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Convergence Analysis of a Block-by-Block Method for Fractional Differential Equations

Published online by Cambridge University Press:  28 May 2015

Jianfei Huang ,
Yifa Tang  and
Luis Vázquez
Jianfei Huang*
Affiliation:
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Yifa Tang*
Affiliation:
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Luis Vázquez*
Affiliation:
Departamento de Matemática Aplicada, Facultad de Informática, Instituto de Matemática Interdisciplinar (IMI), Universidad Complutense de Madrid, 28040-Madrid, Spain
*
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
Corresponding author.Email address:[email protected]
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Abstract

The block-by-block method, proposed by Linz for a kind of Volterra integral equations with nonsingular kernels, and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations (FDEs) with Caputo derivatives, is an efficient and stable scheme. We analytically prove and numerically verify that this method is convergent with order at least 3 for any fractional order index α > 0.

Keywords

26A3365M12Fractional differential equationCaputo derivativeblock-by-block methodconvergence analysis
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 2 , May 2012 , pp. 229 - 241
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Baeumer, B., Meerschaert, M. M., Benson, D. A. and Wheatcraft, S. W., Subordinated advection-dispersion equation for contaminant transport, Water Resour. Res., 37 (2001), pp. 1543–1550.CrossRefGoogle Scholar
[2]Bagley, R. L. and Calico, R. A., Fractional order state equations for the control of viscoelastically damped structures, J. Guid. Contr. Dyn., 14 (1991), pp. 304–311.CrossRefGoogle Scholar
[3]Chang, F.-X., Chen, J. and Huang, W., Anomalous diffusion and fractional advection-diffusion equation, Chin. Phys. Soc., 54 (2005), pp. 1113–1117.Google Scholar
[4]Dison, J. and Mekee, S., Weakly singular discrete Gronwall inequalities, Z. Angew. Math. Mech., 66 (1986), pp. 535–544.CrossRefGoogle Scholar
[5]Diethelm, K., Ford, N. J. and Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam., 29 (2002), pp. 3–22.CrossRefGoogle Scholar
[6]Diethelm, K. and Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265 (2002), pp. 220–248.CrossRefGoogle Scholar
[7]Diethelm, K., Ford, N. J. and Freed, A. D., Deailed error analysis for a fractional Adams method, Numer. Algorithms, 36 (2004), pp. 31–52.CrossRefGoogle Scholar
[8]Gorenflo, R., Mainardi, F., Scalas, E. and Raberto, M., Fractional calculus and continuous-time finace. III, The diffusion limit. Mathematical finance, Trends in Math., Birkhuser, Basel, 2001.Google Scholar
[9]Kumar, P. and Agrawal, O.P., An approximate method for numerical solution of fractional differential equations, Singal Process., 86 (2006), pp. 2602–2610.CrossRefGoogle Scholar
[10]Lubich, Ch., Discretized fractional calculus, SIAM J. Math. Anal., vol. 17, no. 3, (1986), pp. 704–719.CrossRefGoogle Scholar
[11]Linz, P., An method for nonlinear solving Volterra integral equations of the second kind, Math. Comput., vol. 23, no. 107, (1969), pp. 595–599.CrossRefGoogle Scholar
[12]Lin, R. and Liu, F., Fractional high order methods for the nonlinear fractional ordinary differential equation, Nonlinear Dynam., 66 (2007), pp. 856–869.Google Scholar
[13]Miller, K. and Ross, B., An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993.Google Scholar
[14]Oldham, K. B. and Spanier, J., The fractional calculus Academic Press, New York, 1974.Google Scholar
[15]Podlubny, I., Fractional differential equations Academic Press, New York, 1999.Google Scholar
[16]Raberto, M., Scalas, E. and Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A, 314 (2002), pp. 749–755.CrossRefGoogle Scholar
[17]Sabatelli, L., Keating, S., Dudley, J. and Richmond, P., Waiting time distributions in financial markets, Eur. Phys. J. B, 27 (2002), pp. 273–275.CrossRefGoogle Scholar
[18]Schumer, R., Benson, D. A., Meerschaert, M. M. and Baeumer, B., Multiscaling fractional advection-dispersion equations and their solutions, Water Resour. Res., 39 (2003), pp. 1022–1032.CrossRefGoogle Scholar
[19]Vázquez, L., From Newton’s equation to fractional diffusion and wave equations, Adv. Differ. Equ., Article ID 169421 (2011), 13 pages.Google Scholar