A numerical investigation of the time distributed-order diffusion model

Xiuling Hu, Fawang Liu, Vo Anh, Ian Turner

Abstract


Distributed-order differential models are more powerful tools to describe complex dynamical systems than the classical and fractional-order models because of their nonlocal properties. A time distributed-order diffusion model is investigated. By employing some numerical integration techniques, we approximate the distributed-order fractional model with a multi-term fractional model, which is then solved by an implicit numerical method. The stability and convergence of the numerical method is analyzed. Some numerical results are presented to demonstrate the effectiveness of the method and to exhibit the solution behavior of three different diffusion models.

References
  • I. M. Sokolov, A. V. Chechkin and J. Klafter, Distributed-order fractional kinetics, Acta Phys. Pol. B, 35:1323–1341, 2004. http://www.actaphys.uj.edu.pl/_cur/store/vol35/pdf/v35p1323.pdf.
  • F. Liu, P. Zhuang, V. Anh, I. Turner and K. Burrage, Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191:12–20, 2007. doi:10.1016/j.amc.2006.08.162.
  • R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339:1–77, 2000. doi:10.1016/S0370-1573(00)00070-3.
  • A. V. Chechkin, R. Gorenflo and I. M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E, 66:046129, 2002. doi:10.1103/PhysRevE.66.046129.
  • M. M. Meerschaert, E. Nane and P. Vellaisamy, Distributed-order fractional diffusions on bounded domains, J. Math. Anal. Appl., 379:216–228, 2011. doi:10.1016/j.jmaa.2010.12.056.
  • C. F. Lorenz and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dynam., 29:57–98, 2002. http://link.springer.com/article/10.1023/A:1016586905654.
  • M. Caputo, Mean fractional-order-derivatives differential equations and filters. Ann. Univ. Ferrara, 41:73–84, 1995. http://link.springer.com/article/10.1007%2FBF02826009.
  • K. Diethelm and N. J. Ford, Numerical analysis for distributed-order differential equations, J. Comp. Appl. Math., 225:96–104, 2009. doi:10.1016/j.cam.2008.07.018.
  • H. Ye, F. Liu, V. Anh and I. Turner, Maximum principle and numerical method for the multi-term time-space Riesz–Caputo fractional differential equations, Appl. Math. Comput., 227:531–540, 2014. doi:10.1016/j.amc.2013.11.015.
  • F. Liu, M. M. Meerschaert, R. J. McGough, P. Zhuang and Q. Liu, Numerical methods for solving the multi-term time-fractional wave-equation, Fract. Calc. Appl. Anal., 16:9–25, 2013. doi:10.2478/s13540-013-0002-2.
  • Z.-Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56:193–209, 2006. doi:10.1016/j.apnum.2005.03.003.

Keywords


time distributed-order diffusion model; multi-term fractional model;implicit numerical method; Stability and convergence

Full Text:

PDF BIB


DOI: http://dx.doi.org/10.21914/anziamj.v55i0.7888



Remember, for most actions you have to record/upload into this online system
and then inform the editor/author via clicking on an email icon or Completion button.
ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.