Fractional wave equations with attenuation

Fractional Calculus and Applied Analysis

Volumn 16, Issue 1, 2013, Pages 262-272

  Straka, Peter ^a   Meerschaert, Mark M ^b   McGough, Robert J ^c   Zhou, Yuzhen ^b  

^a UNIVERSITY OF MANCHESTER   (United Kingdom)

^b MICHIGAN STATE UNIVERSITY   (United States)

^c Michigan State University   (United States)

Author keywords

attenuation; continuous time random walk; dispersion; fractional derivative; stable law; subordination; wave equation

Indexed keywords

EID: 84871765883     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-013-0016-9     Document Type: Article

References (30)

1. O.P. Agrawal, A general solution for the fourth-order fractional diffusion-wave equation. Fract. Calc. Appl. Anal. 3, No 1 (2000), 1-12.
2. B. Baeumer, D.A. Benson and M.M. Meerschaert, Advection and dispersion in time and space. Phys. A 350, No 2-4 (2005), 245-262.
3. P. Becker-Kern, M.M. Meerschaert and H.P. Scheffler, Limit theorem for continuous time random walks with two time scales. J. Applied Probab. 41, No 2 (2004), 455-466.
4. D.T. Blackstock, Transient solution for sound radiated into a viscous fluid. J. Acoust. Soc. Am. 41 (1967), 1312-1319.
5. M. Caputo. Linear models of dissipation whose Q is almost frequency independent-II. Geophys. J. R. Astron. Soc. 13, No 5 (1967), 529-539; Reprinted in: Fract. Calc. Appl. Anal. 11, No 1 (2008), 3-14.
6. 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.
7. W. Chen and S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Am. 115, No 4 (2004), 1424-1430.
8. F.A. Duck, Physical Properties of Tissue. Academic Press, Boston (1990).
9. R. Gorenflo, A. Iskenderov and Yu. Luchko, Mapping between solutions of fractional diffusion-wave equations. Fract. Calc. Appl. Anal. 3, No 1 (2000), 75-86.
10. N. Jacob, Pseudo Differential Operators and Markov Processes, Vol. I. Imperial College Press, London (2001).
11. J.F. Kelly, R.J. McGough, and M.M. Meerschaert, Time-domain 3D Green's functions for power law media. J. Acoust. Soc. Am. 124, No 5 (2008), 2861-2872.
12. J. Kemppainen, Solvability of a Dirichlet problem for a time fractional diffusion-wave equation in Lipschitz domains. Fract. Calc. Appl. Anal. 15, No 2 (2012), 195-206; DOI:10.2478/s13540-012-0014-3; at http://link.springer.com/ article/10.2478/s13540-012-0014-3.
13. A.A. Kilbas, J.J. Trujillo and A.A. Voroshilov, Cauchy-Type problem for diffusion-wave equation with the Riemann-Liouville partial derivative. Fract. Calc. Appl. Anal. 8, No 4 (2005), 403-430; at http://www.math.bas.bg/~fcaa.
14. V.N. Kolokoltsov, Generalized continuous-time random walks, subordination by hitting times, and fractional dynamics. Th. Probab. Appl. 53, No 4 (2009), 594-609.
15. 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, No 5 (2004), 2742-2750.
16. F. Liu, M.M. Meerschaert, R.J. McGough, P. Zhuang and Q. Liu, Numerical methods for solving the multi-term time-fractional wavediffusion equation. Fract. Calc. Appl. Anal. 16,No 1 (2013), 9-25 (same issue); DOI:10.2478/s13540- 013-0002-2; at http://link.springer.com/journal/13540.
17. F. Mainardi, Fractional Calculus and Waves in Linear Viscoleasticity. Imperial College Press, London (2010).
18. M.M. Meerschaert, D.A. Benson, H.P. Scheffler, and B. Baeumer, Stochastic solution of space-time fractional diffusion equations. Phys. Rev. E 65, No 4 (2002), 1103-1106.
19. M.M. Meerschaert and H.P. Scheffler, Limit theorems for continuous time random walks with infinite mean waiting times. J. Applied Probab. 41, No 3 (2004), 623-638.
20. M.M. Meerschaert and H.-P. Scheffler, Triangular array limits for continuous time random walks. Stoch. Proc. Appl. 118, No 9 (2008), 1606-1633.
21. M.M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus. De Gruyter, Berlin (2012).
22. M.M. Meerschaert, P. Straka, Y. Zhou, and J. McGough, Stochastic solution to a time-fractional attenuated wave equation. Nonlinear Dynamics 70,No. 2 (2012), 1273-1281.
23. Y. Povstenko, Non-central-symmetric solution to timefractional diffusion-wave equation in a sphere under Dirichlet boundary condition. Fract. Calc. Appl. Anal. 15, No 2 (2012), 253-266; DOI:10.2478/s13540-012-0019-y; at http://link.springer.com/article/10.2478/s13540-012-0019-y.
24. A.I. Saichev and G.M. Zaslavsky. Fractional kinetic equations: Solutions and applications. Chaos 7, No 4 (1997), 753-764.
25. G. Samorodnitsky and M. Taqqu, Stable non-Gaussian Random Processes. Chapman and Hall, New York (1994).
26. I. Stakgold, Green's Functions and Boundary Value Problems. Second Edition, John Wiley and Sons, New York (1998).
27. M.N. Stojanović, Well-posedness of diffusion-wave problem with arbitrary finite number of time fractional derivatives in Sobolev spaces Hs. Fract. Calc. Appl. Anal. 13, No 1 (2010), 21-42; at http://www.math.bas.bg/ ~fcaa.
28. T.L. Szabo, Causal theories and data for acoustic attenuation obeying a frequency power-law. J. Acoust. Soc. Am. 97, No 1 (1995), 14-24.
29. 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, No 5 (2010), 2741-2748.
30. M.G. Wismer, Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation. J. Acoust. Soc. Am. 120, No 6 (2006), 3493-3502.