Numerical simulation of anomalous infiltration in porous media

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• 作者：S. Shen (1)
  F. Liu (2)
  Q. Liu (3)
  V. Anh (2)
  
  1. School of Mathematical Sciences ; Huaqiao University ; Quanzhou ; Fujian ; China
  2. School of Mathematical Sciences ; Queensland University of Technology ; GPO Box 2434 ; Brisbane ; Qld. 4001 ; Australia
  3. School of Mathematical Sciences ; Xiamen University ; 361005 ; Xiamen ; China
  
• 关键词：Anomalous infiltration ; Porous media ; Subdiffusion and superdiffusion ; Time ; fractional Boussinesq equation ; Time variable order fractional derivative
• 刊名：Numerical Algorithms
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：68
• 期：3
• 页码：443-454
• 全文大小：195 KB
• 参考文献：1. Chen, C, Liu, F, Anh, V, Turner, I (2010) Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM. J. Sci. Commun. 32: pp. 1740-1760
  2. Chen, J, Liu, F, Anh, V, Shen, S, Liu, Q, Liao, C (2012) The analytical solution and numerical solution of the fractional diffusion-wave equation with damping. Appl. Math. Comput. 219: pp. 1737-1748 CrossRef
  3. Coimbra, CFM (2003) Mechanics with variable-order differential operators. Annalen der Physik 12: pp. 692-703 CrossRef
  4. Diaz, G, Coimbra, CFM (2009) Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation. Nonlinear Dyn. 56: pp. 145-157 CrossRef
  5. Gerasimov, DN, Kondratieva, VA, Sinkevich, OA (2010) An anomalous non-self-similar infiltration and fractional diffusion equation. Physica D: Nonlinear Phenom. 239: pp. 1593-1597 CrossRef
  6. Ingman, D, Suzdalnitsky, J, Zeifman, M (2000) Constitutive dynamic-order model for nonlinear contact phenomena. J. Appl. Mech. 67: pp. 383-390 CrossRef
  7. Ingman, D, Suzdalnitsky, J (2004) Control of damping oscilations by fractional differential operator with time-dependent order. Comput. Methods Appl. Mech. Eng. 193: pp. 5585-5595 CrossRef
  8. Li, C, Deng, W, Wu, Y (2011) Numerical analysis and physical simulations for the time fractional radial diffusion equation. Comput Math. Appl. 62: pp. 1024-1037 CrossRef
  9. Lorenzo, CF, Hartley, TT (2007) Initialization, conceptualization, and application in the generalized (fractional) calculus. Crit. Rev. Biomed. Eng. 35: pp. 447-553 CrossRef
  10. Lorenzo, CF, Hartley, TT (2002) Variable order and distributed order fractional operators. Nonlinear Dyn. 29: pp. 57-98 CrossRef
  11. Pedro, HTC, Kobayashi, MH, Pereira, JMC, Coimbra, CFM (2008) Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere. J. Vib. Control. 14: pp. 1659-1672 CrossRef
  12. Ramirez, LES, Coimbra, CFM (2007) A variable order constitutive relation for viscoelasticity. Ann. Phys 16: pp. 543-552 CrossRef
  13. Samko, SG, Ross, B (1993) Intergation and differentiation to a variable fractional order. Integr. Transforms and Spec. Funct. 1: pp. 277-300 CrossRef
  14. Samko, SG (1995) Fractional integration and differentiation of variable order. Anal. Math. 21: pp. 213-236 CrossRef
  15. Shen, S, Liu, F, Chen, J, Turner, I, Anh, V (2012) Numerical techniques for the variable order time fractional diffusion equation. Appl. Math. Comput. 218: pp. 10861-10870 CrossRef
  16. Soon, CM, Coimbra, CFM, Kobayashi, MH (2005) The variable viscoelasticity oscillator. Annalen der Physik 14: pp. 378-389 CrossRef
  17. Sun, H, Chen, W, Chen, Y (2009) Variable-order fractional differential operators in anomalous diffusion modeling. Physica A 388: pp. 4586-4592 CrossRef
  18. Sun, H, Chen, W, Sheng, H, Chen, Y (2010) On mean square displacement behaviors of anomalous diffusions with variable and random orders. Phys. Lett. A 374: pp. 906-910 CrossRef
  19. Sun, Z, Wu, X (2006) A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56: pp. 193-209 CrossRef
  20. Zhuang, P, Liu, F, Anh, V, Turner, I (2009) Numerical methods for the variable-order fractonal advection-diffusion equation with a nonlinear source term, SIAM. J. Numer. Anal. 47: pp. 1760-1781 CrossRef
  
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing
  Algorithms
  Mathematics
  Algebra
  Theory of Computation
  
• 出版者：Springer U.S.
• ISSN：1572-9265

文摘

Nonlinear time-fractional diffusion equations have been used to describe the liquid infiltration for both subdiffusion and superdiffusion in porous media. In this paper, some problems of anomalous infiltration with a variable-order time-fractional derivative in porous media are considered. The time-fractional Boussinesq equation is also considered. Two computationally efficient implicit numerical schemes for the diffusion and wave-diffusion equations are proposed. Numerical examples are provided to show that the numerical methods are computationally efficient.