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Stability and Convergence of Modified Du Fort–Frankel Schemes for Solving Time-Fractional Subdiffusion Equations

Zeitschrift:: Journal of Scientific Computing > Ausgabe 3/2014

Autoren:: Hong-lin Liao, Ya-nan Zhang, Ying Zhao, Han-sheng Shi

Wichtige Hinweise

Hong-lin Liao and Han-sheng Shi are supported by National Science Foundation for Young Scientists of China (No. 11001271, 11201239), and Ya-nan Zhang is supported by National Science Foundation of China (No. 11326229) and China Postdoctoral Science Foundation (2013M530265).

Abstract

A class of modified Du Fort–Frankel-type schemes is investigated for fractional subdiffusion equations in the Jumarie’s modified Riemann–Liouville form with constant, variable or distributed fractional order. New explicit difference methods are constructed by combining the \(L1\) approximation of the modified fractional derivative with the idea of Du Fort–Frankel scheme, well-known for ordinary diffusion equations. Unconditional stability of the explicit methods is established in the sense of a discrete energy norm. The proposed schemes are shown to be convergent under the time-step (consistency) restriction of the classical Du Fort–Frankel scheme. Numerical examples are included to support our theoretical results.

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Bagley, R.L., Torvik, P.J.: On the existence of the order domain and the solution of distributed order equations (Parts I, II). Int. J. Appl. Mech. 2(865–882), 965–987 (2000) MathSciNetMATH

Brunner, H., Ling, L., Yamamoto, M.: Numerical simulations of 2D fractional subdiffusion problems. J. Comput. Phys. 229, 6613–6622 (2010) MathSciNetCrossRefMATH

Caputo, M.: Mean fractional-order-derivatives differential equations and filters. Annali dell’Universita di Ferrara 41(1), 73–84 (1995) MathSciNetMATH

Caputo, M.: Distributed order differential equations modelling dielectric induction and diffusion. Fract. Calc. Appl. Anal. 4, 421–442 (2001) MathSciNetMATH

Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66(046129), 1–6 (2002)

Chen, C.M., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equation. SIAM J. Sci. Comput. 32, 1740–1760 (2010) MathSciNetCrossRefMATH

Chen, C., Liu, F., Anh, V., Turner, I.: Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation. Math. Comput. 81, 345–366 (2011) MathSciNetCrossRef

Ciesielski, M., Leszczynski, J.: Numerical simulations of anomalous diffusion. In: Proceedings of the 15th Conference on Computer Methods in Mechanics. Wisla, Polonia. (2003). arXiv:math-ph/0309007v1

Coimbra, C.F.M.: Mechanics with variable-order differential operators. Annalen der Physik 12, 692–703 (2003) MathSciNetCrossRefMATH

10.

Diethelm, K., Ford, N.J.: Numerical solution methods for distributed order differential equations. Fract. Calc. Appl. Anal. 4, 531–542 (2001) MathSciNetMATH

11.

Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Yu.: Algorithms for the fractional calculus: a selection of numerical methods. Comput. Methods Appl. Mech. Eng. 194, 743–773 (2005) MathSciNetCrossRefMATH

12.

Diethelm, K., Ford, N.J.: Numerical analysis for distributed-order differential equations. J. Comput. Appl. Math. 225(1), 96–104 (2009) MathSciNetCrossRefMATH

13.

Gorenflo, R., Mainardi, F., et al.: Time-fractional diffusion: a discrete random walk approach. Nonlinear Dyn. 29, 129 (2002) MathSciNetCrossRefMATH

14.

Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheologica Acta (2005). doi: 10.1007/s00397-005-0043-5

15.

Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) MATH

16.

Jumarie, G.: Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results. Comput. Math. Appl. 51, 1367–1376 (2006) MathSciNetCrossRefMATH

17.

Jumarie, G.: Fractional partial differential equations and modified Riemann–Liouville derivative new methods for solution. J. Appl. Math. Comput. 24, 31–48 (2007) MathSciNetCrossRefMATH

18.

Jumarie, G.: Modeling fractional stochastic systems as non-random fractional dynamics driven by Brownian motions. Appl. Math. Model. 32, 836–859 (2008) MathSciNetCrossRefMATH

19.

Jumarie, G.: Table of some basic fractional calculus formulae derived from a modiffed Riemann–Liouville derivative for non-differentiable functions. Appl. Math. Lett. 22, 378–385 (2009) MathSciNetCrossRefMATH

20.

Jumarie, G.: Cauchys integral formula via modified Riemann–Liouville derivative for analytic functions of fractional order. Appl. Math. Lett. 23, 1444–1450 (2010) MathSciNetCrossRefMATH

21.

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) MathSciNetCrossRefMATH

22.

Langlands, T.A.M.: Solution of a modified fractional diffusion equation. Phys. A 367, 136–144 (2006) MathSciNetCrossRef

23.

Lin, Y., Li, X., Xu, C.: Finite difference/spectral approximations for the fractional cable equation. Math. Comput. 80, 1369–1396 (2011) MathSciNetCrossRefMATH

24.

Lorenzo, C.F., Hartley, T.T.: Variable-order and distributed order fractional operators. Nonlinear Dyn. 29, 57–98 (2002) MathSciNetCrossRefMATH

25.

Murillo, J.Q., Yuste, S.B.: On three explicit difference schemes for fractional diffusion and diffusion-wave equations. Phys. Scr. T 136, 014025 (2009) CrossRef

26.

Murillo, J.Q., Yuste, S.B.: On an explicit difference method for fractional diffusion and diffusion-wave equations. In: Proceedings of the ASME 2009 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference. San Diego, California, (2009)

27.

Mustapha, K., AlMutawa, J.: A finite difference method for an anomalous sub-diffusion equation, theory and applications. Numer. Algorithm 61, 525–543 (2012) MathSciNetCrossRefMATH

28.

Mustapha, K., McLean, W.: Discontinuous Galerkin method for an evolution equation with a memory term of positive type. Math. Comput. 78, 1975–1995 (2009) MathSciNetCrossRefMATH

29.

Mustapha, K., McLean, W.: Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations. SIAM J. Numer. Anal. 51, 491–515 (2013) MathSciNetCrossRefMATH

30.

Naber, M.: Distributed order fractional subdiffusion. Fractals 12, 23–32 (2004) MathSciNetCrossRefMATH

31.

Oldham, K., Spanier, J.: The fractional calculus: theory and applications of differentiation and integration to arbitrary order. In: Mathematics in Science and Engineering, vol. 111. Academic Press, New York (1974)

32.

Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999) MATH

33.

Samko, S.G.: Fractional integration and differentiation of variable order. Anal. Math. 21, 213–236 (1995) MathSciNetCrossRefMATH

34.

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) MathSciNetCrossRefMATH

35.

Sokolov, I.M., Chechkin, A.V., Klafter, J.: Distributed-order fractional kinetics. Acta Physica Polonica 35, 1323–1341 (2004)

36.

Sun, H., Chen, W., Chen, Y.: Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A 388, 4586–4592 (2009) CrossRef

37.

Sun, H., Chen, W., Li, C., Chen, Y.: Finite difference schemes for variable-order time fractional diffusion equation. Int. J. Bifurc. Chaos 22(4), 1250085 (2012) MathSciNetCrossRef

38.

Umarov, S., Steinberg, S.: Random walk model sassociated with distributed fractional order differential equations. Lect. Notes Monogr. Ser. 51, 117–127 (2006) MathSciNetCrossRef

39.

Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Neumann-type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42(5), 1862–1874 (2005) MathSciNetCrossRefMATH

40.

Yuste, S.B.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216(1), 264–274 (2006) MathSciNetCrossRefMATH

41.

Zhang, Y.N., Sun, Z.Z., Wu, H.W.: Error estimates of Crank–Nicolson-type difference schemes for the sub-diffusion equation. SIAM J. Numer. Anal. 49, 2302–2322 (2011) MathSciNetCrossRefMATH

42.

Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit schemes for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012) MathSciNetCrossRefMATH

43.

Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical method for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760–1781 (2009) MathSciNetCrossRefMATH

Titel: Stability and Convergence of Modified Du Fort–Frankel Schemes for Solving Time-Fractional Subdiffusion Equations
Autoren:: Hong-lin Liao
Ya-nan Zhang
Ying Zhao
Han-sheng Shi
Publikationsdatum: 01.12.2014
DOI: https://doi.org/10.1007/s10915-014-9841-1
Verlag: Springer US
Zeitschrift: Journal of Scientific Computing
Ausgabe 3/2014
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691

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