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Finite Difference/Element Method for a Two-Dimensional Modified Fractional Diffusion Equation

Published online by Cambridge University Press:  03 June 2015

Na Zhang ,
Weihua Deng  and
Yujiang Wu
Na Zhang*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Weihua Deng*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
Yujiang Wu*
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
*
Email: [email protected]
Corresponding author. Email: [email protected]
Email: [email protected]
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Abstract

We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the L1-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.

Keywords

26A3365M0665M1265M60Modified subdiffusion equationfinite difference methodfinite element methodstabilityconvergence rate
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 4 , August 2012 , pp. 496 - 518
Copyright
Copyright © Global-Science Press 2012

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References

[1]Adams, R. A., Sobolev Spaces, Academic Press, New York, 1975.Google Scholar
[2]Bramble, J. H. and Xu, J. C., Some estimates for a weighted L2 projection, Math. Comput., 56 (1991), pp. 463–476.Google Scholar
[3]Ciarlet, P. G., The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford, 1978.Google Scholar
[4]Chen, C.-M., Liu, F., Anh, V. and Turner, L., A Fourier method for the fractional diffusion equation dscribing sub-diffusion, J. Comput. Phys., 227 (2007), pp. 886–897.CrossRefGoogle Scholar
[5]Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Electr. Trans. Numer. Anal., 5 (1997), pp. 1–6.Google Scholar
[6]Deng, W. H., Numerical algorithm for the time fractional Fokker-Planck equation, J. Comput. Phys., 227 (2007), pp. 1510–1522.Google Scholar
[7]Deng, W. H. and Li, C., Finite difference methods and their physical constraints for the fractional Klein-Kramers equation, Numer. Methods. Partial. Diff. Equations., 27 (2011), pp. 1561–1583.Google Scholar
[8]Guermond, J. L. and Shen, J., Velocity-correction projection method for incompressible flows, SIAM J. Numer. Anal., 41 (2003), pp. 112–134.Google Scholar
[9]Jiang, Y. and Ma, J., High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math., 235 (2011), pp. 3285–3290.Google Scholar
[10]Langlands, T. A. M. and Henry, B. I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys., 205 (2005), pp. 719–736.Google Scholar
[11]Langlands, T. A. M., Solution of a modified fractional diffusion equation, Phys. A., 367 (2006), pp. 136–144.Google Scholar
[12]Li, X. and Xu, C., Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation, Commun. Comput. Phys., 8 (2010), pp. 1016–1051.Google Scholar
[13]Liu, F., Yang, C. and Burrage, K., Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term, J. Comput. Appl. Math., 231 (2009), pp. 160–176.Google Scholar
[14]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys., 225 (2007), pp. 1533–1552.Google Scholar
[15]Lin, Y., Li, X. and Xu, C., Finite difference/spectral approximations for the fractional cable equation, Math. Comput., 80 (2011), pp. 1369–1396.CrossRefGoogle Scholar
[16]Liu, Q., Liu, F., Turner, I. and Anh, V., Finite element approximation for a modified anomalous subdiffusion equation, Appl. Math. Model., 35 (2011), pp. 4103–4116.Google Scholar
[17]Metzler, R. and Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339 (2000), pp. 1–77.Google Scholar
[18]Metzler, R. and Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractonal dynamics, J. Phys. A. Math. Gen., 37 (2004), pp. 161–208.Google Scholar
[19]Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, 1999.Google Scholar
[20]Quarteroni, A. and Valli, A., Numerical Approximation of Partial Differential Equations, Springer-Verlag, New York, 1997.Google Scholar
[21]Sokolov, I. M., Chechkin, A. V. and Klafter, J., Distributed-order fractional kinetics, Acta. Phys. Pol. B., 35 (2004), pp. 1323–1341.Google Scholar
[22]Sokolov, I. M. and Klafter, J., From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion, Chaos., 15 (2005), 026103.Google Scholar
[23]Gao, G. H. and Sun, Z. Z., A compact difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), pp. 586–595.Google Scholar
[24]Yuste, S. B. and Acedo, L., An explicit finite difference method and a new Von Neumanntype stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42 (2005), pp. 1862–1874.Google Scholar
[25]Zhuang, P., Liu, F., Anh, V. and Turner, I., New solution and analytical techniques of the implticit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal., 46 (2008), pp. 1079–1095.CrossRefGoogle Scholar