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Numerical techniques for the variable-order time fractional diffusion equation. (English) Zbl 1280.65089

Summary: We consider the variable-order time fractional diffusion equation. We adopt the Coimbra variable-order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable-order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R11 Fractional partial differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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[1] Anh, V. V.; Leonenko, N. N., Spectral analysis of fractional kinetic equations with random data, J. Statis. Phys., 104, 5-6, 1349-1387 (2001) · Zbl 1034.82044
[2] Blaszczyk, T.; Ciesielski, M.; Klimek, M.; Leszczynski, J., Numerical solution of fractional oscillator equation, Appl. Math. Comput., 218, 2480-2488 (2011) · Zbl 1243.65090
[3] Chen, C.; Liu, F.; Anh, V.; Turner, I., Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation, SIAM J. Sci. Comput., 32, 4, 1740-1760 (2010) · Zbl 1217.26011
[4] Coimbra, C. F.M., Mechanics with variable-order differential operators, Ann. Phys., 12, 11-12, 692-703 (2003) · Zbl 1103.26301
[5] Diaz, G.; Coimbra, C. F.M., Nonlinear dynamics and control of a variable order oscillator with application to the van der Pol equation, Nonlinear Dynam., 56, 145-157 (2009) · Zbl 1170.70012
[6] Galue, Leda; Kalla, S. L.; Al-Saqabi, B. N., Fractional extensions of the temperature field problems in oil strata, Appl. Math. Comput., 186, 35-44 (2007) · Zbl 1110.76050
[7] J.M. Holte, Discrete Gronwall lemma and applications, MAA-NCS Meeting at the University of North Dakota, 24 October 2009.; J.M. Holte, Discrete Gronwall lemma and applications, MAA-NCS Meeting at the University of North Dakota, 24 October 2009.
[8] Ingman, D.; Suzdalnitsky, J.; Zeifman, M., Constitutive dynamic-order model for nonlinear contact phenomena, J. Appl. Mech., 67, 383-390 (2000) · Zbl 1110.74493
[9] Ingman, D.; Suzdalnitsky, J., Control of damping oscilations by fractional differential operator with time-dependent order, Comput. Methods Appl. Mech. Eng., 193, 5585-5595 (2004) · Zbl 1079.70020
[10] Langlands, T. A.M.; Henry, B. I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205, 719-736 (2005) · Zbl 1072.65123
[11] Lin, R.; Liu, F.; Anh, V.; Turner, I., Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractonal diffusion equation, Appl. Math. Comput., 212, 435-445 (2009) · Zbl 1171.65101
[12] Lin, Y.; Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225, 1533-1552 (2007) · Zbl 1126.65121
[13] Liu, F.; Anh, V.; Turner, I., Numerical solution of space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, 209-219 (2004) · Zbl 1036.82019
[14] Liu, F.; Zhuang, P.; Anh, V.; Turner, I.; Burrage, K., Stability and convergence of difference methods for the space-time fractional advection-diffusion equation, Appl. Math. Comput., 191, 1, 12-20 (2007) · Zbl 1193.76093
[15] Liu, F.; Yang, C.; Burrage, K., Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math., 231, 160-176 (2009) · Zbl 1170.65107
[16] Liu, F.; Burrage, K., Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl., 62, 822-833 (2011) · Zbl 1228.93114
[17] Liu, F.; Yang, Q.; Turner, I., Two new implicit numerical methods for the fractional cable equation, J. Comput. Nonlinear Dynam., 6, 1, 01108 (2011)
[18] F. Liu, P. Zhuang, K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models, Comput. Math. Appl., in press. <http://dx.doi.org/10.1016/j.camwa.2012.01.020>; F. Liu, P. Zhuang, K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models, Comput. Math. Appl., in press. <http://dx.doi.org/10.1016/j.camwa.2012.01.020> · Zbl 1268.65124
[19] Lorenzo, C. F.; Hartley, T. T., Initialization, conceptualization, and application in the generalized fractional calculus, NASA Technical Publication 98-208415 (1998), NASA Lewis Reseach Center
[20] Lorenzo, C. F.; Hartley, T. T., Variable order and distributed order fractional operators, Nonlinear Dynam., 29, 57-98 (2002) · Zbl 1018.93007
[21] Odibat, Zaid, Approximations of fractional integrals and Caputo fractional derivatives, Appl. Math. Comput., 178, 527-533 (2006) · Zbl 1101.65028
[22] 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, 1659-1672 (2008) · Zbl 1229.76099
[23] Ramirez, E. S.; Coimbra, F. M., On the selection and meaning of variable order operators for dynamic modeling, Int. J. Differ. Equat. Vol. (2010) · Zbl 1207.34011
[24] Samko, S. G.; Ross, B., Intergation and differentiation to a variable fractional order, Integral Trans. Special Func., 1, 4, 277-300 (1993) · Zbl 0820.26003
[25] Samko, S. G., Fractional integration and differentiation of variable order, Anal. Math., 21, 213-236 (1995) · Zbl 0838.26006
[26] Soon, C. M.; Coimbra, F. M.; Kobayashi, M. H., The variable viscoelasticity oscillator, Ann. Phys., 14, 6, 378-389 (2005) · Zbl 1125.74316
[27] Sun, H.; Chen, W.; Chen, Y., Variable-order fractional differential operators in anomalous diffusion modeling, Physica A, 388, 4586-4592 (2009)
[28] 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
[29] Zhuang, P.; Liu, F.; Anh, V.; Turner, I., Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47, 1760-1781 (2009) · Zbl 1204.26013
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