Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-28T01:59:55.910Z Has data issue: false hasContentIssue false
  • English
  • Français

High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary Conditions

Published online by Cambridge University Press:  28 May 2015

Seakweng Vong  and
Zhibo Wang
Seakweng Vong*
Affiliation:
Department of Mathematics, University of Macau, Av. Padre Tomás Pereira Taipa, Macau, China
Zhibo Wang*
Affiliation:
Department of Mathematics, University of Macau, Av. Padre Tomás Pereira Taipa, Macau, China
*
Corresponding author. Email address: [email protected]
Corresponding author. Email address: [email protected]
Get access

Abstract

A compact finite difference scheme is derived for a time fractional differential equation subject to Neumann boundary conditions. The proposed scheme is second-order accurate in time and fourth-order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. The stability and convergence of the schemes are analysed using their matrix forms.

Keywords

65M0665M1265M1535R11Time fractional differential equationNeumann boundary conditionscompact ADI schemeweighted and shifted Grünwald difference operatorconvergence
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 3 , August 2014 , pp. 222 - 241
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Podlubny, I., Fractional Differential Equations, Academic Press, New York (1999).Google Scholar
[2]Kilbas, A., Srivastava, H. and Trujillo, J., Theory and Applications of Fractional Differential Equations, Elsevier Science and Technology (2006).Google Scholar
[3]Lubich, C., Discretized fractional calculus, SIAM J. Math. Anal. 17, 704–719 (1986).Google Scholar
[4]Meerschaert, M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172, 65–77 (2004).Google Scholar
[5]Yuste, S., Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216, 264–274 (2006).Google Scholar
[6]Ervin, V.J. and Roop, J.P., Variational formulation for the stationary fractional advection dispersion equation, Numer. Meth. Part. Differ. Eq. 22, 558–576 (2006).Google Scholar
[7]Sun, Z. and Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math. 56, 193–209 (2006).Google Scholar
[8]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225, 1533–1552 (2007).Google Scholar
[9]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. Numer. Anal. 46, 1079–1095 (2008).CrossRefGoogle Scholar
[10]Li, X. and Xu, C., A spacetime spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal. 47, 2018–2131 (2009).CrossRefGoogle Scholar
[11]Du, R., Cao, W. and Sun, Z., A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model. 34, 2998–3007 (2010).Google Scholar
[12]Sousa, E. and Li, C., A weighted finite difference method for the fractional diffusion equation based on the Riemann-Liouville derivative, arXiv:1109.2345 [math.NA].Google Scholar
[13]Tian, W., Zhou, H. and Deng, W., A class of second-order difference approximations for solving space fractional diffusion equations, Math. Comput. arXiv:1201.5949 [math.NA].Google Scholar
[14]Zhou, H., Tian, W. and Deng, W., Quasi-compact finite difference schemes for space fractional diffusion equations, J. Sci. Comput. 56, 45–66 (2013).Google Scholar
[15]Huang, J., Tang, Y., Vázquez, L. and Yang, J., Two finite difference schemes for time fractional diffusion-wave equation, Numer. Algor. 64, 707–720 (2013).Google Scholar
[16]Wang, Z. and Vong, S., Compact difference schemes for the modified anomalous fractional subdiffusion equation and the fractional diffusion-wave equation, arXiv:1310.5298 [math.NA].Google Scholar
[17]Wang, Z. and Vong, S., A high order ADI scheme for the two-dimensional time fractional diffusion-wave equation, Int. J. Comput. Math. doi:10.1080/00207160.2014.915960.CrossRefGoogle Scholar
[18]Vong, S. and Wang, Z., Compact finite difference scheme for the fourth-order fractional subdiffusion system, Adv. Appl. Math. Mech. (to appear).Google Scholar
[19]Langlands, T. and Henry, B., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205, 719–736 (2005).CrossRefGoogle Scholar
[20]Gao, G. and Sun, Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys. 230, 586–595 (2011).CrossRefGoogle Scholar
[21]Yuste, S. and Acedo, L., An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal. 42, 1862–1874 (2005).Google Scholar
[22]Huang, J., Tang, Y., Wang, W. and Yang, J., A compact difference scheme for time fractional diffusion equation with Neumann boundary conditions, AsiaSim 2012, 273–284, Springer, Berlin Heidelberg (2012).CrossRefGoogle ScholarPubMed
[23]Zhao, X. and Sun, Z., A box-type scheme for fractional sub-diffusion equation with Neumann boundaryconditions, J. Comput. Phys. 230, 6061–6074 (2011).CrossRefGoogle Scholar
[24]Ren, J., Sun, Z. and Zhao, X., Compact difference scheme for the fractional sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys. 232, 456–467 (2013).CrossRefGoogle Scholar
[25]Ren, J. and Sun, Z., Numerical algorithm with high spatial accuracy for the fractional diffusion-wave equation with Neumann boundary conditions, J. Sci. Comput. 56, 381–408 (2013).Google Scholar
[26]Quarteroni, A. and Valli, A., Numerical Approximation of Partial Differential Equations, Springer, Berlin (1997).Google Scholar
[27]Zhang, Y., Sun, Z. and Zhao, X., Compact alternating direction implicit scheme for the two-dimensional fractional diffusion-wave equation, SIAM J. Numer. Anal. 50, 1535–1555 (2012).Google Scholar