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Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. (English) Zbl 1171.65101

The authors construct an explicit finite difference scheme for the approximate solution of the nonlinear diffusion equation of fractional order \[ \frac{\partial {u(x,t)}}{\partial{t}}=B(x,t)_{x}{R^{\alpha(x,t)}u(x,t)+f(u,x,t),\quad {{X_{a}}<X<{X_{b}}},0<t<T} \] with the initial and boundary conditions of usual form. The derivative of fractional order is considered in the generalized sense of Riesz. The approximate scheme can be written in matrix form
\[ U^{j+1}=P^{j}U^{j}+B^{j}+F^{j}. \]
The convergence and stability of this scheme are proved and some numerical examples are presented.

MSC:

65R20 Numerical methods for integral equations
45K05 Integro-partial differential equations
45G10 Other nonlinear integral equations
35K57 Reaction-diffusion equations
26A33 Fractional derivatives and integrals
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] B. Baeumer, M. Kovacs, M.M. Meerschaert, Fractional reaction-diffusion equation for species growth and dispersal, J. Math. Biology (2007). Available from: <http://www.maths.otago.ac.nz/ mcubed/JMBseed.pdf>; B. Baeumer, M. Kovacs, M.M. Meerschaert, Fractional reaction-diffusion equation for species growth and dispersal, J. Math. Biology (2007). Available from: <http://www.maths.otago.ac.nz/ mcubed/JMBseed.pdf> · Zbl 1296.92195
[2] Chen, Chang-Ming; Liu, F.; Turner, I.; Anh, V., Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comput. Phys., 227, 886-897 (2007) · Zbl 1165.65053
[3] Davis, L. C., Model of magnetorheological elastomers, J. Appl. Phys., 85, 6, 3342-3351 (1999)
[4] del Castillo-Negrete, D.; Carreras, B. A.; Lynch, V. E., Front dynamics in reaction-diffusion systems with levy flghts: a fractional diffusion approach, Phys. Rev. Lett., 91, 1, 018302 (2003)
[5] Evans, K. P.; Jacob, N., Feller semigroups obtained by variable order subordination, Rev. Mat. Complut., 20, 2, 293-307 (2007) · Zbl 1153.47033
[6] Glockle, W. G.; Nonnenmacher, T. F., A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68, 46-53 (1995)
[7] Henry, B. I.; Wearne, S. L., Fractional reaction-diffusion, Physica A, 276, 448-455 (2000)
[8] Klass, D. L.; Martinek, T. W., Electroviscous fluids. I. Rheological properties, J. Appl. Phys., 38, 1, 67-74 (1967)
[9] Leopold, H. G., Embedding of function spaces of variable order of differentiation, Czech. Math. J., 49, 633-644 (1999) · Zbl 1008.46015
[10] Lin, R.; Liu, F., Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear Anal., 6, 856-869 (2006) · Zbl 1118.65079
[11] Liu, F.; Anh, V.; Turne, I., Numerical solution of space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, 209-219 (2004) · Zbl 1036.82019
[12] Liu, F.; Anh, V.; Turner, I.; Zhuang, P., Numerical simulation for solute transport in fractal porous media, ANZIAM J., 45, E, 461-473 (2004) · Zbl 1123.76363
[13] Liu, Q.; Liu, F.; Turner, I.; Anh, V., Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method, J. Phys. Comput., 222, 57-70 (2007) · Zbl 1112.65006
[14] Liu, F.; Shen, S.; Anh, V.; Turner, I., Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46, E, 488-504 (2005) · Zbl 1082.60511
[15] C.F. Lorenzo, T.T. Hartley, Initialization, conceptualization and application in the generalized fractional calculus, NASA/TP-1998-208-208415, 1999.; C.F. Lorenzo, T.T. Hartley, Initialization, conceptualization and application in the generalized fractional calculus, NASA/TP-1998-208-208415, 1999.
[16] Lorenzo, C. F.; Hartley, T. T., Variable-order and distributed order fractional operators, Nonlinear Dyn., 29, 57-98 (2002) · Zbl 1018.93007
[17] Meerschaert, M.; Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 65-77 (2004) · Zbl 1126.76346
[18] Miller, K. S.; Ross, B., An Introduction to the Fractional Calculus and Fractional Differential Equations (1993), John Wiley: John Wiley New York · Zbl 0789.26002
[19] Oldham, K. B.; Spanier, J., The Fractional Calculus (1974), Academic Press · Zbl 0428.26004
[20] Podlubny, I., Fractional Differential Equations (1999), Academic Press · Zbl 0918.34010
[21] Roop, J. P., Computational aspects of FEM approximation of fractional advection dispersion equation on bounded domains in \(R {}^2\), J. Comput. Appl. Math., 193, 1, 243-268 (2006) · Zbl 1092.65122
[22] Ruiz-Medina, M. D.; Anh, V. V.; Angulo, J. M., Fractional generalized random fields of variable order, Stochastic Anal. Appl., 22, 2, 775-799 (2004) · Zbl 1069.60040
[23] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives: Theory and Applications (1993), Gordon and Breach: Gordon and Breach Newark, NJ · Zbl 0818.26003
[24] Seki, K.; Wojcik, M.; Tachiya, M., Fractional reaction-diffusion equation, J. Chem. Phys., 119, 2165-2174 (2003)
[25] Shen, S.; Liu, F., Error analysis of an explicit finite difference approximation for the space fractional diffusion, ANZIAM J., 46, E, 871-887 (2005)
[26] Shiga, T., Deformation and viscoelastic behavior of polymer gel in electric fields, Proceedings of the Japanese Academy, Series B, Phys. Biol. Sci., 74, 6-11 (1998)
[27] Yu, Q.; Liu, F.; Anh, V.; Turner, I., Solving linear and nonlinear space-time fractional reaction-diffusion equations by Adomian decomposition method, Int. J. Numer. Meth. Eng., 74, 138-158 (2008) · Zbl 1159.76367
[28] Zhuang, P.; Liu, F., Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comput., 22, 3, 87-99 (2006) · Zbl 1140.65094
[29] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., New solution and analytical techniques of the implicit numerical methods for the anomalous sub-diffusion equation, SIAM J. Numer. Anal., 46, 2, 1079-1095 (2008) · Zbl 1173.26006
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