A. Sunarto*, J. Sulaiman, & A. Saudi

School of Science and Technology,
Universiti Malaysia Sabah, 88400 Kota Kinabalu, Sabah.
School of Engineering and Information Technology
Universiti Malaysia Sabah, 88400 Kota Kinabalu, Sabah.
Email: andang99@gmail.com

ABSTRACT. In this study, we derive an unconditionally implicit finite difference approximation equation from the discretization of the one-dimensional linear time fractional diffusion equations by using the Caputo’s time fractional derivative. Then this approximation equation hence will be used to generate the corresponding system of linear equations. The approximation solution of the linear system is described via the implementation of Successive Over-Relaxation (SOR) iterative method. An example of the problem is presented to illustrate the effectiveness of SOR method. The findings of this study show that the proposed iterative method is superior compared with the Gauss-Seidel iterative method.

KEYWORDS. Caputo’s fractional derivative; Implicit Finite Difference Scheme; SOR Method

REFERENCES

• Abdullah, A. R. 1991. The Four Point Explicit Decoupled Group (EDG) Method: A Fast Poisson Solver. International Journal of Computer Mathematics, 38: 61-70.

• Agrawal, O. P. 2002. Solution for a Fractional Diffusion-wave Equation Defined in a Bounded Domain. Nonlinear Dynamic, 29: 145-155.

• Ali, Erman, S., Ozgur, B., & Korkmaz, E. 2013. Analysis of Fractional Partial Differential Equations by Taylor Series Expansion, Boundary Value Problem A Springer Open Journal, 2: 68-80.

• Chaves, A. 1998. Fractional Diffusion Equation to Describe Levy Flight. Physic Letter A, 239: 13-16.

• Diethelm, K., & Freed, A. D. 1999. On the Solution of Nonlinear Fractional Order Differential Equation used in the Modeling of Viscoplasticity, in Scientific Computing in Chemical engineering II Computational Fluid Dynamic, Reaction Engineering and Molecular Properties, Heidelberg: Springer Verlag, 217-224.

• El-Kahlout, A., Salim, T. O., & El-Azab, S. 2008. Exact Solution of Time-Fractional Partial Differential Equation. Applied Mathematical Science, 52: 2577-2590.

• Evans, D. J., & Sahimi, M. S. 1988. The Alternating Group Explicit Iterative Method (AGE) to Solve Parabolic and Hyperbolic Partial Differential Equations. Annual Review of Fluid Mechanics and Heat Transfer, 2: 283-389.

• Evans, D. J. 1985. Group Explicit Iterative Methods for Solving Large Linear Systems. International Journal of Computer Mathematics, 17: 81-108.

• Hackbusch, W. 1995. Iterative Solution of Large Sparse Systems of Equations. New York: Springer-Verlag.

• Hadjidimos, A. 2000. Successive Overrelaxations and Relative Methods. Journal of Computational and Applied Mathematics, 123: 177-199.

• Liu, F., Anh, V. & Turner, I. 2004. Numerical Solution of the Space Fractional Fokker-Planck equation. Journal of Computational and Applied Mathematics, 166: 209-219.

• Liu, F., Zhuang, P., Anh, V., & Turner, I. 2006. A Fractional-order Implicit Difference Approximation for the Space-time Fractional Diffusion Equation. ANZIAM Journal, 47: C871-C887.

• Mainardi, F. 1997. Fractals and Fractional Calculus Continum Mechanics. Heidelberg: Springer Verlag. 291-348.

• Martins, M. M., Yousif, W. S., & Evans, D. J. 2002. Explicit Group AOR Method for Solving Elliptic Partial Differential Equations. Neura Parallel, and Science Computation, 10 (4): 411-422.

• Meerschaert, M. M., & Tadjeran, C. 2004. Finite Difference Approximation for Fractional Advection-dispersion Flow Equations. Journal of Computational and Applied Mathematics, 172: 145-155.

• Murio, Diego A. 2008. Implicit Finite Difference Approximation for Time-Fractional Diffusion Equations. An International Journal Computer & Mathematics with Applications, 56: 1138-1145.

• Othman, M. & Abdullah, A. R. 2000. An Efficient Four Points Modified Explicit Group Poisson Solver. International Journal Computer Mathematics, 76: 203-217.

• Saad, Y. 1996. Iterative Method for Sparse Linear Systems. Boston: International Thomas Publishing.

• Shen, S. & Liu, F. 2005. Error Analysis of an Explicit Finite Difference Approximation for the Space Fractional Diffusion Equation with Insulated Ends. ANZIAM Journal, 46 (E): C871-C887.

• Starke, G. & Niethammer, W. 1991. SOR for AX – XB = C. Linear Algebra and Its Applications, 154-156: 355- 375.

• Li-ying, S. 2005. A Comparison Theorem for the SOR Iterative Method. Journal of Computational and Applied Mathematics, 181: 336-341.

• Sweilam, N. H., Khader, M. M., & Mahdy, A. M. S. 2012. Crank-Nicolson Finite Difference Method for Solving Time-fractional Diffusion Equation. Journal of Fractional Calculus and Applications, 2: 1-9.

• Young, D. M. 1954. Iterative Methods for solving Partial Difference Equations of Elliptic Type. Transactions of the AMS – American Mathematical Society, 76: 92-111.

• Young, D. M. 1971. Iterative Solution of Large Linear Systems. London: Academic Press.

• Young, D. M. 1972. Second-degree Iterative Methods for the Solution of Large Linear Systems. Journal of Approximation Theory, 5: 137-148.

• Yousif, W. S., & Evans, D. J. 1995. Explicit De-coupled Group Iterative Methods and Their Implementations. Parallel Algorithms and Applications, 7: 53-71.

• Youssef, I. K. 2012. On the Successive Overrelaxation Method. Journal of Mathematics and Statistics, 2: 176-184.

• Yuste, S. B., & Acedo, L. 2005. An Explicit Finite Difference Method and a New Von Neumann-type Stability Analysis for Fractional Diffusion Equations. SIAM Journal on Numerical Analysis, 42 (5): 1862-1874.

• Yuste, S. B. 2006. Weighted Average Finite Difference Method for Fractional Diffusion Equations. Journal of Computational Physics, 216: 264-274.

• Zhang, Y. 2009. A Finite Difference Method for Fractional Partial Differential Equation. Applied Mathematics and Computation, 215: 524-529.

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