10.1007/s40096-021-00397-2

Gaussian radial basis function and quadrature Sinc method for two-dimensional space-fractional diffusion equations

  • Nafiseh Noghrei1,
  • Asghar Kerayechian*2,
  • Ali R. Soheili2,
  1. Department of Applied Mathematics, Ferdowsi University of Mashhad, Mashhad, IR
  2. Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, IR

Published in Issue 2021-04-11

How to Cite

Noghrei, N., Kerayechian, A., & Soheili, A. R. (2021). Gaussian radial basis function and quadrature Sinc method for two-dimensional space-fractional diffusion equations. Mathematical Sciences, 16(1 (March 2022). https://doi.org/10.1007/s40096-021-00397-2
Crossref
Scopus
Google Scholar
Europe PMC

Abstract

Abstract The combination of Sinc quadrature method and double exponential transformation (DE) is a powerful tool to approximate the singular integrals, and radial basis functions (RBFs) are useful for the higher-dimensional space problem. In this study, we develop a numerical method base on Gaussian-RBF combined with QR-factorization of arising matrix and DE-quadrature Sinc method to approximate the solution of two-dimensional space-fractional diffusion equations. When the number of central nodes increases, the ill-conditioning of resultant matrix can be eliminated by using GRBF-QR method. Two numerical examples have been presented to test the efficiency and accuracy of the method.

Keywords

  • Space-fractional diffusion equations,
  • Riemann–Liouville fractional derivatives,
  • DE-Sinc quadrature method,
  • Gaussian-RBF

References

  1. Jin et al. (2015) Preconditioned iterative methods for two-dimensional space-fractional diffusion equations (pp. 469-488) https://doi.org/10.4208/cicp.120314.230115a
  2. Chen et al. (2018) A Kronecker product splitting preconditioner for two-dimensional space-fractional diffusion equations (pp. 1-14) https://doi.org/10.1016/j.jcp.2018.01.034
  3. Moghaderi et al. (2017) Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations (pp. 992-1011) https://doi.org/10.1016/j.jcp.2017.08.064
  4. Fukunaga, M., Masataka, J.C.: Application of Fractional Diffusion Equation to Amorphous Semiconductors, pp. 389–400 (2005)
  5. Blackledge (2010) Application of the fractional diffusion equation for predicting market behaviour 40(3) (pp. 130-158)
  6. Bai and Feng (2007) Fractional-order anisotropic diffusion for image denoising (pp. 2492-2502) https://doi.org/10.1109/TIP.2007.904971
  7. Carreras et al. (2001) Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model (pp. 5096-5103) https://doi.org/10.1063/1.1416180
  8. Raberto et al. (2002) Waiting-times and returns in high-frequency financial data: an empirical study (pp. 749-755) https://doi.org/10.1016/S0378-4371(02)01048-8
  9. Podlubny, I.: Fractional Differential Equations, Math. Sci. Engrg, 198, Academic Press, San Diego (1999)
  10. Benson et al. (2006) Application of a fractional advection-dispersion equation (pp. 1403-1412)
  11. Lei and Sun (2013) A circulant preconditioner for fractional diffusion equations (pp. 715-725) https://doi.org/10.1016/j.jcp.2013.02.025
  12. Chen (2014) A splitting preconditioner for the iterative solution of implicit Runge–Kutta and boundary value methods (pp. 607-621) https://doi.org/10.1007/s10543-014-0467-3
  13. Lin et al. (2014) Preconditioned iterative methods for fractional diffusion equation (pp. 109-117) https://doi.org/10.1016/j.jcp.2013.07.040
  14. Pan et al. (2014) Preconditioning techniques for diagonal-times-Toeplitz matrices in fractional diffusion equations (pp. 2698-2719) https://doi.org/10.1137/130931795
  15. Simmons et al. (2015) A preconditioned numerical solver for stiff nonlinear reaction–diffusion equations with fractional Laplacians that avoids dense matrices (pp. 254-268) https://doi.org/10.1016/j.jcp.2015.02.012
  16. Murio (2008) Implicit finite difference approximation for time fractional diffusion equations (pp. 1138-1145) https://doi.org/10.1016/j.camwa.2008.02.015
  17. Wang et al. (2010) A direct O(N log2 N) finite difference method for fractional diffusion equations (pp. 8095-8104) https://doi.org/10.1016/j.jcp.2010.07.011
  18. Wang and Wang (2011) An O(N log2 N) alternating-direction finite difference method for two-dimensional fractional diffusion equations (pp. 7830-7839) https://doi.org/10.1016/j.jcp.2011.07.003
  19. Meerschaert et al. (2006) Finite difference methods for two-dimensional fractional dispersion equation (pp. 249-261) https://doi.org/10.1016/j.jcp.2005.05.017
  20. Meerschaert and Tadjeran (2006) Finite difference approximations for two-sided space-fractional partial differential equations (pp. 80-90) https://doi.org/10.1016/j.apnum.2005.02.008
  21. Tian et al. (2015) A class of second order difference approximation for solving space-fractional diffusion equations (pp. 1703-1727) https://doi.org/10.1090/S0025-5718-2015-02917-2
  22. Chen (2015) Generalized Kronecker product splitting iteration for the solution of implicit Runge–Kutta and boundary value methods (pp. 357-370) https://doi.org/10.1002/nla.1960
  23. Lei and Sun (2013) A circulant preconditioner for fractional diffusion equation (pp. 715-725) https://doi.org/10.1016/j.jcp.2013.02.025
  24. Pang and Sun (2012) Multigrid method for fractional diffusion equations (pp. 693-703) https://doi.org/10.1016/j.jcp.2011.10.005
  25. Ruge, J. W., Stüben, K.: Algebraic multigrid. In: S.F. McCormick (Ed.), Multigrid Methods, in Front. Math. Appl, 3, SIAM, Philadelphia, pp. 73–130 (1987)
  26. Chan et al. (1998) Multigrid method for ill-conditioned symmetric Toeplitz systems (pp. 516-529) https://doi.org/10.1137/S1064827595293831
  27. Serra-Capizzano (2003) Generalized locally Toeplitz sequences: spectral analysis and applications to discretized differential equations (pp. 371-402) https://doi.org/10.1016/S0024-3795(02)00504-9
  28. Tilli (1998) A note on the spectral distribution of Toeplitz matrices 45(2–3) (pp. 147-159) https://doi.org/10.1080/03081089808818584
  29. Zhu et al. (2018) Differential quadrature method for space-fractional diffusion equations on 2D irregular domains (pp. 853-877) https://doi.org/10.1007/s11075-017-0464-0
  30. Fasshouer (2007) World Scientific https://doi.org/10.1142/6437
  31. Wenlland (2005) Cambridg University Press
  32. Fedoseyer et al. (2002) Improved multiquadrics method for elliptic partial differential equations via PDE collocation on the boundary (pp. 439-455) https://doi.org/10.1016/S0898-1221(01)00297-8
  33. Fornberg et al. (2011) Stable computations with Gaussian radial basis functions (pp. 869-892) https://doi.org/10.1137/09076756X
  34. Piret and Hanert (2013) A radial basis functions method for fractional diffusion equations (pp. 71-81) https://doi.org/10.1016/j.jcp.2012.10.041
  35. Okayama et al. (2010) Approximate formulae for fractional derivatives by means of sinc methods
  36. Lund and Bowers (1992) SIAM https://doi.org/10.1137/1.9781611971637
  37. Stenger (1993) Springer https://doi.org/10.1007/978-1-4612-2706-9
  38. Tanaka et al. (2009) Function classes for double exponential integration formulas (pp. 631-655) https://doi.org/10.1007/s00211-008-0195-1
  39. Mori and Sugihara (2001) The double-exponential transformation in numerical analysis (pp. 287-296) https://doi.org/10.1016/S0377-0427(00)00501-X