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, 56:36 | Cite as

Efficient solution of time-fractional differential equations with a new adaptive multi-term discretization of the generalized Caputo–Dzherbashyan derivative

  • Fabio DurastanteEmail author
Article
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Abstract

In this paper, we deal with both the discretization and the efficient solution of initial and initial-boundary value problems with a time derivative of distributed order. A new discretization based on an adaptive Gauss quadrature and product integral formulas is introduced and analyzed. The efficient solution of the resulting sequence of linear systems by Krylov iterative methods and approximate inverse preconditioning is discussed, together with the spectral analysis of the relative matrix sequences. Several numerical examples showing the effectiveness of the approach are included.

Keywords

Time-fractional derivatives Gauss quadrature Approximate inverse preconditioners 

Mathematics Subject Classification

65M22 65F10 65F08 35R11 

Notes

Acknowledgements

The author would like to thank the referees for their valuable comments.

References

  1. 1.
    Benzi, M.: Preconditioning techniques for large linear systems: a survey. J. Comput. Phys. 182, 418–477 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Benzi, M., Cullum, J.K., Tüma, M.: Robust approximate inverse preconditioning for the conjugate gradient method. SIAM J. Sci. Comput. 22(4), 1318–1332 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Benzi, M., Meyer, C.D., Tüma, M.: A sparse approximate inverse preconditioner for the conjugate gradient method. SIAM J. Sci. Comput. 17(5), 1135–1149 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benzi, M., Tüma, M.: A sparse approximate inverse preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 19(3), 968–994 (1998)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bertaccini, D., Donatelli, M., Durastante, F., Serra-Capizzano, S.: Optimizing a multigrid Runge–Kutta smoother for variable-coefficient convection–diffusion equations. Linear Algebra Appl. 533, 507–535 (2017).  https://doi.org/10.1016/j.laa.2017.07.036 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bertaccini, D., Durastante, F.: Interpolating preconditioners for the solution of sequence of linear systems. Comput. Math. Appl. 72(4), 1118–1130 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bertaccini, D., Durastante, F.: Solving mixed classical and fractional partial differential equations using short-memory principle and approximate inverses. Numer. Algor. 74(4), 1061–1082 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bertaccini, D., Durastante, F.: Iterative Methods and Preconditioning for Large and Sparse Linear Systems with Applications. Chapman & Hall/CRC Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2018)zbMATHGoogle Scholar
  9. 9.
    Bertaccini, D., Durastante, F.: Block structured preconditioners in tensor form for the all-at-once solution of a finite volume fractional diffusion equation. Appl. Math. Lett. 95, 92–97 (2019).  https://doi.org/10.1016/j.aml.2019.03.028 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bertaccini, D., Filippone, S.: Sparse approximate inverse preconditioners on high performance GPU platforms. Comput. Math. Appl. 71, 693–711 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Calabrò, F., Manni, C., Pitolli, F.: Computation of quadrature rules for integration with respect to refinable functions on assigned nodes. Appl. Numer. Math. 90, 168–189 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Canuto, C., Simoncini, V., Verani, M.: On the decay of the inverse of matrices that are sum of Kronecker products. Linear Algebra Appl. 452, 21–39 (2014)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chechkin, A., Gorenflo, R., Sokolov, I.: Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys. Rev. E 66(4), 046129 (2002)CrossRefGoogle Scholar
  14. 14.
    Chechkin, A.V., Gorenflo, R., Sokolov, I.M., Gonchar, V.Y.: Distributed order time fractional diffusion equation. Fract. Calc. Appl. Anal. 6(3), 259–280 (2003)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Chechkin, A.V., Klafter, J., Sokolov, I.M.: Fractional Fokker–Planck equation for ultraslow kinetics. Europhys. Lett. 63(3), 326 (2003)CrossRefGoogle Scholar
  16. 16.
    Demko, S., Moss, W.F., Smith, P.W.: Decay rates for inverses of band matrices. Math. Comput. 43(168), 491–499 (1984)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Diethelm, K., Ford, N.J.: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 154(3), 621–640 (2004)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and structure preserving preconditioners for fractional diffusion equations. J. Comput. Phys. 307, 262–279 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Donatelli, M., Mazza, M., Serra-Capizzano, S.: Spectral analysis and multigrid methods for finite volume approximations of space-fractional diffusion equations. SIAM J. Sci. Comput. 40(6), A4007–A4039 (2018).  https://doi.org/10.1137/17M115164X MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Edwards, J.T., Ford, N.J., Simpson, A.C.: The numerical solution of linear multi-term fractional differential equations: systems of equations. J. Comput. Appl. Math. 148(2), 401–418 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Fukuda, H., Katuya, M., Alt, E., Matveenko, A.: Gaussian quadrature rule for arbitrary weight function and interval. Comput. Phys. Commun. 167(2), 143–150 (2005)CrossRefGoogle Scholar
  22. 22.
    Garoni, C., Manni, C., Pelosi, F., Serra-Capizzano, S., Speleers, H.: On the spectrum of stiffness matrices arising from isogeometric analysis. Numer. Math. 127(4), 751–799 (2014).  https://doi.org/10.1007/s00211-013-0600-2 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Garoni, C., Serra-Capizzano, S.: The Theory of Generalized Locally Toeplitz Sequences: Theory and Applications. Springer Monographs, vol. I. ISBN: 978-3-319-53678-1. http://www.springer.com/gp/book/9783319536781 (2017) CrossRefGoogle Scholar
  24. 24.
    Garoni, C., Serra-Capizzano, S., Sesana, D.: Spectral analysis and spectral symbol of \(d\)-variate \(\mathbb{Q}_p\) Lagrangian FEM stiffness matrices. SIAM J. Matrix Anal. Appl. 36(3), 1100–1128 (2015).  https://doi.org/10.1137/140976480 MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Garrappa, R.: Stability-preserving high-order methods for multiterm fractional differential equations. Int. J. Bifurc. Chaos 22(04), 1250073 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Garrappa, R.: Numerical solution of fractional differential equations: a survey and a software tutorial. Mathematics 6(2), 16 (2018)CrossRefGoogle Scholar
  27. 27.
    Gautschi, W.: Orthogonal Polynomials: Computation and Approximation. Oxford University, Oxford (2004)zbMATHGoogle Scholar
  28. 28.
    Jin, B., Lazarov, R., Sheen, D., Zhou, Z.: Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data. Fract. Calc. Appl. Anal. 19(1), 69–93 (2016)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Katsikadelis, J.T.: Numerical solution of distributed order fractional differential equations. J. Comput. Phys. 259, 11–22 (2014)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Liao, H., Lyu, P., Vong, S., Zhao, Y.: Stability of fully discrete schemes with interpolation-type fractional formulas for distributed-order subdiffusion equations. Numer. Algorithm 75, 1–34 (2016)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17(3), 704–719 (1986)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Mainardi, F., Pagnini, G., Gorenflo, R.: Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 187(1), 295–305 (2007)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Moghaderi, H., Dehghan, M., Donatelli, M., Mazza, M.: Spectral analysis and multigrid preconditioners for two-dimensional space-fractional diffusion equations. J. Comput. Phys. 350, 992–1011 (2017)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Morgado, M.L., Rebelo, M., Ferrás, L.L., Ford, N.J.: Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method. Appl. Numer. Math. 114, 108–123 (2017).  https://doi.org/10.1016/j.apnum.2016.11.001 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Popolizio, M.: Numerical solution of multiterm fractional differential equations using the matrix Mittag–Leffler functions. Mathematics 6(1), 7 (2018)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Wheeler, J.C.: Modified moments and Gaussian quadratures. Rocky Mt. J. Math. 4(2), 287–296 (1974)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Ye, H., Liu, F., Anh, V.: Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains. J. Comput. Phys. 298, 652–660 (2015)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Young, A.: Approximate product-integration. Proc. R. Soc. Lond. A 224(1159), 552–561 (1954)MathSciNetCrossRefGoogle Scholar

Copyright information

© Istituto di Informatica e Telematica (IIT) 2019

Authors and Affiliations

  1. 1.Dipartimento di InformaticaUniversità di PisaPisaItaly

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