Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-24T02:50:20.842Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Solution of the Time-Fractional Sub-Diffusion Equation on an Unbounded Domain in Two-Dimensional Space

Part of: Partial differential equations, boundary value problems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  07 September 2017

Hongwei Li ,
Xiaonan Wu  and
Jiwei Zhang
Hongwei Li*
Affiliation:
School of Mathematics and Statistics, Shandong Normal University, Ji'nan 250014, China
Xiaonan Wu*
Affiliation:
Department of Mathematics, Hong Kong Baptist University, Hong Kong, China
Jiwei Zhang*
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, China
*
*Corresponding author. Email addresses:[email protected] (H. Li), [email protected] (X. Wu), [email protected] (J. Zhang)
*Corresponding author. Email addresses:[email protected] (H. Li), [email protected] (X. Wu), [email protected] (J. Zhang)
*Corresponding author. Email addresses:[email protected] (H. Li), [email protected] (X. Wu), [email protected] (J. Zhang)
Get access

Abstract

The numerical solution of the time-fractional sub-diffusion equation on an unbounded domain in two-dimensional space is considered, where a circular artificial boundary is introduced to divide the unbounded domain into a bounded computational domain and an unbounded exterior domain. The local artificial boundary conditions for the fractional sub-diffusion equation are designed on the circular artificial boundary by a joint Laplace transform and Fourier series expansion, and some auxiliary variables are introduced to circumvent high-order derivatives in the artificial boundary conditions. The original problem defined on the unbounded domain is thus reduced to an initial boundary value problem on a bounded computational domain. A finite difference and L1 approximation are applied for the space variables and the Caputo time-fractional derivative, respectively. Two numerical examples demonstrate the performance of the proposed method.

Keywords

Fractional sub-diffusion equationunbounded domainlocal artificial boundary conditionsfinite difference methodCaputo time-fractional derivative

MSC classification

Secondary: 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods 65N06: Finite difference methods
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 3 , August 2017 , pp. 439 - 454
Copyright
Copyright © Global-Science Press 2017 

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] West, B.J., Bologna, M. and Grigolini, P., Physics of Fractal Operators, Springer, New York (2003).Google Scholar
[2] Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).Google Scholar
[3] Metzler, R. and Klafter, J., The random walks guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339, 1–77 (2000).Google Scholar
[4] Herrmann, R., Fractional Calculus: An Introduction for Physicists, World Scientific, Hackensack, New Jersey (2011).Google Scholar
[5] Henry, B.I. and Wearne, S.L., Fractional reaction-diffusion, Phys. A 276, 448–455 (2000).Google Scholar
[6] Yuste, S.B. 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
[7] Schneider, W.R. and Wyss, W., Fractional diffusion and wave equations, J. Math. Phys. 30, 134–144 (1989).Google Scholar
[8] Wyss, W., The fractional diffusion equation, J. Math. Phys. 27, 2782–2785 (1986).Google Scholar
[9] Gorenflo, R., Luchko, Y. and Mainardi, F., Wright function as scale-invariant solutions of the diffusion-wave equation, J. Comput. Appl. Math. 118, 175–191 (2000).CrossRefGoogle Scholar
[10] 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).Google Scholar
[11] Zhuang, P. and Liu, F., Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comput. 22, 87–99 (2006).Google Scholar
[12] Liu, Q., Gu, Y.T., Zhuang, P., Liu, F. and Nie, Y.F., An implicit RBF meshless approach for time fractional diffusion equations, Comput. Mech. 48, 1–12 (2011).Google Scholar
[13] Zhang, Y.N. and Sun, Z., Alternating direction implicit schemes for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys. 230, 8713–8728 (2011).Google Scholar
[14] Wang, Y.M., Maximum norm error estimates of ADI methods for a two dimensional fractional sub-diffusion equation, Adv. Math. Phys. 1-10, Article ID 293706 (2013).CrossRefGoogle Scholar
[15] Cui, M., Convergence analysis of high-order compact alternating direction implicit schemes for the two dimensional time fractional diffusion equation, Numer. Algorithms 62, 383–409 (2013).Google Scholar
[16] Sun, H., Chen, W. and Sze, K.Y., A semi-discrete finite element method for a class of time-fractional diffusion equations, Phil. Trans. R. Soc. A 371, 20120268 (2013).CrossRefGoogle ScholarPubMed
[17] Guo, B., Xu, Q. and Zhu, A., A second-order finite difference method for two-dimensional fractional percolation equations, Commun. Comput. Phys. 19, 733–757 (2016).Google Scholar
[18] Li, G., Sun, C., Jia, X. and Du, D., Numerical solution to the multi-term time fractional diffusion equation in a finite domain, Numer. Math. Theor. Meth. Appl. 9, 337–357 (2016).CrossRefGoogle Scholar
[19] Yang, X., Zhang, H. and Xu, D., Orthogonal spline collocation method for the two-dimensional fractional sub-diffusion equation, J. Comput. Phys. 256, 824–837 (2014).Google Scholar
[20] Yan, L. and Yang, F., A kansa-type MFS scheme for two dimensional time fractional diffusion equations, Eng. Anal. Bound. Elem. 37, 1426–1435 (2013).CrossRefGoogle Scholar
[21] Jin, B., Lazarov, R. and Zhou, Z., Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal. 51, 445–466 (2013).Google Scholar
[22] Gao, G., Sun, Z. and Zhang, H., A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications, J. Comput. Phys. 259, 33–50 (2014).Google Scholar
[23] Li, D., Zhang, C., Ran, M., A linear finite difference scheme for generalized time fractional Burgers equation, Appl. Math. Model. 40, 6069–6081 (2016).Google Scholar
[24] Gao, G. and Sun, Z., The finite difference approximation for a class of fractional sub-diffusion equations on a space unbounded domain, J. Comput. Phys. 236, 443–460 (2013).Google Scholar
[25] Brunner, H., Han, H. and Yin, D., Artificial boundary conditions and finite difference approximations for a time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain, J. Comput. Phys. 276, 541–562 (2014).Google Scholar
[26] Ghaffari, R. and Hosseini, S.M., Obtaining artificial boundary conditions for fractional sub-diffusion equation on space two-dimensional unbounded domains, Comput. Math. Appl. 68, 13–26 (2014).Google Scholar
[27] Dea, J.R., Absorbing boundary conditions for the fractional wave equation, Appl. Math. Comput. 219, 9810–9820 (2013).Google Scholar
[28] Awotunde, A.A., Ghanam, R.A. and Tatar, N., Artificial boundary condition for a modified fractional diffusion problem, Bound. Value Probl. 1, 1–17 (2015).Google Scholar
[29] Arnold, A., Ehrhardt, M., Schulte, M. and Sofronov, I., Discrete transparent boundary conditions for the Schrödinger equation on circular domains, Commun. Math. Sci. 10, 889–916 (2012).Google Scholar
[30] Han, H. and Huang, Z., Exact artificial boundary conditions for Schrödinger equation in ℝ2 , Commun. Math. Sci. 2, 79–94 (2004).Google Scholar
[31] Li, D. and Zhang, J., Efficient implementation to numerically solve the nonlinear time fractional parabolic problems on unbounded spatial domain, J. Comput. Phys. 322, 415–428 (2016).Google Scholar
[32] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego (1999).Google Scholar
[33] Han, H. and Wu, X., Artificial Boundary Method, Springer-Verlag, Berlin, Heidelberg and Tsinghua University Press, Beijing (2013).Google Scholar
[34] Han, H. and Bao, W., High-order local artificial boundary conditions for problems in unbounded domains, Comput. Methods Appl. Mech. Eng. 188, 455–471 (2000).Google Scholar
[35] Li, H., Wu, X. and Zhang, J., Local artificial boundary conditions for Schrödinger and heat equations by using high-order azimuth derivatives on circular artificial boundary, Comput. Phys. Commun. 185, 1606–1615 (2014).CrossRefGoogle Scholar
[36] Oldham, K.B. and Spanier, J., The Fractional Calculus, Academic Press, New York (1974).Google Scholar
[37] Diethelm, K., Ford, N.J., Freed, A.D. and Yu. Luchko, A selection of numerical methods, Comput. Methods Appl. Mech. Eng. 194, 743–773 (2005).Google Scholar
[38] Zhang, J., Han, H. and Brunner, H., Numerical blow-up of semilinear parabolic PDEs on unbounded domains in ℝ2 , J. Sci. Comput. 49, 367–382 (2011).CrossRefGoogle Scholar
[39] Zheng, C., Approximation, stability and fast evaluation of exact artificial boundary condition for one-dimensional heat equation, J. Comput. Math. 25, 730–745 (2007).Google Scholar
[40] Li, J., A fast time stepping method for evaluating fractional integrals, SIAM J. Sci. Comput. 31, 4696–4714 (2010).Google Scholar
[41] Jiang, S., Zhang, J., Zhang, Q. and Zhang, Z., Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations, Commun. Comput. Phys. 21, 650–678 (2017).Google Scholar
[42] Jin, X., Lin, F. and Zhao, Z., Preconditioned iterative methods for two-dimensional space-fractional diffusion equations, Commun. Comput. Phys. 18, 469–488 (2015).Google Scholar
[43] Wu, S. and Zhou, T., Fast parareal iterations for fractional diffusion equations, J. Comput. Phys. 329, 210–226 (2017).Google Scholar
[44] Zhang, Q., Zhang, J., Jiang, S. and Zhang, Z., Numerical solution to a linearized time fractional KdV equation on unbounded domains, Math. Comput., to appear (2017).Google Scholar