Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A PREDICTOR-CORRECTOR METHOD FOR FRACTIONAL EVOLUTION EQUATIONS

  • Choi, Hong Won (Department of Mathematics Seoul Science High School) ;
  • Choi, Young Ju (Department of Mathematics Education Seoul National University) ;
  • Chung, Sang Kwon (Department of Mathematics Education Seoul National University)
  • Received : 2015.11.03
  • Published : 2016.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

Abstract. Numerical solutions for the evolutionary space fractional order differential equations are considered. A predictor corrector method is applied in order to obtain numerical solutions for the equation without solving nonlinear systems iteratively at every time step. Theoretical error estimates are performed and computational results are given to show the theoretical results.

Keywords

References

  1. K. Adolfsson, M. Enelund, and S. Larsson, Adaptive discretization of fractional order viscoelasticity using sparse time history, Comput. Methods Appl. Mech. Engrg. 193 (2004), no. 42-44, 4567-4590. https://doi.org/10.1016/j.cma.2004.03.006
  2. B. Baeumer, M. Kovacs, and M. M. Meerschaert, Fractional reproduction-dispersal equations and heavy tail dispersal kernels, Bull. Math. Biol. 69 (2007), no. 7, 2281-2297. https://doi.org/10.1007/s11538-007-9220-2
  3. Z. Chen, Finite Element Methods and Their Applications, Springer Verlag, Berlin Heidelberg, 2005.
  4. H. W. Choi, S. K. Chung, and Y. J. Lee, Numerical solutions for space fractional dispersion equations with nonlinear source terms, Bull. Korean Math. Soc. 47 (2010), no. 6, 1225-1234. https://doi.org/10.4134/BKMS.2010.47.6.1225
  5. Y. J. Choi and S. K. Chung, Finite element solutions for the space fractional diffusion equations with a nonlinear source term, Abstr. Appl. Anal. 2012 (2012), Art. ID 596184, 25 pp.
  6. Z. Q. Deng, V. P. Singh, and L. Bengtsson, Numerical solution of fractional advection-dispersion equation, J. Hydraulic Engineering 130 (2004), 422-431. https://doi.org/10.1061/(ASCE)0733-9429(2004)130:5(422)
  7. K. Diethlem, N. J. Ford, and A. D. Freed, A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam. 29 (2002), no. 1-4, 3-22. https://doi.org/10.1023/A:1016592219341
  8. V. J. Ervin, N. Heuer, and J. P. Roop, Numerical approximation of a time dependent, nonlinear space-fractional diffusion equation, SIAM J. Numer. Anal. 45 (2007), no. 2, 572-591. https://doi.org/10.1137/050642757
  9. M. G. Herrick, D. A. Benson, M. M. Meerschaert, and K. R. McCall, Hydraulic conductivity, velocity, and the order of the fractional dispersion derivative in a high heterogeneous system, Water Resources Research 38 (2002), 1227, doi:10.1029/2001WR000914.
  10. F. Liu, A. Anh, and I. Turner, Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math. 166 (2004), no. 1, 209-219. https://doi.org/10.1016/j.cam.2003.09.028
  11. V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, and H. R. Hicks, Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys. 192 (2003), no. 2, 406-421. https://doi.org/10.1016/j.jcp.2003.07.008
  12. M. M. Meerschaert, H-P. Scheffler, and C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys. 211 (2006), no. 1, 249-261. https://doi.org/10.1016/j.jcp.2005.05.017
  13. M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), no. 1, 65-77. https://doi.org/10.1016/j.cam.2004.01.033
  14. M. M. Meerschaert and C. Tadjeran, Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56 (2006), no. 1, 80-90. https://doi.org/10.1016/j.apnum.2005.02.008
  15. K. B. Oldham and J. Spanier, The Fractional Calculus, Dover Publications, New York, 2002.
  16. J. P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in ${\mathbb{R}}^2$, J. Comput. Appl. Math. 193 (2006), no. 1, 243-268. https://doi.org/10.1016/j.cam.2005.06.005
  17. C. Tadjeran, M. M. Meerschaert, and H.-P. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006), no. 1, 205-213. https://doi.org/10.1016/j.jcp.2005.08.008