Distributed-order wave equations with composite time fractional derivative

References

• I. Ali and N.A. Malik, Hilfer fractional advection–diffusion equations with power-law initial condition: A numerical study using variational iteration method, Comput. Math. Appl. 68 (2014), pp. 1161–1179. doi: 10.1016/j.camwa.2014.08.021
   Web of Science ®Google Scholar
• V.M. Bulavatsky, Mathematical modeling of fractional differential filtration dynamics based on models with Hilfer-Prabhakar derivative, Cybern. Syst. Anal. 53 (2017), pp. 204–216. doi: 10.1007/s10559-017-9920-z
   Web of Science ®Google Scholar
• W. Cai, W. Chen, and W. Xu, Fractional modeling of Pasternak-type viscoelastic foundation, Mech. Time-Depend. Mater. 21 (2017), pp. 119–131. doi: 10.1007/s11043-016-9321-0
   Web of Science ®Google Scholar
• M. Caputo, Elasticità e Dissipazione, Zanichelli, Bologna, 1969.
   Google Scholar
• A.V. Chechkin, R. Gorenflo, and I.M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E 66 (2002), p. 046129. doi: 10.1103/PhysRevE.66.046129
   Web of Science ®Google Scholar
• A.V. Chechkin, R. Gorenflo, I.M. Sokolov, and V.Yu. Gonchar, Distributed order fractional diffusion equation, Fract. Calc. Appl. Anal. 6 (2003), pp. 259–279.
   Google Scholar
• K. Diethelm and N.J. Ford, Numerical analysis for distributed-order differential equations, J. Comput. Appl. Math. 225 (2009), pp. 96–104. doi: 10.1016/j.cam.2008.07.018
   Web of Science ®Google Scholar
• J.L.A. Dubbeldam, Z. Tomovski, and T. Sandev, Space–time fractional Schrödinger equation with composite time fractional derivative, Fract. Calc. Appl. Anal. 18 (2015), pp. 1179–1200. doi: 10.1515/fca-2015-0068
   Web of Science ®Google Scholar
• F. Durbin, Numerical inversion of Laplace transforms: An efficient improvement to Dubner and Abate's method, Comput. J. 17 (1974), pp. 371–376. doi: 10.1093/comjnl/17.4.371
   Web of Science ®Google Scholar
• A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcedential Functions, Vol. 3, McGraw-Hill, New York, 1955.
   Google Scholar
• W. Feller, An Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York, 1968.
   Google Scholar
• K.M. Furati, M.D. Kassim, and N. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl. 64 (2012), pp. 1616–1626. doi: 10.1016/j.camwa.2012.01.009
   Web of Science ®Google Scholar
• R. Garra, R. Gorenflo, F. Polito, and Z. Tomovski, Hilfer–Prabhakar derivatives and some applications, Appl. Math. Comput. 242 (2014), pp. 576–589.
   Web of Science ®Google Scholar
• R.A. Ghanam, N. Malik, and N. Tatar, Transparent boundary conditions for a diffusion problem modified by Hilfer derivative, J. Math. Sci. Univ. Tokyo 21 (2014), pp. 129–152.
   Google Scholar
• R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
   Google Scholar
• R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, Chem. Phys. 284 (2002), pp. 399–408. doi: 10.1016/S0301-0104(02)00670-5
   Web of Science ®Google Scholar
• R. Hilfer, On fractional relaxation, Fractals 11 (2003), pp. 251–257. doi: 10.1142/S0218348X03001914
   Web of Science ®Google Scholar
• R. Hilfer, Y. Luchko, and Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009), pp. 299–318.
   Google Scholar
• J. Liang and Y.Q. Chen, Hybrid symbolic and numerical simulation studies of time-fractional order wave-diffusion systems, Int. J. Control 79 (2006), pp. 1462–1470. doi: 10.1080/00207170600726493
   Web of Science ®Google Scholar
• A. Liemert, T. Sandev, and H. Kantz, Generalized Langevin equation with tempered memory kernel, Physica. A. 466 (2017), pp. 356–369. doi: 10.1016/j.physa.2016.09.018
   Web of Science ®Google Scholar
• F. Mainardi, Fractional Calculus and Waves in Linear Viscoelesticity: An introduction to Mathematical Models, Imperial College Press, London, 2010.
   Google Scholar
• F. Mainardi and P. Paradisi, Fractional diffusive waves, J. Comput. Acoust. 9 (2001), p. 1417–1436. doi: 10.1142/S0218396X01000826
   Web of Science ®Google Scholar
• A. Mallet, Numerical inversion of Laplace transform, Wolfram Library Archive, Item 0210–968 (2000).
   Google Scholar
• A.M. Mathai, R.K. Saxena, and H.J. Haubold, The H-function: Theory and Applications, Springer, New York, 2010.
   Google Scholar
• R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), pp. 1–77. doi: 10.1016/S0370-1573(00)00070-3
   Web of Science ®Google Scholar
• R.R. Nigmatullin, Fractional integral and its physical interpretation, Theor. Math. Phys. 90 (1992), pp. 242–251. doi: 10.1007/BF01036529
   Web of Science ®Google Scholar
• J. Paneva-Konovska, Convergence of series in three parametric Mittag–Leffler functions, Math. Slovaca 64 (2014), pp. 73–84. doi: 10.2478/s12175-013-0188-0
   Web of Science ®Google Scholar
• J. Paneva-Konovska, From Bessel to Multi-Index Mittag–Leffler Functions: Enumerable Families, Series in them and Convergence, World Scientific, London, 2016.
   Google Scholar
• J. Paneva-Konovska, Overconvergence of series in generalized Mittag–Leffler functions, Fract. Calc. Appl. Anal. 20 (2017), pp. 506–520. doi: 10.1515/fca-2017-0026
   Web of Science ®Google Scholar
• F. Polito and Z. Tomovski, Some properties of Prabhakar-type fractional calculus operators, Frac. Diff. Calc. 6 (2016), pp. 73–94.
   Google Scholar
• T.R. Prabhakar, A singular integral equation with a generalized Mittag–Leffler function in the kernel, Yokohama Math. J. 19 (1971), pp. 7–15.
   Google Scholar
• T. Sandev and Z. Tomovski, The general time fractional wave equation for a vibrating string, J. Phys. A: Math. Theoret. 43 (2010), p. 055204. doi: 10.1088/1751-8113/43/5/055204
   Web of Science ®Google Scholar
• T. Sandev, R. Metzler, and Z. Tomovski, Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative, J. Phys. A: Math. Theoret. 44 (2011), p. 255203. doi: 10.1088/1751-8113/44/25/255203
   Web of Science ®Google Scholar
• T. Sandev, Z. Tomovski, and J.L.A. Dubbeldam, Generalized Langevin equation with a three parameter Mittag–Leffler noise, Physica. A. 390 (2011), pp. 3627–3636. doi: 10.1016/j.physa.2011.05.039
   Web of Science ®Google Scholar
• T. Sandev, Z. Tomovski, and B. Crnkovic, Generalized distributed order diffusion equations with composite time fractional derivative, Comput. Math. Appl. 73 (2017), pp. 1028–1040. doi: 10.1016/j.camwa.2016.07.009
   Web of Science ®Google Scholar
• R.K. Saxena, A.M. Mathai, and H.J. Haubold, Space–time fractional reaction-diffusion equations associated with a generalized Riemann–Liouville fractional derivative, Axioms 3 (2014), pp. 320–334. doi: 10.3390/axioms3030320
   Google Scholar
• R.K. Saxena, Z. Tomovski, and T. Sandev, Fractional Helmholtz and fractional wave equations with Riesz-Feller and generalized Riemann–Liouville fractional derivatives, Eur. J. Pure Appl. Math. 7 (2014), pp. 312–334.
   Google Scholar
• R.K. Saxena, A.M. Mathai, and H.J. Haubold, Computable solutions of fractional reaction-diffusion equations associated with generalized Riemann–Liouville fractional derivatives of fractional orders, preprint (2015). Available at arXiv:1502.06127v1.
   Google Scholar
• R.K. Saxena, R. Garra, and E. Orsingher, Analytical solution of space-time fractional telegraph-type equations involving Hilfer and Hadamard derivatives, Integral Transform. Spec. Funct. 27 (2015), pp. 30–42. doi: 10.1080/10652469.2015.1092142
   Web of Science ®Google Scholar
• H.M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009), pp. 198–210.
   Web of Science ®Google Scholar
• H.G. Sun, Y. Zhang, W. Chen, and D.M. Reeves, Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media, J. Contaminant Hydrol. 157 (2014), pp. 47–58. doi: 10.1016/j.jconhyd.2013.11.002
   PubMed Web of Science ®Google Scholar
• Z. Tomovski, Generalized Cauchy type problems for nonlinear fractional differential equations with composite fractional derivative operator, Nonlinear Anal. 75 (2012), pp. 3364–3384. doi: 10.1016/j.na.2011.12.034
   Web of Science ®Google Scholar
• Z. Tomovski and T. Sandev, Effects of a fractional friction with power-law memory kernel on string vibrations, Comput. Math. Appl. 62 (2011), pp. 1554–1561. doi: 10.1016/j.camwa.2011.04.042
   Web of Science ®Google Scholar
• Z. Tomovski, R. Hilfer, and H.M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag–Leffler type functions, Integral Transform. Spec. Funct. 21 (2010), pp. 797–814. doi: 10.1080/10652461003675737
   Web of Science ®Google Scholar
• Z. Tomovski, T. Sandev, R. Metzler, and J. Dubbeldam, Generalized space-time fractional diffusion equation with composite fractional time derivative, Physica. A. 391 (2012), pp. 2527–2542. doi: 10.1016/j.physa.2011.12.035
   Web of Science ®Google Scholar
• S. Wei, W. Chen, and J. Zhang, Time-fractional derivative model for chloride ions sub-diffusion in reinforced concrete, Eur. J. Environ. Civil Eng. 21 (2017), pp. 319–331. doi: 10.1080/19648189.2015.1116467
   Web of Science ®Google Scholar