Caputo Finite Difference Solution for solving Time-Fractional Diffusion Equations via weighted point iteration

Mohd Usran Alibubin; Jumat Sulaiman; Fatihah Anas Muhiddin; Andang Sunarto.

Transactions on Science and Technology, 11(3), 165 - 174.

Back to main issue

ABSTRACT
Time-fractional diffusion equations (TFDEs) are widely used in modeling anomalous diffusion processes, which occur in various fields such as physics, engineering, and economics. These equations offer a more accurate representation of systems where classical diffusion models fall short, particularly in capturing memory and hereditary properties of materials. In this paper, we employ the Caputo finite difference approximation equation for TFDEs by applying a discretization scheme based on the second-order implicit finite difference and Caputo fractional derivative operator. To solve these equations numerically, the one-dimensional TFDEs are discretized using Caputo’s implicit finite difference approximation. The corresponding system of linear approximation equations is then solved using weighted point iteration methods, specifically Successive Overrelaxation (SOR) and Gauss-Seidel (GS). Three examples are provided to evaluate the performance of these iterative methods. The numerical results demonstrate that the SOR method requires fewer iterations and reduces computational time, proving to be more efficient compared to the Gauss-Seidel method.

KEYWORDS: Finite Difference Scheme; Caputo Derivative Operator; Time-Fractional Diffusion Equations; Weighted point iteration.



Download this PDF file

REFERENCES
  1. Ali, U., Abdullah, F. A. & Ismail, A. I. 2017. Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation. Journal of Interpolation and Approximation in Scientific Computing, 2017(2), 18–29.
  2. Alibubin, M. U., Sunarto, A., Akhir, M. K. M. & Sulaiman, J. (2016). Performance analysis of half-sweep SOR iteration with rotated nonlocal arithmetic mean scheme for 2D nonlinear elliptic problems. Global Journal of Pure and Applied Mathematics, 12(4), 3415–3424.
  3. Alibubin, M.U., Sunarto, A. & Sulaiman, J. 2016. Quarter-sweep Nonlocal Discretization Scheme with QSSOR Iteration for Nonlinear Two-point Boundary Value Problems. Journal of Physics: Conference Series, 710(1), 12023
  4. Alibubin, M.U., Sulaiman, J., Muhiddin, F. A. & Sunarto, A. 2024. Implementation of the KSOR Method for Solving One-Dimensional Time-Fractional Parabolic Partial Differential Equations with the Caputo Finite Difference Scheme Title of Manuscript. Journal of Advanced Research in Applied Sciences and Engineering Technology, 48(1), 168–179.
  5. Almeida, R., Bastos, N. R. & Monteiro, M. T. 2015. Modeling some real phenomena by fractional differential equations. Mathematical Methods in the Applied Sciences, 39(16), 4846–4855.
  6. Baharuddin, S., Sunarto, A. & Dalle, J. 2017. KSOR iterative method for solving Fredholm integral equations of second kind. Journal of Engineering and Applied Sciences, 12, 3220–3224.
  7. Basiron, Y. 2007. Palm oil production through sustainable plantations. European Journal of Lipid Science and Technology, 109(3), 289–295.
  8. Cen, Z., Huang, J. & Xu, A. 2018. An efficient numerical method for a two-point boundary value problem with a Caputo fractional derivative. Journal of Computational and Applied Mathematics, 336, 1–7.
  9. Chen, W., Sun, H., Zhang, X. & Chen, W. 2015. Fractional calculus in anomalous diffusion and fractional mechanics. World Scientific.
  10. Diethelm, K. 2010. The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer Science & Business Media.
  11. Diethelm, K. & Ford, N. J. 2002. Analysis of fractional differential equations. Numerical Algorithms, 36(1), 31–52.
  12. Edeki, S., Ugbebor, O. & Owoloko, E. 2017. Analytical solution of the time-fractional order Black-Scholes model for stock option valuation on dividend yield basis. International Journal of Applied Mathematics, 47(4), 435–447.
  13. Evans, D. J. 1985. Group explicit iterative methods for solving large linear systems. International Journal of Computer Mathematics, 17(1), 81–108.
  14. Ford, J. N., Xiao, J. & Yan, Y. 2011. A finite element method for time fractional partial differential equations. Fractional Calculus and Applied Analysis, 8(3), 454–474.
  15. Gaspar, F. J. & Rodrigo, C. 2017. Multigrid waveform relaxation for the time-fractional heat equation. SIAM Journal on Scientific Computing, 39(4), A1201–A1224.
  16. Ghaffari, R. & Ghoreishi, F. 2019. Reduced spline method based on a proper orthogonal decomposition technique for fractional sub-diffusion equations. Applied Numerical Mathematics, 137, 62–79.
  17. Gunzburger, M. & Wang, J. A. 2019. Second-order Crank-Nicolson method for time-fractional PDES. International Journal of Numerical Analysis and Modeling, 16(2), 225–239.
  18. Karatay, I., Bayramoglu, S. R. & Sahin, A. 2011. Implicit difference approximation for the time fractional heat equation with the nonlocal condition. Applied Numerical Mathematics, 61, 1281–1288.
  19. Khalid, M., Ahmad, A. & Qasim, M. 2020. Fractional derivatives in image processing. Advances in Mathematics: Scientific Journal, 9(1), 413–421.
  20. Kilbas, A. A., Saigo, M. & Trujillo, J. J. 2004. Theory and Applications of Fractional Differential Equations. Elsevier.
  21. Kurulay, M. & Bayram, M. 2012. Solutions to fractional partial differential equations via the generalized Mittag-Leffler function. Mathematical and Computer Modelling, 55(3–4), 303–311.
  22. Miller, K. S. & Ross, B. 1993. An Introduction to Fractional Calculus and Fractional Differential Equations. Wiley.
  23. Mohammad, T., Neeraj, D., Deependra, N. & Anand, C. 2021. Approximation of Caputo time-fractional diffusion equation using redefined cubic exponential B-spline collocation technique. AIMS Mathematics, 6(4), 3805–3820.
  24. Muhiddin, F. A., Sulaiman, J. & Sunarto, A. 2019. Numerical performance of half-sweep SOR iteration with the Grünwald implicit finite difference for time-fractional parabolic equations. Journal of Advanced Research in Dynamical and Control Systems, 11(12 Special Issue), 119–125.
  25. Paliivets, S., Shpak, I. & Gromov, V. 2021. Fractional-order fluid mechanics modeling. Applied Mathematical Modelling, 97, 201–214.
  26. Paradisi, P. 2015. Fractional calculus in statistical physics: The case of time fractional diffusion equation. Communications in Applied and Industrial Mathematics, e-530, 1–25.
  27. Podlubny, I. 1999. Fractional Differential Equations. Academic Press.
  28. Radzuan, N. Z., Suardi, F. M. M. & Sulaiman, J. 2017. KSOR iterative method with quadrature scheme for solving system of Fredholm integral equations of second kind. Journal of Fundamental and Applied Sciences, 9(5S), 609–623.
  29. Rashid, S., Khalid, M. & Mansoor, S. 2021. Numerical simulation of time-fractional diffusion models using wavelet methods. Mathematical Methods in the Applied Sciences, 44(10), 7458–7476.
  30. Sunarto, A., Sulaiman, J. & Saudi, A. 2014. SOR method for the implicit finite difference solution of time-fractional diffusion equations. Borneo Science, 34, 34–42.
  31. Sunarto, A., Sulaiman, J. & Saudi, A. 2016. Caputo’s implicit solution of time-fractional diffusion equation using half sweep AOR iteration. Global Journal of Pure and Applied Mathematics, 12(4), 3469–3479.
  32. Wang, Y. M. & Ren, L. A. 2019. High-order L2-compact difference method for Caputo-type time fractional sub-diffusion equations with variable coefficients. Applied Mathematics and Computation, 342, 71–93.
  33. Wright, E. M. 1935. The asymptotic expansion of the generalized Bessel function. Proceedings of the London Mathematical Society, 38(1), 257–270.
  34. Wu, L., Zhao, Y. & Yang, X. 2018. Alternating segment explicit-implicit and implicit-explicit parallel difference method for time fractional sub-diffusion equation. Journal of Applied Mathematics and Physics, 6(5), 1017–1033.
  35. Xu, Q. & Xu, Y. 2018. Extremely low order time-fractional differential equation and application in combustion process. Communications in Nonlinear Science and Numerical Simulation, 64, 135–148.
  36. Young, D. M. 1971. Iterative Solution of Large Linear Systems. Academic Press.
  37. Youssef, I. K. 2012. On the Successive Overrelaxation method. Journal of Mathematics and Statistics, 8(2), 176–184.
  38. Youssef, I. K. & Taha, A. A. 2013. A generalization of the Successive Overrelaxation method. Applied Mathematics and Computation, 219(9), 4601–4613.
  39. Yuste, S. B. 2006. Weighted average finite difference method for fractional diffusion equations. Journal of Computational Physics, 216, 264–274.
  40. Zahra, W. K. & Elkholy, S. M. 2013. Cubic spline solution of fractional Bagley-Torvik equation. Electronic Journal of Mathematical Analysis and Applications, 1(2), 230–241.
  41. Zhang, Y. 2009. A finite difference method for fractional partial differential equation. Applied Mathematics and Computation, 215, 524–529.