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Lubich Second-Order Methods for Distributed-Order Time-Fractional Differential Equations with Smooth Solutions

Published online by Cambridge University Press:  12 May 2016

Rui Du ,
Zhao-peng Hao  and
Zhi-zhong Sun
Rui Du*
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
Zhao-peng Hao
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
Zhi-zhong Sun
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, China
*
*Corresponding author. Email addresses:[email protected] (R. Du), [email protected] (Z. Hao), [email protected] (Z. Sun)
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Abstract

This article is devoted to the study of some high-order difference schemes for the distributed-order time-fractional equations in both one and two space dimensions. Based on the composite Simpson formula and Lubich second-order operator, a difference scheme is constructed with convergence in the L1(L∞)-norm for the one-dimensional case, where τ,h and σ are the respective step sizes in time, space and distributed-order. Unconditional stability and convergence are proven. An ADI difference scheme is also derived for the two-dimensional case, and proven to be unconditionally stable and convergent in the L1(L∞)-norm, where h1 and h2 are the spatial step sizes. Some numerical examples are also given to demonstrate our theoretical results.

Keywords

Distributed-order time-fractional equationsLubich operatorcompact difference schemeADI schemeconvergencestability
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 2 , May 2016 , pp. 131 - 151
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Podlubny,, I.Fractional Differential Equations, Academic Press, New York (1999).Google Scholar
[2]Oldham, K.B. and Spanier, J., The Fractional Calculus, Academic Press, New York (1974).Google Scholar
[3]Kilbas, A., Srivastava, H. and Trujillo, J., Theory and Applications of Fractional Differential Equations, Elsevier, Boston (2006).Google Scholar
[4]Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London (2010).CrossRefGoogle Scholar
[5]Huang, F. and Liu, F., The time fractional diffusion equation and the advection-dispersion equation, ANZIAM J. 46, 317–330 (2005).CrossRefGoogle Scholar
[6]Gorenflo, R., Luchko, Y.and Mainardi, F.Wright functions as scale-invariant solutions of the diffusion-wave equation, J. Comp. Appl. Math. 118, 175–191 (2000).CrossRefGoogle Scholar
[7]Povstenko, Y., Non-central-symmetric solution to time-fractional diffusion-wave equation in a sphere under Dirichlet boundary condition, Fract. Calc. Appl. Anal. 15, 253–266 (2012).CrossRefGoogle Scholar
[8]Ray, S.S. and Bera, R.K., Analytical solution of a fractional diffusion equation by Adomian decomposition method, Appl. Math. Comp. 174, 329–336 (2006).Google Scholar
[9]Yuste, S. and Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Num. Anal. 42, 1862–1874 (2005).CrossRefGoogle Scholar
[10]Langlands, T. and Henry, B., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comp. Phys. 205, 719–736 (2005).CrossRefGoogle Scholar
[11]Chen, C., Liu, F., Turner, I. and Anh, V., A Fourier method for the fractional diffusion equation describing sub-diffusion, J. Comp. Phys. 227, 886–897 (2007).CrossRefGoogle Scholar
[12]Zhuang, P., Liu, F., Anh, V. and Turner, I., New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Num. Anal. 46, 1079–1095 (2008).CrossRefGoogle Scholar
[13]Cao, J. and Xu, C., A high order schema for the numerical solution of the fractional ordinary differential equations, J. Comp. Phys. 238, 154–168 (2013).CrossRefGoogle Scholar
[14]Li, C. and Ding, H., Higher order finite difference method for the reaction and anomalous-diffusion equation, Appl. Math. Model. 38, 3802–3821 (2014).CrossRefGoogle Scholar
[15]Sun, Z.Z. and Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Num. Math. 56, 193–209 (2006).CrossRefGoogle Scholar
[16]Wang, Z. and Vong, S., Compact difference schemes for the modified anomalous fractional subdiffusion equation and the fractional diffusion-wave equation, J. Comp. Phys. 277, 1–15 (2014).CrossRefGoogle Scholar
[17]Li, C. and Zeng, F., Finite difference methods for fractional differential equations, Int. J. Bifurcation Chaos 22(04), 1230014 (2012).CrossRefGoogle Scholar
[18]Cui, M., Compact alternating direction implicit method for two-dimensional time fractional diffusion equation, J. Comp. Phys. 231, 2621–2633 (2012).CrossRefGoogle Scholar
[19]Cui, M., Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation, Num. Algor. 62, 383–409 (2013).CrossRefGoogle Scholar
[20]Zhang, Y.N. and Sun, Z.Z., Alternating direction implicit schemes for the two-dimensional fractional subdiffusion equation, J. Comp. Phys. 230, 8713–8728 (2011).CrossRefGoogle Scholar
[21]Zhang, Y.N. and Sun, Z.Z., Error analysis of a compact ADI scheme for the 2D fractional subdiffusion equation, J. Sci. Comp. 59, 104–128 (2014).CrossRefGoogle Scholar
[22]Dimitrov, Y., Numerical approximations for fractional differential equations, J. Fract. Calc. Appl. 5, suppl. 3S, No. 22, 45 pp. (2014).Google Scholar
[23]Podlubny, I., Skovranek, T., Jara, B.M., Petras, J., Verbitsky, V. and Chen, Y.Q., Matrix approach to discrete fractional calculus III: Non-equidistant grids, variable step length and distributed orders, Phil. Trans. R. Soc. A 371, 20120153 (2012).Google Scholar
[24]Diethelm, K. and Ford, N.J., Numerical analysis for distributed order differential equations, J. Comp. Appl. Math. 225, 96–104 (2009).CrossRefGoogle Scholar
[25]Liu, F., Meerschaert, M.M., McGough, R.J., Zhuang, P. and Liu, Q., Numerical methods for solving the multi-term time-fractional wave-diffusion equation, Fract. Calc. Appl. Anal. 16, 9–25 (2013).CrossRefGoogle ScholarPubMed
[26]Ye, H., Liu, F., Anh, V. and Turner, I., Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains, IMA J. Appl. Math. 80, 825–838 (2015).CrossRefGoogle Scholar
[27]Ford, N.J., Morgado, M.L. and Rebelo, M., A numerical method for the distributed order time-fractional diffusion equation, ICFDA'14 Catania, ISBN 978-1-4799-2590-2, pp. 23–25 (2014).Google Scholar
[28]Morgado, M.L. and Rebelo, M., Numerical approximation of distributed order reaction-diffusion equations, J. Comp. Appl. Math. 275, 216–227 (2014).CrossRefGoogle Scholar
[29]Katsikadelis, J.T., Numerical solution of distributed order fractional differential equations, J. Comp. Phys. 259, 11–22 (2014).CrossRefGoogle Scholar
[30]Lubich, C., Discretizedfractional calculus, SIAM J. Math. Anal. 17, 704–719 (1986).CrossRefGoogle Scholar
[31]Brunner, H. and Van der Houwen, P.J., The Numerical Solution of Volterra Equations, CWI Monographs, Vol. 3. North-Holland, Amsterdam, (1986).Google Scholar
[32]Zhou, H., Tian, W.Y. and Deng, W., Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comp. 56, 45–66 (2013).CrossRefGoogle Scholar
[33]Sun, Z.Z., Numerical Methods of Partial Differential Equations, Science Press, Beijing (2012).Google Scholar
[34]Liao, H.L. and Sun, Z.Z., Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations, Num. Meth. Partial Differential Eq. 26, 37–60 (2010).CrossRefGoogle Scholar
[35]Henrici, P., Fast Fourier methods in computational complex analysis, SIAM Rev. 21, 481–527 (1979).CrossRefGoogle Scholar