Investigation of space-fractional diffusion equations via PGS iterative method

Authors

  • Howard Crane Wayne State University, Michigan, United States

Keywords:

Matrix, Implicit, PGS method, GS iterative method, linear

Abstract

To speed up the convergence rate in solving the linear system iteratively, we construct the corresponding preconditioned linear system. Then we formulate and implement the Preconditioned Gauss-Seidel (PGS) iterative method for solving the generated linear system. One example of the problem is presented to illustrate the effectiveness of PGS method. The numerical results of this study show that the proposed iterative method is superior to the basic GS iterative method. In this paper, we deal with the application of an unconditionally implicit finite difference approximation equation of the one-dimensional linear space-fractional diffusion equations via the Caputo’s space-fractional derivative. Based on this implicit approximation equation, the corresponding linear system can be generated in which its coefficient matrix is large scale and sparse.

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References

Baeumer, B., Kovács, M., & Meerschaert, M. M. (2008). Numerical solutions for fractional reaction– diffusion equations. Computers & Mathematics with Applications, 55(10), 2212-2226.

Bu, W., Tang, Y., & Yang, J. (2014). Galerkin finite element method for two-dimensional Riesz space fractional diffusion equations. Journal of Computational Physics, 276, 26-38.

Chen, S., Liu, F., Jiang, X., Turner, I., & Anh, V. (2015). A fast semi-implicit difference method for a nonlinear two-sided space-fractional diffusion equation with variable diffusivity coefficients. Applied Mathematics and Computation, 257, 591-601.

Churbanov, A. G., & Vabishchevich, P. N. (2016). Numerical investigation of a space-fractional model of turbulent fluid flow in rectangular ducts. Journal of Computational Physics, 321, 846-859.

Dai, P. F., Wu, Q. B., & Zhu, S. F. (2019). Quasi Toeplitz splitting iteration methods for unsteady space fractional diffusion equations. Numerical Methods for Partial Differential Equations, 35(2), 699-715.

Huang, J., & Yang, D. (2017). A unified difference-spectral method for time–space fractional diffusion equations. International Journal of Computer Mathematics, 94(6), 1172-1184.

Jia, J., & Wang, H. (2015). Fast finite difference methods for space-fractional diffusion equations with fractional derivative boundary conditions. Journal of Computational Physics, 293, 359-369.

Lei, S. L., & Sun, H. W. (2013). A circulant preconditioner for fractional diffusion equations. Journal of Computational Physics, 242, 715-725.

Li, X. Y., & Wu, B. Y. (2018). Iterative reproducing kernel method for nonlinear variable-order space fractional diffusion equations. International Journal of Computer Mathematics, 95(6-7), 12101221.

Lin, F. R., Yang, S. W., & Jin, X. Q. (2014). Preconditioned iterative methods for fractional diffusion equation. Journal of Computational Physics, 256, 109-117.

Luo, W. H., Li, C., Huang, T. Z., Gu, X. M., & Wu, G. C. (2018). A high-order accurate numerical scheme for the caputo derivative with applications to fractional diffusion problems. Numerical functional analysis and optimization, 39(5), 600-622.

Pang, H. K., & Sun, H. W. (2012). Multigrid method for fractional diffusion equations. Journal of Computational Physics, 231(2), 693-703.

Saadatmandi, A., & Dehghan, M. (2011). A tau approach for solution of the space fractional diffusion equation. Computers & Mathematics with Applications, 62(3), 1135-1142.

Singh, J., Kumar, D., & Swroop, R. (2016). Numerical solution of time-and space-fractional coupled Burgers’ equations via homotopy algorithm. Alexandria Engineering Journal, 55(2), 1753-1763.

Sunarto, A., Sulaiman, J., & Saudi, A. (2014). Full-Sweep SOR Iterative Method to Solve SpaceFractional Diffusion Equations. Australian Journal of Basic and Applied Science, 8(24), 153-158.

Tadjeran, C., & Meerschaert, M. M. (2007). A second-order accurate numerical method for the twodimensional fractional diffusion equation. Journal of Computational Physics, 220(2), 813-823.

Wang, H., & Du, N. (2013). A superfast-preconditioned iterative method for steady-state space-fractional diffusion equations. Journal of Computational Physics, 240, 49-57.

Wang, H., & Du, N. (2014). Fast alternating-direction finite difference methods for three-dimensional space-fractional diffusion equations. Journal of Computational Physics, 258, 305-318.

Wang, H., & Du, N. (2014). Fast solution methods for space-fractional diffusion equations. Journal of Computational and Applied Mathematics, 255, 376-383.

Wang, H., & Wang, K. (2011). An O (N log2N) alternating-direction finite difference method for twodimensional fractional diffusion equations. Journal of Computational Physics, 230(21), 78307839.

Wang, H., Wang, K., & Sircar, T. (2010). A direct O (N log2 N) finite difference method for fractional diffusion equations. Journal of Computational Physics, 229(21), 8095-8104.

Wu, G. C., & Lee, E. W. M. (2010). Fractional variational iteration method and its application. Physics Letters A, 374(25), 2506-2509.

Yang, F., Zhang, Y., & Li, X. X. (2020). Landweber iterative method for identifying the initial value problem of the time-space fractional diffusion-wave equation. Numerical Algorithms, 83(4), 15091530.

Yang, Q., Turner, I., Moroney, T., & Liu, F. (2014). A finite volume scheme with preconditioned Lanczos method for two-dimensional space-fractional reaction–diffusion equations. Applied Mathematical Modelling, 38(15-16), 3755-3762.

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Published

2020-11-09

How to Cite

Crane, H. (2020). Investigation of space-fractional diffusion equations via PGS iterative method. Tennessee Research International of Social Sciences, 2(2), 40–47. Retrieved from http://triss.org/index.php/journal/article/view/24

Issue

Vol. 2 No. 2: July-December 2020

Section

Research Articles

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