메뉴 건너뛰기




Volumn 16, Issue 1, 2013, Pages 9-25

Numerical methods for solving the multi-term time-fractional wave-diffusion equation

Author keywords

a power law wave equation; Caputo derivative; finite difference method; fractional predictor corrector method; multi term time fractional wave diffusion equations

Indexed keywords


EID: 84871790575     PISSN: 13110454     EISSN: 13142444     Source Type: Journal    
DOI: 10.2478/s13540-013-0002-2     Document Type: Article
Times cited : (303)

References (38)
  • 1
    • 0001311725 scopus 로고    scopus 로고
    • Stochastic solutions for fractional Cauchy problems
    • 1874479 1057.35102
    • B. Baeumer and M.M. Meerschaert, Stochastic solutions for fractional Cauchy problems. Fract. Calc. Appl. Anal. 4, No 4 (2001), 481-500.
    • (2001) Fract. Calc. Appl. Anal. , vol.4 , Issue.4 , pp. 481-500
    • Baeumer, B.1    Meerschaert, M.M.2
  • 2
    • 48349133740 scopus 로고    scopus 로고
    • Inhomogeneous fractional diffusion eqautions
    • 2268419 1202.86005
    • B. Baeumer, S. Kurita and M.M. Meerschaert, Inhomogeneous fractional diffusion eqautions. Fract. Calc. Appl. Anal., 8, No 4 (2005), 371-376; at http://www.math.bas.bg/~fcaa.
    • (2005) Fract. Calc. Appl. Anal. , vol.8 , Issue.4 , pp. 371-376
    • Baeumer, B.1    Kurita, S.2    Meerschaert, M.M.3
  • 3
    • 84869179719 scopus 로고    scopus 로고
    • Product rule for vector fractional derivatives
    • 2944111
    • D. Bolster, M.M. Meerschaert and A. Sikorskii, Product rule for vector fractional derivatives. Fract. Calc. Appl. Anal. 15, No 3 (2012), 463-478; DOI:10.2478/s13540-012-0033-0; at http://link.springer.com/article/10.2478/ s13540-012-0033-0.
    • (2012) Fract. Calc. Appl. Anal. , vol.15 , Issue.3 , pp. 463-478
    • Bolster, D.1    Meerschaert, M.M.2    Sikorskii, A.3
  • 4
    • 0242693197 scopus 로고    scopus 로고
    • Modified Szabo's wave equation models for lossy media obeying frequency power law
    • 10.1121/1.1621392
    • W. Chen, S. Holm, Modified Szabo's wave equation models for lossy media obeying frequency power law. J. Acoust. Soc. Am. 114 (2003), 2570-2754.
    • (2003) J. Acoust. Soc. Am. , vol.114 , pp. 2570-2754
    • Chen, W.1    Holm, S.2
  • 5
    • 34250899368 scopus 로고    scopus 로고
    • Analysis of fractional differential equations with multi-orders
    • 2332373 1176.34008 10.1142/S0218348X07003472
    • W. Deng, C. Li, Q. Guo, Analysis of fractional differential equations with multi-orders. Fractals 15, No 2 (2007), 173-182.
    • (2007) Fractals , vol.15 , Issue.2 , pp. 173-182
    • Deng, W.1    Li, C.2    Guo, Q.3
  • 6
    • 33947133956 scopus 로고    scopus 로고
    • Stability analysis of linear fractional differential system with multiple time-delays
    • 2312588 1185.34115 10.1007/s11071-006-9094-0
    • W. Deng, C. Li, J. Lu, Stability analysis of linear fractional differential system with multiple time-delays. Nonlinear Dynamics 48, No 4 (2007), 409-416.
    • (2007) Nonlinear Dynamics , vol.48 , Issue.4 , pp. 409-416
    • Deng, W.1    Li, C.2    Lu, J.3
  • 9
    • 77951184169 scopus 로고    scopus 로고
    • An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation
    • 2667749 1231.65178
    • Y. Gu, P. Zhuang, F. Liu, An advanced implicit meshless approach for the non-linear anomalous subdiffusion equation. Computer Modeling in Eng. & Sciences 56 (2010), 303-334.
    • (2010) Computer Modeling in Eng. & Sciences , vol.56 , pp. 303-334
    • Gu, Y.1    Zhuang, P.2    Liu, F.3
  • 10
    • 64249135201 scopus 로고    scopus 로고
    • Numerical approximation of a fractional-in-space diffusion equation (I)
    • 2252038 1126.26009
    • M. Ilic, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation (I). Fract. Calc. Appl. Anal., 8, No 3 (2005), 323-341; at http://www.math.bas.bg/~fcaa.
    • (2005) Fract. Calc. Appl. Anal. , vol.8 , Issue.3 , pp. 323-341
    • Ilic, M.1    Liu, F.2    Turner, I.3    Anh, V.4
  • 11
    • 64249104309 scopus 로고    scopus 로고
    • Numerical approximation of a fractional-in-space diffusion equation (II) - With nonhomogeneous boundary conditions
    • 2300467 1132.35507
    • M. Ilic, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation (II) - with nonhomogeneous boundary conditions. Fract. Calc. Appl. Anal., 9, No 4 (2006), 333-349; at http://www.math.bas.bg/~fcaa.
    • (2006) Fract. Calc. Appl. Anal. , vol.9 , Issue.4 , pp. 333-349
    • Ilic, M.1    Liu, F.2    Turner, I.3    Anh, V.4
  • 12
    • 84862824195 scopus 로고    scopus 로고
    • Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain
    • 2879284 1234.35300 10.1016/j.jmaa.2011.12.055
    • H. Jiang, F. Liu, I. Turner, K. Burrage, Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. J. Math. Anal. Appl. 389 (2012), 1117-1127.
    • (2012) J. Math. Anal. Appl. , vol.389 , pp. 1117-1127
    • Jiang, H.1    Liu, F.2    Turner, I.3    Burrage, K.4
  • 13
    • 56749132453 scopus 로고    scopus 로고
    • Analytical time-domain Green's functions for power-law media
    • 10.1121/1.2977669
    • J.K. Kelly, R.J. McGough, M.M. Meerschaert, Analytical time-domain Green's functions for power-law media. J. Acoust. Soc. Am. 124 (2008), 2861-2872.
    • (2008) J. Acoust. Soc. Am. , vol.124 , pp. 2861-2872
    • Kelly, J.K.1    McGough, R.J.2    Meerschaert, M.M.3
  • 14
    • 84868198807 scopus 로고    scopus 로고
    • Spectral approximations to the fractional integral and derivative
    • 2944106
    • C. Li, F. Zeng, F. Liu, Spectral approximations to the fractional integral and derivative. Fract. Calc. Appl. Anal. 15, No 3 (2012), 383-406; DOI:10.2478/s13540-012-0028-x; at http://link.springer.com/article/10.2478/ s13540-012-0028-x
    • (2012) Fract. Calc. Appl. Anal. , vol.15 , Issue.3 , pp. 383-406
    • Li, C.1    Zeng, F.2    Liu, F.3
  • 15
    • 9644281076 scopus 로고    scopus 로고
    • Full wave modeling of therapeutic ultrasound: Efficient time-domain implementation of the frequency power-law attenuation
    • 10.1121/1.1798355
    • M. Liebler, S. Ginter, T. Dreyer, 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.
    • (2004) J. Acoust. Soc. Am. , vol.116 , pp. 2742-2750
    • Liebler, M.1    Ginter, S.2    Dreyer, T.3    Riedlinger, R.E.4
  • 16
    • 1542425102 scopus 로고    scopus 로고
    • Numerical solution of the space fractional Fokker-Planck equation
    • 2057973 1036.82019 10.1016/j.cam.2003.09.028
    • F. Liu, V. Anh, I. Turner, Numerical solution of the space fractional Fokker-Planck equation. J. Comp. Appl. Math. 166 (2004), 209-219.
    • (2004) J. Comp. Appl. Math. , vol.166 , pp. 209-219
    • Liu, F.1    Anh, V.2    Turner, I.3
  • 17
    • 34547673244 scopus 로고    scopus 로고
    • Stability and convergence of the difference methods for the space-time fractional advectiondiffusion equation
    • 1193.76093 10.1016/j.amc.2006.08.162
    • F. Liu, P. Zhuang, V. Anh, I. Turner, K. Burrag, Stability and convergence of the difference methods for the space-time fractional advectiondiffusion equation. J. Comp. Appl. Math. 191 (2007), 12-20.
    • (2007) J. Comp. Appl. Math. , vol.191 , pp. 12-20
    • Liu, F.1    Zhuang, P.2    Anh, V.3    Turner, I.4    Burrag, K.5
  • 18
    • 67349231192 scopus 로고    scopus 로고
    • Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term
    • 2532659 1170.65107 10.1016/j.cam.2009.02.013
    • F. Liu, C. Yang, K. Burrage, Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comp. Appl. Math. 231 (2009), 160-176.
    • (2009) J. Comp. Appl. Math. , vol.231 , pp. 160-176
    • Liu, F.1    Yang, C.2    Burrage, K.3
  • 19
    • 84871742819 scopus 로고    scopus 로고
    • Numerical methods and analysis for a class of fractional advection-dispersion models
    • 2989329
    • F. Liu, P. Zhuang, K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models. Computers and Math. with Appl. 63 (2012), 1-22.
    • (2012) Computers and Math. with Appl. , vol.63 , pp. 1-22
    • Liu, F.1    Zhuang, P.2    Burrage, K.3
  • 20
    • 33846798041 scopus 로고    scopus 로고
    • Approximation of the Lévy- Feller advection-dispersion process by random walk and finite difference method
    • 2298036 1112.65006 10.1016/j.jcp.2006.06.005
    • Q. Liu, F. Liu, I. Turner, V. Anh, Approximation of the Lévy- Feller advection-dispersion process by random walk and finite difference method. J. Comp. Phys. 222 (2007), 57-70.
    • (2007) J. Comp. Phys. , vol.222 , pp. 57-70
    • Liu, Q.1    Liu, F.2    Turner, I.3    Anh, V.4
  • 21
    • 79251616666 scopus 로고    scopus 로고
    • An implicit RBF meshless approach for time fractional diffusion equations
    • 2812854 06032194 10.1007/s00466-011-0573-x
    • Q. Liu, Y. Gu, P. Zhuang, F. Liu, Y. Nie, An implicit RBF meshless approach for time fractional diffusion equations. Comput. Mech. 48 (2011), 1-12.
    • (2011) Comput. Mech. , vol.48 , pp. 1-12
    • Liu, Q.1    Gu, Y.2    Zhuang, P.3    Liu, F.4    Nie, Y.5
  • 22
    • 77957822720 scopus 로고    scopus 로고
    • Initial-boundary-value problems for the generalized multiterm time-fractional diffusion equation
    • 2729240 1202.35339 10.1016/j.jmaa.2010.08.048
    • Y. Luchko, Initial-boundary-value problems for the generalized multiterm time-fractional diffusion equation. J. Math. Anal. Appl. 374 (2011), 538-548.
    • (2011) J. Math. Anal. Appl. , vol.374 , pp. 538-548
    • Luchko, Y.1
  • 25
    • 4444368867 scopus 로고    scopus 로고
    • Finite difference approximations for fractional advection-dispersion flow equations
    • 2091131 1126.76346 10.1016/j.cam.2004.01.033
    • M.M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations. J. Comp. Appl. Math. 172 (2004), 65-77.
    • (2004) J. Comp. Appl. Math. , vol.172 , pp. 65-77
    • Meerschaert, M.M.1    Tadjeran, C.2
  • 26
    • 84871013921 scopus 로고    scopus 로고
    • Stochastic solution to a time-fractional attenuated wave equation
    • 10.1007/s11071-012-0532-x
    • M.M. Meerschaert, P. Straka, Y. Zhou, R.J. McGough, Stochastic solution to a time-fractional attenuated wave equation. Nonlinear Dynamics 70 (2012), 1273-1281.
    • (2012) Nonlinear Dynamics , vol.70 , pp. 1273-1281
    • Meerschaert, M.M.1    Straka, P.2    Zhou, Y.3    McGough, R.J.4
  • 27
    • 0002641421 scopus 로고    scopus 로고
    • The random walk's guide to anomalous diffusion: A fractional dynamics approach
    • 1809268 0984.82032 10.1016/S0370-1573(00)00070-3
    • R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1-77.
    • (2000) Phys. Rep. , vol.339 , pp. 1-77
    • Metzler, R.1    Klafter, J.2
  • 29
    • 33646191893 scopus 로고    scopus 로고
    • Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2
    • 2228717 1092.65122 10.1016/j.cam.2005.06.005
    • J.P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in R2. J. Comp. Appl. Math. 193 (2006), 243-268.
    • (2006) J. Comp. Appl. Math. , vol.193 , pp. 243-268
    • Roop, J.P.1
  • 31
    • 79951851714 scopus 로고    scopus 로고
    • Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation
    • 2774121 1214.65046 10.1007/s11075-010-9393-x
    • S. Shen, F. Liu, V. Anh, Numerical approximations and solution techniques for the space-time Riesz-Caputo fractional advection-diffusion equation. Numerical Algorithm 56 (2011), 383-404.
    • (2011) Numerical Algorithm , vol.56 , pp. 383-404
    • Shen, S.1    Liu, F.2    Anh, V.3
  • 32
    • 79952190735 scopus 로고    scopus 로고
    • Numerical method for solving diffusion-wave phenomena
    • 2773298 1213.65133 10.1016/j.cam.2010.12.010
    • M. Stojanovic, Numerical method for solving diffusion-wave phenomena. J. Comp. Appl. Math. 235 (2011), 3121-3137.
    • (2011) J. Comp. Appl. Math. , vol.235 , pp. 3121-3137
    • Stojanovic, M.1
  • 33
    • 84871765883 scopus 로고    scopus 로고
    • Fractional wave equations with attenuation
    • P. Straka, M.M. Meerschaert, R.J. McGough, and Y. Zhou, Fractional wave equations with attenuation. Fract. Calc. Appl. Anal. 16, No 1 (2013), 262-272 (same issue); DOI:10.2478/s13540-013-0016-9; at http://link.springer.com/ journal/13540.
    • (2013) Fract. Calc. Appl. Anal. , vol.16 , Issue.1 , pp. 262-272
    • Straka, P.1    Meerschaert, M.M.2    McGough, R.J.3    Zhou, Y.4
  • 34
    • 0028292355 scopus 로고
    • Time domain wave equations for lossy media obeying a frequency power law
    • 10.1121/1.410434
    • T.L. Szabo, Time domain wave equations for lossy media obeying a frequency power law. J. Acoust. Soc. Am. 96 (1994), 491-500.
    • (1994) J. Acoust. Soc. Am. , vol.96 , pp. 491-500
    • Szabo, T.L.1
  • 35
    • 62249116725 scopus 로고    scopus 로고
    • A computationally effective predictor-corrector method for simulating fractional order dynamical control system
    • C. Yang, F. Liu, A computationally effective predictor-corrector method for simulating fractional order dynamical control system. ANZIAM J. 47 (2006), 168-184.
    • (2006) ANZIAM J. , vol.47 , pp. 168-184
    • Yang, C.1    Liu, F.2
  • 36
    • 62349097511 scopus 로고    scopus 로고
    • Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications
    • 10.1016/j.advwatres.2009.01.008
    • Y. Zhang, D.A. Benson, D.M. Reeves, Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications. Advances in Water Resources 32 (2009), 561-581.
    • (2009) Advances in Water Resources , vol.32 , pp. 561-581
    • Zhang, Y.1    Benson, D.A.2    Reeves, D.M.3
  • 37
    • 79959196930 scopus 로고    scopus 로고
    • Stability analysis of fractional differential systems with order lying in (1,2)
    • ID 213485
    • F. Zhang, C. Li, Stability analysis of fractional differential systems with order lying in (1,2). Advances in Difference Equations (2011), ID 213485.
    • (2011) Advances in Difference Equations
    • Zhang, F.1    Li, C.2
  • 38
    • 84907893973 scopus 로고    scopus 로고
    • Numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term
    • 2505873 1204.26013 10.1137/080730597
    • P. Zhuang, F. Liu, V. Anh, I. Turner, Numerical methods for the variable order fractional advection diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47 (2009), 1760-1781.
    • (2009) SIAM J. Numer. Anal. , vol.47 , pp. 1760-1781
    • Zhuang, P.1    Liu, F.2    Anh, V.3    Turner, I.4


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.