Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T19:07:57.273Z Has data issue: false hasContentIssue false
  • English
  • Français

Compact Finite Difference Scheme for the Fourth-Order Fractional Subdiffusion System

Published online by Cambridge University Press:  03 June 2015

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

Abstract

In this paper, we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system. We consider the situation in which the unknown function and its first-order derivative are given at the boundary. The scheme is shown to have high order convergence. Numerical examples are given to verify the theoretical results.

Keywords

26A3335R1165M0665M1265M15Fourth-order fractional subdiffusion equationcompact difference schemeenergy methodstabilityconvergence
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 6 , Issue 4 , August 2014 , pp. 419 - 435
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] Metzler, R. and Klafter, J., The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), pp. R161–R208.Google Scholar
[2] Sokolov, I. M. and Klafter, J., From diffusion to anomalous diffusion: a century after Ein-stein’s Brownian motion, Chaos, 15 (2005), 026103.Google Scholar
[3] Machado, J. A. Tenreiro, Analysis and design offractional-order digital control systems, Syst. Anal. Model. Simul., 27 (1997), pp. 107–122.Google Scholar
[4] Baleanu, D., Defterli, O. and Agrawal, O. P., A central difference numerical scheme for fractional optimal control problems, J. Vip. Control, 15 (2009), pp. 583–597.Google Scholar
[5] Magin, R. L., Fractional calculus in the bioengineering, Crit. Rev. Biomed. Eng., 32 (2004), pp. 1–104.Google Scholar
[6] Rossikhin, Y. A. and Hitikova, M. V. S., Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids, Appl. Mech. Rev., 50 (1997), pp. 15–67.Google Scholar
[7] Tadjeran, C., Meerschaert, M. M. and Scheffler, H. P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys., 213 (2006), pp. 205–213.Google Scholar
[8] Tadjeran, C. and Meerschaert, M. M., A second-order accurate numerical method for the two-dimensional fractional diffusion equation, J. Comput. Phys., 220 (2007), pp. 813–823.CrossRefGoogle Scholar
[9] Sun, Z. and Wu, X., A fully discrete difference scheme for a diffusion-wave system, Appl. Numer. Math., 56 (2006), pp. 193–209.Google Scholar
[10] Du, R., Cao, W. and Sun, Z., A compact difference scheme for the fractional diffusion-wave equa-tion, Appl. Math. Model., 34 (2010), pp. 2998–3007.Google Scholar
[11] Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comput. Phys., 228 (2009), pp. 7792–7804.CrossRefGoogle Scholar
[12] Gao, G. and Sun, Z., A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys., 230 (2011), pp. 586–595.Google Scholar
[13] Vong, S., Pang, H. and Jin, X., A high-order difference scheme for the generalized Cattaneo equation, East Asian J. Appl. Math., 2 (2012), pp. 170–184.Google Scholar
[14] Ren, J., Sun, Z. and Zhao, X., Compact difference scheme for the fractinoal sub-diffusion equation with Neumann boundary conditions, J. Comput. Phys., 232 (2013), pp. 456–467.Google Scholar
[15] Oldhan, K. B. and Spainer, J., The Fractional Calculus, Academic Press, New York, 1974.Google Scholar
[16] Sneddon, I. N., Fourier Transforms, McGraw Hill, New York, 1951.Google Scholar
[17] Jafari, H., Dehghan, M. AND Sayevand, K., Solving a fourth-order fractional diffusion wave equation in a bounded domain by decomposition method, Numer. Methods Partial Differential Equations, 24 (2008), pp. 1115–1126.Google Scholar
[18] Agrawal, O. P., A general solution for the fourth-order fractional diffusion-wave equation, Fract. Calculus Appl. Anal., 3 (2000), pp. 1–12.Google Scholar
[19] Agrawal, O. P., A general solution for the fourth-order fractional diffusion-wave equation defined in bounded domain, Comput. Struct., 79 (2001), pp. 1497–1501.Google Scholar
[20] Golbabai, A. and Sayevand, K., Fractional calculus-a new approach to the analysis of general-ized fourth-order diffusion-wave equations, Comput. Math. Appl., 67 (2011), pp. 2227–2231.Google Scholar
[21] Hu, X. and Zhang, L., On finite difference methods for fourth-order fractional diffusion-wave and subdiffusion systems, Appl. Math. Comput., 218 (2012), pp. 5019–5034.Google Scholar
[22] Hu, X. and Zhang, L., A compact finite difference scheme for the fourth-order fractional diffusion- wave system, Comput. Phys. Commun., 182 (2011), pp. 1645–1650.Google Scholar