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Subordination in a Class of Generalized Time-Fractional Diffusion-Wave Equations

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Abstract

Motivated by recently proposed generalizations of the diffusion-wave equation with the Caputo time fractional derivative of order α ∈ (1, 2), in the present survey paper a class of generalized time-fractional diffusion-wave equations is introduced. Its definition is based on the subordination principle for Volterra integral equations and involves the notion of complete Bernstein function. Various members of this class are surveyed, including the distributed-order time-fractional diffusion-wave equation and equations governing wave propagation in viscoelastic media with completely monotone relaxation moduli.

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References

  1. W. Arendt, C. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel (2011).

    MATH  Google Scholar 

  2. T. Atanacković, On a distributed derivative model of a viscoelastic body. C. R. Mécanique, 331 (2003), 687–692.

    MATH  Google Scholar 

  3. T. Atanacković, S. Konjik, L. Oparnica, D. Zorica, Thermodynamical restrictions and wave propagation for a class of fractional order viscoelastic rods. Abstr. Appl. Anal. 2011 (2011), Art. # 975694-1-32.

  4. T. Atanacković, S. Pilipović, D. Zorica, Diffusion wave equation with two fractional derivatives of different order. J. Phys. A: Math. Theor. 40 (2007), 5319–5333.

    MathSciNet  MATH  Google Scholar 

  5. T. Atanacković, S. Pilipović, D. Zorica, Time distributed-order diffusion-wave equation. I. Volterra-type equation. Proc. R. Soc. A 465 (2009), 1869–1891.

    MathSciNet  MATH  Google Scholar 

  6. T. Atanacković, S. Pilipović, D. Zorica, Time distributed-order diffusion-wave equation. II. Applications of Laplace and Fourier transformations. Proc. R. Soc. A 465 (2009), 1893–1917.

    MathSciNet  MATH  Google Scholar 

  7. T. Atanacković, S. Pilipović, B. Stanković, D. Zorica, Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes. John Wiley & Sons, London (2014).

    MATH  Google Scholar 

  8. R.L. Bagley, P.J. Torvik, On the fractional calculus model of viscoelastic behavior. J. Rheol. 30, No 198, 137–148.

  9. E. Bajlekova, Fractional Evolution Equations in Banach Spaces. PhD Thesis, Eindhoven University of Technology, Eindhoven (2001); available at: https://pure.tue.nl/ws/files/2442305/200113270.pdf.

    MATH  Google Scholar 

  10. E. Bazhlekova, Subordination principle for fractional evolution equations. Fract. Calc. Appl. Anal. 3, No 3 (2000), 213–230.

    MathSciNet  MATH  Google Scholar 

  11. E. Bazhlekova, Series solution of a nonlocal problem for a timefractional diffusion-wave equation with damping. C. R. Acad. Bulg. Sci. 66, No 8 (2013), 1091–1096.

    MathSciNet  MATH  Google Scholar 

  12. E. Bazhlekova, On a nonlocal boundary value problem for the two-term time-fractional diffusion-wave equation. AIP Conf. Proc. 1561 (2013), 172–183.

    Google Scholar 

  13. E. Bazhlekova, I. Bazhlekov, Viscoelastic flows with fractional derivative models: computational approach via convolutional calculus of Dimovski. Fract. Calc. Appl. Anal. 17, No 4 (2014), 954–976; DOI: 10.2478/s13540-014-0209-x; https://www.degruyter.com/view/j/fca.2014.17.issue-4/issue-files/fca.2014.17.issue-4.xml.

    MathSciNet  MATH  Google Scholar 

  14. E. Bazhlekova, Completely monotone functions and some classes of fractional evolution equations. Integr. Transf. Spec. Funct. 26 (2015), 737–752.

    MathSciNet  MATH  Google Scholar 

  15. E. Bazhlekova, K. Tsocheva, Fractional Burgers’ model: thermodynamic constraints and completely monotonic relaxation function. C. R. Acad. Bulg. Sci. 69, No 7 (2016), 825–834.

    MathSciNet  MATH  Google Scholar 

  16. E. Bazhlekova, I. Bazhlekov, Unidirectional flows of fractional Jeffreys fluids: Thermodynamic constraints and subordination. Comput. Math. Appl. 73 (2017), 1363–1376.

    MathSciNet  MATH  Google Scholar 

  17. E. Bazhlekova, I. Bazhlekov, Stokes’ first problem for viscoelastic fluids with a fractional Maxwell model. Fractal Fract. 1, No 1 (2017), Art. # 7; DOI:10.3390/fractalfract1010007.

    Google Scholar 

  18. E. Bazhlekova, I. Bazhlekov, Subordination approach to multi-term time-fractional diffusion-wave equation. J. Comput. Appl. Math. 339 (2018), 179–192.

    MathSciNet  MATH  Google Scholar 

  19. E. Bazhlekova, I. Bazhlekov, Complete monotonicity of the relaxation moduli of distributed-order fractional Zener model. AIP Conf. Proc. (2018) - AMEE 2018, To appear.

    Google Scholar 

  20. L. Boyadjiev, Yu. Luchko, The neutral-fractional telegraph equation. Math. Model. Nat. Phenom. 12 (2017), 51–67.

    MathSciNet  MATH  Google Scholar 

  21. M. Caputo, Mean fractional order derivatives: Differential equations and filters. Annals Univ. Ferrara-Sez. VII-Sc. Mat. XLI (1995), 73–84.

    MathSciNet  MATH  Google Scholar 

  22. M. Caputo, J.M. Carcione, Wave simulation in dissipative media described by distributed-order fractional time derivatives. J. Vib. Control 17 (2010), 1121–1130.

    MATH  Google Scholar 

  23. R.C. Cascaval, E.C. Eckstein, C.L. Frota, J.A. Goldstein, Fractional telegraph equations. J. Math. Anal. Appl. 276 (2002), 145–159.

    MathSciNet  MATH  Google Scholar 

  24. J. Chen, F. Liu, V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338 (2008), 1364–1377.

    MathSciNet  MATH  Google Scholar 

  25. J. Chen, F. Liu, V. Anh, S. Shen, Q. Liu, C. Liao, The analytical solution and numerical solution of the fractional diffusion-wave equation with damping. Appl. Math. Comput. 219 (2012), 1737–1748.

    MathSciNet  MATH  Google Scholar 

  26. I. Colombaro, A. Giusti, F. Mainardi, A class of linear viscoelastic models based on Bessel functions. Meccanica 52 (2017), 825–832.

    MathSciNet  MATH  Google Scholar 

  27. D.O. Craiem, R.L. Armentano, A fractional derivative model to describe arterial viscoelasticity. Biorheology 44 (2007), 251–263.

    Google Scholar 

  28. D.O. Craiem, F.J. Rojo, J.M. Atienza, R.L. Armentano, G.V. Guinea, Fractional-order viscoelasticity applied to describe uniaxial stress relaxation of human arteries. Phys. Med. Biol. 53 (2008), 4543.

    Google Scholar 

  29. X. Ding, G. Zhang, B. Zhao, Y. Wang, Unexpected viscoelastic deformation of tight sandstone: Insights and predictions from the fractional Maxwell model. Sci. Rep. 7 (2017), 11336; DOI: 10.1038/s41598-017- 11618-x.

    Google Scholar 

  30. V.A. Ditkin, A.P. Prudnikov, Integral Transforms and Operational Calculus. Pergamon Press, Oxford, New York (1965).

    MATH  Google Scholar 

  31. V.D. Djordjević, J. Jarić, B. Fabry, J.J. Fredberg, D. Stamenović, Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng. 31 (2003), 692–699.

    Google Scholar 

  32. W. Feller, An Introduction to Probability Theory and its Applications. Vol. 2, Wiley, New York (1971).

    MATH  Google Scholar 

  33. M. Ferreira, M.M. Rodrigues, N. Vieira, Fundamental solution of the multi-dimensional time fractional telegraph equation. Fract. Calc. Appl. Anal. 20 No 4 (2017), 868–894; DOI: 10.1515/fca-2017-0046; https://www.degruyter.com/view/j/fca.2017.20.issue-4/issue-files/fca.2017.20.issue-4.xml.

    MathSciNet  MATH  Google Scholar 

  34. A. Fitouhi, I. Jebabli, E.L. Shishkina, S.M. Sitnik, Applications of integral transforms composition method to wave-type singular differential equations and index shift transmutations. Electron. J. Differ. Eq. 2018, No 130 (2018), 1–27.

    MathSciNet  MATH  Google Scholar 

  35. Y. Fujita, Integrodifferential equations which interpolate the heat equation and the wave equation. Osaka J. Math. 27 (1990), 309–321.

    MathSciNet  MATH  Google Scholar 

  36. R. Garrappa, F. Mainardi, G. Maione, Models of dielectric relaxation based on completely monotone functions. Fract. Calc. Appl. Anal. 19, No 5 (2016), 1105–1160; DOI: 10.1515/fca-2016-0060; https://www.degruyter.com/view/j/fca.2016.19.issue-5/issue-files/fca.2016.19.issue-5.xml.

    MathSciNet  MATH  Google Scholar 

  37. R. Gorenflo, Stochastic processes related to time-fractional diffusionwave equation. Commun. Appl. Ind. Math. 6, No 2 (2015), e-531; DOI: 10.1685/journal.caim.531.

    Google Scholar 

  38. R. Gorenflo, A. Iskenderov, Yu. Luchko, Mapping between solutions of fractional diffusion-wave equations. Fract. Calc. Appl. Anal. 3 (2000), 75–86. 894 E. Bazhlekova

    MathSciNet  MATH  Google Scholar 

  39. R. Gorenflo, A. Kilbas, F. Mainardi, S. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin, Heidelberg (2014).

    MATH  Google Scholar 

  40. R. Gorenflo, Y. Luchko, M. Stojanović, Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density. Fract. Calc. Appl. Anal. 16, No 2 (2013), 297–316; DOI: 10.2478/s13540-013-0019-6; https://www.degruyter.com/view/j/fca.2013.16.issue-2/issue-files/fca.2013.16.issue-2.xml.

    MathSciNet  MATH  Google Scholar 

  41. R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order. In: A. Carpinteri, F. Mainardi (Eds.) Fractals and Fractional Calculus in Continuum Mechanics, Springer- Verlag, Wien/ New York (1997), 223–276.

    MATH  Google Scholar 

  42. R. Gorenflo, F. Mainardi, Subordination pathways to fractional diffusion. Eur. Phys. J. Special Topics 193 (2011), 119–132.

    Google Scholar 

  43. R. Gorenflo, R. Rutman, On ultraslow and intermediate processes. In: “Transform Methods and Special Functions, Sofia’ 94” (Proc. 1st Int. Workshop), P. Rusev, I. Dimovski, V. Kiryakova (Eds.) SCTP, Singapore (1995), 61–81.

    Google Scholar 

  44. A. Hanyga, Multidimensional solutions of time-fractional diffusionwave equations. Proc. R. Soc. Lond. A 458 (2002), 933–957.

    MathSciNet  MATH  Google Scholar 

  45. A. Hanyga, Physically acceptable viscoelastic models. In: K. Hutter and Y. Wang (Eds.), Trends in Applications of Mathematics to Mechanics, Shaker Verlag GmbH, Aachen (2005), 125–136.

    Google Scholar 

  46. A. Hanyga, Wave propagation in linear viscoelastic media with completely monotonic relaxation moduli. Wave Motion 50 (2013), 909–928.

    MathSciNet  MATH  Google Scholar 

  47. A. Hernández-Jiménez, J. Hernández-Santiago, A. Macias-García, J. Sánchez-González, Relaxation modulus in PMMA and PTFE fitting by fractional Maxwell model. Polym. Test. 21 (2002), 325–331.

    Google Scholar 

  48. R. Hilfer, Applications of Fractional Calculus in Physics. World Sci. Publ., Singapore (2000).

    MATH  Google Scholar 

  49. S. Holm, S.P. Näsholm, F. Prieur, R. Sinkus, Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations. Comput. Math. Appl. 66 (2013), 621–629.

    MathSciNet  MATH  Google Scholar 

  50. S. Holm, S.P. Näsholm, Comparison of fractional wave equations for power law attenuation in ultrasound and elastography. Ultrasound Med. Biol. 40, No 4 (2014), 695–703.

    Google Scholar 

  51. S. Holm, M.B. Holm, Restrictions on wave equations for passive media. J. Acoust. Soc. Am. 142 (2017), 1888–1896.

    Google Scholar 

  52. A. Jaishankar, G.H. McKinley, A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids. J. Rheol. 58 (2014), 1751–1788.

    Google Scholar 

  53. H. Jiang, F. Li, I. Turner, K. Burrage, Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Comput. Math. Appl. 64 (2012), 3377–3388.

    MathSciNet  MATH  Google Scholar 

  54. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. North-Holland Mathematics studies, Elsevier, Amsterdam (2006).

    MATH  Google Scholar 

  55. A. Kochubei, General fractional calculus, evolution equations, and renewal processes. Integr. Equ. Oper. Theory 71 (2011), 583–600.

    MathSciNet  MATH  Google Scholar 

  56. A. Kochubei, Cauchy problem for fractional diffusion-wave equations with variable coefficients. Applicable Analysis 93 (2014), 2211–2242.

    MathSciNet  MATH  Google Scholar 

  57. A. Kochubei, Asymptotic properties of solutions of the fractional diffusion-wave equation. Fract. Calc. Appl. Anal. 17, No 3 (2014), 881–896; DOI: 10.2478/s13540-014-0203-3; https://www.degruyter.com/view/j/fca.2014.17.issue-3/issue-files/fca.2014.17.issue-3.xml.

    MathSciNet  MATH  Google Scholar 

  58. S. Konjik, L. Oparnica, D. Zorica, Distributed order fractional constitutive stress-strain relation in wave propagation modeling. arXiv Preprint, arXiv:1709.01339v2 (2018).

    Google Scholar 

  59. M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20, No 1 (2017), 7–51; DOI: 10.1515/fca- 2017-0002; https://www.degruyter.com/view/j/fca.2017.20.issue-1/issue-files/fca.2017.20.issue-1.xml.

    MathSciNet  MATH  Google Scholar 

  60. C.G. Li, M. Kostić, M. Li, S. Piskarev, On a class of time-fractional differential equations. Fract. Calc. Appl. Anal. 15, No 4 (2012), 639–668; DOI: 10.2478/s13540-012-0044-x; https://www.degruyter.com/view/j/fca.2012.15.issue-4/issue-files/fca.2012.15.issue-4.xml.

    MathSciNet  MATH  Google Scholar 

  61. Y. Liu, L. Zheng, X. Zhang, Unsteady MHD Couette flow of a generalized Oldroyd-B fluid with fractional derivative. Comput. Math. Appl. 61 (2011), 443–450.

    MathSciNet  MATH  Google Scholar 

  62. J. Lubliner, V.P. Panoskaltsis, The modified Kuhn model of linear viscoelasticity. Int. J. Solids Struct. 29, No 24 (1992), 3099–3112.

    MATH  Google Scholar 

  63. Yu. Luchko, Fractional wave equation and damped waves. J. Math. Phys. 54 (2013), Art. # 031505.

  64. Yu. Luchko, Multi-dimensional fractional wave equation and some properties of its fundamental solution. Commun. Appl. Ind. Math. 6 (2014), e–485.

    MathSciNet  Google Scholar 

  65. Yu. Luchko, Subordination principles for the multi-dimensional space-time-fractional diffusion-wave equation. arXiv Preprint, arXiv:1802.04752 (2018).

    Google Scholar 

  66. Yu. Luchko, R. Gorenflo, An operational method for solving fractional differential equations with the Caputo derivatives. Acta Math. Vietnamica 24 (1999), 207–233.

    MathSciNet  MATH  Google Scholar 

  67. Yu. Luchko, F. Mainardi, Y. Povstenko, Propagation speed of the maximum of the fundamental solution to the fractional diffusion-wave equation. Comput. Math. Appl. 66 (2013), 774–784; DOI:10.1016/j.camwa.2013.01.005.

    MathSciNet  MATH  Google Scholar 

  68. Yu. Luchko, M. Yamamoto, General time fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems. Fract. Calc. Appl. Anal. 19, No 3 (2016), 676–695; DOI: 10.1515/fca-2016-0036; https://www.degruyter.com/view/j/fca.2016.19.issue-3/issue-files/fca.2016.19.issue-3.xml.

    MathSciNet  MATH  Google Scholar 

  69. J.A.T. Machado, V. Kiryakova, The chronicles of fractional calculus. Fract. Calc. Appl. Anal. 20, No 2 (2017), 307–336; DOI: 10.1515/fca- 2017-0017; https://www.degruyter.com/view/j/fca.2017.20.issue-2/issue-files/fca.2017.20.issue-2.xml.

    MathSciNet  MATH  Google Scholar 

  70. F. Mainardi, Fractional relaxation-oscillation and fractional diffusionwave phenomena. Chaos Soliton. Fract. 7 (1996), 1461–1477.

    MATH  Google Scholar 

  71. F. Mainardi, The fundamental solutions for the fractional diffusionwave equation, Appl. Math. Lett. 9 (1996), 23–28.

    MathSciNet  MATH  Google Scholar 

  72. F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010).

    MATH  Google Scholar 

  73. 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; https://www.degruyter.com/view/j/fca.2012.15.issue-4/issue-files/fca.2012.15.issue-4.xml.

    MathSciNet  MATH  Google Scholar 

  74. F. Mainardi, Yu. Luchko, G. Pagnini, The fundamental solution of the space-time fractional diffusion equation. Fract. Calc. Appl. Anal. 4 (2001), 153–192.

    MathSciNet  MATH  Google Scholar 

  75. F. Mainardi, G. Pagnini, R. Gorenflo, Mellin transform and subordination laws in fractional diffusion processes. Fract. Calc. Appl. Anal. 6, No 4 (2003), 441–459.

    MathSciNet  MATH  Google Scholar 

  76. F. Mainardi, P. Paradisi, Fractional diffusive waves. J. Comput. Acoust. 9 (2001), 1417–1436.

    MathSciNet  MATH  Google Scholar 

  77. F. Mainardi, G. Spada, Creep, relaxation and viscosity properties for basic fractional models in rheology. Eur. Phys. J. Special Topics 193 (2011), 133–160.

    Google Scholar 

  78. S. Majumdar, S. Hazra, M. Choudhury, S. Sinha, S. Das, T. Middya, S. Tarafdar, T. Dutta, A study of the rheological properties of visco-elastic materials using fractional calculus. Colloid Surface A 516 (2017), 181–189.

    Google Scholar 

  79. M.O. Mamchuev, Solutions of the main boundary value problems for the time-fractional telegraph equation by the Green function method. Fract. Calc. Appl. Anal. 20, No 1 (2017), 190–211; DOI: 10.1515/fca2017-0010; https://www.degruyter.com/view/j/fca.2017.20.issue-1/issue-files/fca.2017.20.issue-1.xml.

    MathSciNet  MATH  Google Scholar 

  80. M. Meerschaert, R. Schilling, A. Sikorskii, Stochastic solutions for fractional wave equations. Nonlinear Dynam. 80 (2015), 1685–1695.

    MathSciNet  MATH  Google Scholar 

  81. M. Meerschaert, B. Toaldo, Relaxation patterns and semi-Markov dynamics. arXiv Preprint, arXiv:1506.02951 (2015).

    Google Scholar 

  82. L. Miller, M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation. Inverse Probl. 29 (2013), Art. # 075013.

  83. S.P. Näsholm, S. Holm, On a fractional Zener elastic wave equation. Fract. Calc. Appl. Anal. 16, No 1 (2013), 26–50; DOI: 10.2478/s13540- 013-0003-1; https://www.degruyter.com/view/j/fca.2013.16.issue-1/issue-files/fca.2013.16.issue-1.xml.

    MathSciNet  MATH  Google Scholar 

  84. E. Orsingher, L. Beghin, Time-fractional telegraph equations and telegraph processes with brownian time. Probab. Theory Relat. Fields 128 (2004), 141–160.

    MathSciNet  MATH  Google Scholar 

  85. E. Orsingher, X. Zhao, The space-fractional telegraph equation and the related fractional telegraph processes. Chin. Ann. Math., Ser. B 45 (2003), 24–45.

    MATH  Google Scholar 

  86. J. Paneva-Konovska, From Bessel to Multi-Index Mittag-Leffler Functions: Enumerable Families, Series in them and Convergence. World Sci. Publ., London (2016).

    MATH  Google Scholar 

  87. J. Paneva-Konovska, Overconvergence of series in generalized Mittag- Leffler functions. Fract. Calc. Appl. Anal. 20, No 2 (2017), 506–520; DOI: 10.1515/fca-2017-0026; https://www.degruyter.com/view/j/fca.2017.20.issue-2/issue-files/fca.2017.20.issue-2.xml.

    MathSciNet  MATH  Google Scholar 

  88. P. Perdikaris, G. Karniadakis, Fractional-order viscoelasticity in onedimensional blood flow models. Ann. Biomed. Eng. 42, No 5 (2014), 1012–1023; DOI: 10.1007/s10439-014-0970-3.

    Google Scholar 

  89. L.M. Petrovic, D.T. Spasic, T.M. Atanackovic, On a mathematical model of a human root dentin. Dent. Mater. 21, No 2 (2005), 125–128.

    Google Scholar 

  90. L. Preziosi, D.D. Joseph, Stokes’ first problem for viscoelastic fluids. J. Non Newtonian Fluid Mech. 25 (1987), 239–259.

    MATH  Google Scholar 

  91. J. Prüss, Evolutionary Integral Equations and Applications. Birkhäuser, Basel, Boston, Berlin (1993).

    MATH  Google Scholar 

  92. H. Qi, M. Xu, Some unsteady unidirectional flows of a generalized Oldroyd-B fluid with fractional derivative. Appl. Math. Model. 33 (2009), 4184–4191.

    MathSciNet  MATH  Google Scholar 

  93. H. Qi, X. Guo, Transient fractional heat conduction with generalized Cattaneo model. Int. J. Heat Mass Transfer. 76 (2014), 535–539.

    Google Scholar 

  94. H. Qi, H. Xu, X. Guo, The Cattaneo-type time fractional heat conduction equation for laser heating. Comput. Math. Appl. 66 (2013), 824–831.

    MathSciNet  MATH  Google Scholar 

  95. S. Rogosin, F. Mainardi, George William Scott Blair–the pioneer of factional calculus in rheology. Commun. Appl. Ind. Math. 6, No 1 (2014), e-481; DOI: 10.1685/journal.caim.481.

    Google Scholar 

  96. Y.A. Rossikhin, M.V. Shitikova, Analysis of rheological equations involving more than one fractional parameter by the use of the simplest mechanical systems based on these equations. Mech. Time-Depend. Mat. 5 (2001), 131–175.

    Google Scholar 

  97. Y.A. Rossikhin, M.V. Shitikova, Analysis of the viscoelastic rod dynamics via models involving fractional derivatives or operators of two different orders. Shock Vib. Digest 36, No 1 (2004), 3–26.

    Google Scholar 

  98. Y.A. Rossikhin, M.V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results. Appl. Mech. Rev. 63 (2010), Art. # 010801-1-25.

  99. K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusionwave equations and applications to some inverse problems. J. Math. Anal. Apl. 382 (2012), 426–447.

    MATH  Google Scholar 

  100. T. Sandev, A. Chechkin, N. Korabel, H. Kantz, I.M. Sokolov, R. Metzler, Distributed-order diffusion equations and multifractality: Models and solutions. Phys. Rev. E 92 (2015) Art. # 042117; DOI:10.1103/PhysRevE.92.042117.

  101. T. Sandev, I.M. Sokolov, R. Metzler, A. Chechkin, Beyond monofractional kinetics. Chaos Soliton. Fact. 102 (2017), 210–217.

    MathSciNet  MATH  Google Scholar 

  102. T. Sandev, R. Metzler, A. Chechkin, From continuous time random walks to the generalized diffusion equation. Fract. Calc. Appl. Anal. 21, No 1 (2018), 10–28; DOI: 10.1515/fca-2018-0002; https://www.degruyter.com/view/j/fca.2018.21.issue-1/issue-files/fca.2018.21.issue-1.xml.

    MathSciNet  MATH  Google Scholar 

  103. W.R. Schneider, W. Wyss, Fractional diffusion and wave equations. J. Math. Phys. 30 (1989), 134–144.

    MathSciNet  MATH  Google Scholar 

  104. H. Schiessel, R. Metzler, A. Blumen, T.F. Nonnenmacher, Generalized viscoelastic models: Their fractional equations with applications. J. Phys. A 28 (1995), 6567–6584.

    MATH  Google Scholar 

  105. R. Schilling, R. Song, Z. Vondraček, Bernstein Functions: Theory and Applications. De Gruyter, Berlin (2010).

    MATH  Google Scholar 

  106. D. Song, T. Jiang, Study on the constitutive equation with fractional derivative for the viscoelastic fluids - Modified Jeffreys model and its application. Rheol. Acta 37 (1998), 512–517.

    Google Scholar 

  107. N. Su, P.N. Nelson, S. Connor, The distributed-order fractional diffusion-wave equation of groundwater flow: Theory and application to pumping and slug tests. J. Hydrol. 529, No 3 (2015), 1262–1273.

    Google Scholar 

  108. W. Tan, W. Pan, M. Xu, A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates. Int. J. Non Linear Mech. 38 (2003), 645–650.

    MATH  Google Scholar 

  109. W. Tan, M. Xu, Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model. Acta Mech. Sin. 18 (2002), 342–349.

    MathSciNet  Google Scholar 

  110. Ž Tomovski, T. Sandev, Effects of a fractional friction with power-law memory kernel on string vibrations. Comput. Math. Appl. 62 (2011), 1554–1561.

    MathSciNet  MATH  Google Scholar 

  111. V. Vergara, R. Zacher, Optimal decay estimates for time-fractional and other non-local subdiffusion equations via energy methods. SIAM J. Math. Anal. 47, No 1 (2015), 210–239.

    MathSciNet  MATH  Google Scholar 

  112. D. Yang, K. Zhu, Start-up flow of a viscoelastic fluid in a pipe with a fractional Maxwell’s model. Comput. Math. Appl. 60 (2010), 2231–2238.

    MathSciNet  MATH  Google Scholar 

  113. P. Yang, K. Zhu, Thermodynamic compatibility and mechanical analogue of the generalized Jeffreys and generalized Oldroyd-B fluids with fractional derivatives. Sci. China - Phys. Mech. Astron. 54 (2011), 737–742.

    Google Scholar 

  114. Y. Yin, K. Zhu, Oscillating flow of a viscoelastic fluid in a pipe with the fractional Maxwell model. Appl. Math. Comput. 173 (2006), 231–242.

    MathSciNet  MATH  Google Scholar 

  115. C. Zhao, C. Yang, Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels. Appl. Math. Comput. 211, No 2 (2009), 502–509.

    MathSciNet  MATH  Google Scholar 

  116. L. Zheng, Z. Guo, X. Zhang, 3D flow of a generalized Oldroyd-B fluid induced by a constant pressure gradient between two side walls perpendicular to a plate. Nonlinear Anal. RWA 12 (2011), 3499–3508.

    MathSciNet  MATH  Google Scholar 

  117. L. Zheng, X. Zhang, Modeling and Analysis of Modern Fluid Problems. Academic Press, Cambridge, MA, USA (2017).

    MATH  Google Scholar 

  118. Y. Zhou, J.R. Wang, L. Zhang, Basic Theory of Fractional Differential Equations. World Sci. Publ., London (2016).

    Google Scholar 

  119. D. Zorica, S. Cvetićanin, Fractional telegrapher’s equation as a consequence of Cattaneo’s heat conduction law generalization. Theoretical and Applied Mechanics 45, No 1 (2018), 35–51.

    MATH  Google Scholar 

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  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad. G. Bontchev” Str., Block 8, Sofia, 1113, Bulgaria

    Bazhlekova Emilia

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Emilia, B. Subordination in a Class of Generalized Time-Fractional Diffusion-Wave Equations. FCAA 21, 869–900 (2018). https://doi.org/10.1515/fca-2018-0048

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