Kurup, and I. Korovajchuk, Rheological representation of fractional order viscoelastic material models. Rheol. Acta 49 (2010), 381–400. http://dx.doi.org/10.1007/s00397-010-0436-yCrossrefGoogle Scholar

  • [70] H. Pauly and H. P. Schwan, Mechanism of absorption of ultrasound in liver tissue. J. Acoust. Soc. Am. 50, No 2B (1971), 692–699. http://dx.doi.org/10.1121/1.1912685CrossrefGoogle Scholar

  • [71] L. M. Petrovic, D. T. Spasic, and T. M. Atanackovic, On a mathematical model of a human root dentin. Dent. Mater. 21, No 2 (2005), 125–128. http://dx.doi.org/10.1016/j.dental.2004.01.004CrossrefGoogle Scholar

  • [72] I. Podlubny, Fractional Differential Equations, Ch. 10.2. Academic Press, New York, 1999. Google Scholar

  • [73] I. Podlubny, Fractional Differential Equations, Chs. 1–2. Academic Press, New York, 1999. Google Scholar

  • [74] F. Prieur and S. Holm, Nonlinear acoustic wave equations with fractional loss operators. J. Acoust. Soc. Am. 130, No 3 (2011), 1125–1132. http://dx.doi.org/10.1121/1.3614550CrossrefGoogle Scholar

  • [75] F. Prieur, G. Vilenskiy, and S. Holm, A more fundamental approach to the derivation of nonlinear acoustic wave equations with fractional loss operators. J. Acoust. Soc. Am. 132 (2012), 2169–2172. http://dx.doi.org/10.1121/1.4751540CrossrefGoogle Scholar

  • [76] T. Pritz, Analysis of four-parameter fractional derivative model of real solid materials. J. Sound. Vib. 195, No 1 (1996), 103–115. http://dx.doi.org/10.1006/jsvi.1996.0406CrossrefGoogle Scholar

  • [77] T. Pritz, Loss factor peak of viscoelastic materials: Magnitude to width relations. J. Sound. Vib. 246, No 2 (2001), 265–280. http://dx.doi.org/10.1006/jsvi.2001.3636CrossrefGoogle Scholar

  • [78] T. Pritz, Five-parameter fractional derivative model for polymeric damping materials. J. Sound. Vib. 265, No 5 (2003), 935–952. http://dx.doi.org/10.1016/S0022-460X(02)01530-4CrossrefGoogle Scholar

  • [79] H. Roitner, J. Bauer-Marschallinger, T. Berer, and P. Burgholzer, Experimental evaluation of time domain models for ultrasound attenuation losses in photoacoustic imaging. J. Acoust. Soc. Am. 131 (2012), 3763–3774. http://dx.doi.org/10.1121/1.3699194CrossrefGoogle Scholar

  • [80] Y. A. Rossikhin, Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Applied Mech. Rev. 63, No 1 (2010), 010701-1–010701-12. CrossrefGoogle Scholar

  • [81] Y. A. Rossikhin and M. V. Shitikova, Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50 (1997), 15–67. http://dx.doi.org/10.1115/1.3101682CrossrefGoogle Scholar

  • [82] Y. A. Rossikhin and M. V. Shitikova, Analysis of rheological equations involving more than one fractional parameters by the use of the simplest mechanical systems based on these equations. Mech. Time-Depend. Mat. 5, No 2 (2001), 131–175. http://dx.doi.org/10.1023/A:1011476323274CrossrefGoogle Scholar

  • [83] Y. A. Rossikhin and M. V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results. Appl. Mech. Rev. 63 (2010), 010801–1-25. http://dx.doi.org/10.1115/1.4000563CrossrefGoogle Scholar

  • [84] D. Royer and E. Dieulesaint, Elastic Waves in Solids, Vol. I. Springer, Berlin, 2000. http://dx.doi.org/10.1007/978-3-662-06938-7CrossrefGoogle Scholar

  • [85] I. Sack, B. Beierbach, J. Wuerfel, D. Klatt, U. Hamhaber, S. Papazoglou, P. Martus, and J. Braun, The impact of aging and gender on brain viscoelasticity. NeuroImage 46, No 3 (2009), 652–657. http://dx.doi.org/10.1016/j.neuroimage.2009.02.040CrossrefGoogle Scholar

  • [86] M. Sasso, G. Palmieri, and D. Amodio, Application of fractional derivative models in linear viscoelastic problems. Mech. Time-Depend. Mat. 15 (2011), 367–387. http://dx.doi.org/10.1007/s11043-011-9153-xCrossrefGoogle Scholar

  • [87] H. Schiessel and A. Blumen, Hierarchical analogues to fractional relaxation equations. J. Phys. A 26, No 19 (1993), 5057–5069. http://dx.doi.org/10.1088/0305-4470/26/19/034CrossrefGoogle Scholar

  • [88] H. Schiessel and A. Blumen, Mesoscopic pictures of Sol-Gel transition: Ladder models and fractal networks. Macromolecules 28 (1995), 4013–4019. http://dx.doi.org/10.1021/ma00115a038CrossrefGoogle Scholar

  • [89] M. Seredyńska and A. Hanyga, Relaxation, dispersion, attenuation, and finite propagation speed in viscoelastic media. J. Math. Phys. 51, No 9 (2010), 092901. http://dx.doi.org/10.1063/1.3478299CrossrefGoogle Scholar

  • [90] R. Sinkus, J.-L. Daire, V. Vilgrain, and B. E. Van Beers, Elasticity imaging via MRI: Basics, overcoming the waveguide limit, and clinical liver results. Curr. Med. Imaging Rev. 8, No 1 (2012), 56–63. http://dx.doi.org/10.2174/157340512799220544CrossrefGoogle Scholar

  • [91] R. Sinkus, J. Lorenzen, D. Schrader, M. Lorenzen, M. Dargatz, and D. Holz, High-resolution tensor MR elastography for breast tumour detection. Phys. Med. Biol. 45, No 6 (2000), 1649–1664. http://dx.doi.org/10.1088/0031-9155/45/6/317CrossrefGoogle Scholar

  • [92] R. Sinkus, K. Siegmann, T. Xydeas, M. Tanter, C. Claussen, and M. Fink, MR elastography of breast lesions: Understanding the solid/liquid duality can improve the specificity of contrast-enhanced MR mammography. Magn. Res. in Med. 58, No 6 (2007), 1135–1144. http://dx.doi.org/10.1002/mrm.21404CrossrefGoogle Scholar

  • [93] A. A. Stanislavsky, The stochastic nature of complexity evolution in the fractional systems. Chaos Soliton Fract. 34, No 1 (2007), 51–61. http://dx.doi.org/10.1016/j.chaos.2007.01.049CrossrefGoogle Scholar

  • [94] T. L. Szabo and J. Wu, A model for longitudinal and shear wave propagation in viscoelastic media. J. Acoust. Soc. Am. 107 (2000), 2437–2446. http://dx.doi.org/10.1121/1.428630CrossrefGoogle Scholar

  • [95] M. Tabei, T. D. Mast, and R. C. Waag, Simulation of ultrasonic focus aberration and correction through human tissue. J. Acoust. Soc. Am. 113, No 2 (2003), 1166–1176. http://dx.doi.org/10.1121/1.1531986CrossrefGoogle Scholar

  • [96] B. E. Treeby and B. T. Cox, Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian. J. Acoust. Soc. Am. 127 (2010), 2741–2748. http://dx.doi.org/10.1121/1.3377056CrossrefGoogle Scholar

  • [97] B. E. Treeby and B. T. Cox, Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian. J. Acoust. Soc. Am. 127 (2010), 2741–2748, Section IIB. http://dx.doi.org/10.1121/1.3377056CrossrefGoogle Scholar

  • [98] B. E. Treeby, J. Jaros, A. P. Rendell, and B. T. Cox, Modeling nonlinear ultrasound propagation in heterogeneous media with power law absorption using a k-space pseudospectral method. J. Acoust. Soc. Am. 131, No 6 (2012), 4324–4336. http://dx.doi.org/10.1121/1.4712021CrossrefGoogle Scholar

  • [99] B. E. Treeby, E. Z. Zhang, and B. T. Cox, Photoacoustic tomography in absorbing acoustic media using time reversal. Inverse Probl. 26, No 11 (2010), 115003. http://dx.doi.org/10.1088/0266-5611/26/11/115003CrossrefGoogle Scholar

  • [100] G. Vilensky, G. ter Haar, and N. Saffari, A model of acoustic absorption in fluids based on a continuous distribution of relaxation times. Wave Motion 49, No 1 (2012), 93–108. http://dx.doi.org/10.1016/j.wavemoti.2011.07.005CrossrefGoogle Scholar

  • [101] K. R. Waters, J. Mobley, and J. G. Miller, Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion. IEEE Trans. Ultrason. Ferroelectr., Freq. Control, 52, No 5 (2005), 822–833. http://dx.doi.org/10.1109/TUFFC.2005.1503968CrossrefGoogle Scholar

  • [102] R. L. Weaver and Y. H. Pao, Dispersion relations for linear wave propagation in homogeneous and inhomogeneous media. Journ. Math. Phys. 22 (1981), 1909–1918. http://dx.doi.org/10.1063/1.525164CrossrefGoogle Scholar

  • [103] K. Weron and A. Klauzer, Probabilistic basis for the Cole-Cole relaxation law. Ferroelectrics 236, No 1 (2000), 59–69. http://dx.doi.org/10.1080/00150190008016041CrossrefGoogle Scholar

  • [104] D. Widder, An Introduction to Transform Theory, Ch. 5.13. Pure and Applied Mathematics Ser., Academic Press, 1971. Google Scholar

  • [105] A. Wiman, Über den Fundamentalsatz in der Theorie der Funktionen E α(x) (About the fundamental theorem in the theory of the function E α(x)). Acta Mathematica 29 (1905), 191–201. http://dx.doi.org/10.1007/BF02403202CrossrefGoogle Scholar

  • [106] M. G. Wismer, Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation. J. Acoust. Soc. Am. 120 (2006), 3493–3502. http://dx.doi.org/10.1121/1.2354032CrossrefGoogle Scholar

  • [107] M. G. Wismer and R. Ludwig, An explicit numerical time domain formulation to simulate pulsed pressure waves in viscous fluids exhibiting arbitrary frequency power law attenuation. IEEE Trans. Ultrason. Ferroelectr., Freq. Control 42, No 6 (1995), 1040–1049. http://dx.doi.org/10.1109/58.476548CrossrefGoogle Scholar

  • [108] X. Yang and R. O. Cleveland, Time domain simulation of nonlinear acoustic beams generated by rectangular pistons with application to harmonic imaging. J. Acoust. Soc. Am. 117 (2005), 113–123. http://dx.doi.org/10.1121/1.1828671CrossrefGoogle Scholar

  • [109] T. K. Yasar, T. J. Royston, and R. L. Magin, Wideband MR elastography for viscoelasticity model identification. Magnet. Reson. Med., 2012, Online Version of Record published before inclusion in an issue. Google Scholar

About the article

Published Online: 2012-12-27

Published in Print: 2013-03-01


Citation Information: Fractional Calculus and Applied Analysis, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-013-0003-1.

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