Abstract
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.
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Chen, C., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM. J. Sci. Commun. 32(4), 1740–1760 (2010)
Chen, J., Liu, F., Anh, V., Shen, S., Liu, Q., Liao, C.: The analytical solution and numerical solution of the fractional diffusion-wave equation with damping. Appl. Math. Comput. 219, 1737–1748 (2012)
Coimbra, C.F.M.: Mechanics with variable-order differential operators. Annalen der Physik 12(11-12), 692–703 (2003)
Diaz, G., Coimbra, C.F.M.: Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation. Nonlinear Dyn. 56(1-2), 145–157 (2009)
Gerasimov, D.N., Kondratieva, V.A., Sinkevich, O.A.: An anomalous non-self-similar infiltration and fractional diffusion equation. Physica D: Nonlinear Phenom. 239(16), 1593–1597 (2010)
Ingman, D., Suzdalnitsky, J., Zeifman, M.: Constitutive dynamic-order model for nonlinear contact phenomena. J. Appl. Mech. 67(2), 383–390 (2000)
Ingman, D., Suzdalnitsky, J.: Control of damping oscilations by fractional differential operator with time-dependent order. Comput. Methods Appl. Mech. Eng. 193(52), 5585–5595 (2004)
Li, C., Deng, W., Wu, Y.: Numerical analysis and physical simulations for the time fractional radial diffusion equation. Comput Math. Appl. 62(3), 1024–1037 (2011)
Lorenzo, C.F., Hartley, T.T.: Initialization, conceptualization, and application in the generalized (fractional) calculus. Crit. Rev. Biomed. Eng. 35(6), 447–553 (2007)
Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlinear Dyn. 29(1-4), 57–98 (2002)
Pedro, H.T.C., Kobayashi, M.H., Pereira, J.M.C., Coimbra, C.F.M.: Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere. J. Vib. Control. 14(9-10), 1659–1672 (2008)
Ramirez, L.E.S., Coimbra, C.F.M.: A variable order constitutive relation for viscoelasticity. Ann. Phys 16(7-8), 543–552 (2007)
Samko, S.G., Ross, B.: Intergation and differentiation to a variable fractional order. Integr. Transforms and Spec. Funct. 1(4), 277–300 (1993)
Samko, S.G.: Fractional integration and differentiation of variable order. Anal. Math. 21(3), 213–236 (1995)
Shen, S., Liu, F., Chen, J., Turner, I., Anh, V.: Numerical techniques for the variable order time fractional diffusion equation. Appl. Math. Comput. 218(22), 10861–10870 (2012)
Soon, C.M., Coimbra, C.F.M., Kobayashi, M.H.: The variable viscoelasticity oscillator. Annalen der Physik 14(6), 378–389 (2005)
Sun, H., Chen, W., Chen, Y.: Variable-order fractional differential operators in anomalous diffusion modeling. Physica A 388, 4586–4592 (2009)
Sun, H., Chen, W., Sheng, H., Chen, Y.: On mean square displacement behaviors of anomalous diffusions with variable and random orders. Phys. Lett. A 374(7), 906–910 (2010)
Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56(2), 193–209 (2006)
Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable-order fractonal advection-diffusion equation with a nonlinear source term, SIAM. J. Numer. Anal. 47(3), 1760–1781 (2009)
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Shen, S., Liu, F., Liu, Q. et al. Numerical simulation of anomalous infiltration in porous media. Numer Algor 68, 443–454 (2015). https://doi.org/10.1007/s11075-014-9853-9
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DOI: https://doi.org/10.1007/s11075-014-9853-9