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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)
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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 

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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