The Journal of Supercomputing

, Volume 75, Issue 3, pp 1014–1025 | Cite as

HPC optimal parallel communication algorithm for the simulation of fractional-order systems

  • C. Bonchiş
  • E. Kaslik
  • F. RoşuEmail author


A parallel numerical simulation algorithm is presented for fractional-order systems involving Caputo-type derivatives, based on the Adams–Bashforth–Moulton predictor–corrector scheme. The parallel algorithm is implemented using several different approaches: a pure MPI version, a combination of MPI with OpenMP optimization and a memory saving speedup approach. All tests run on a BlueGene/P cluster, and comparative improvement results for the running time are provided. As an applied experiment, the solutions of a fractional-order version of a system describing a forced series LCR circuit are numerically computed, depicting cascades of period-doubling bifurcations which lead to the onset of chaotic behavior.


Fractional-order system Parallel numerical algorithm HPC processing 



This work was supported by a grant of the Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, Project No. PN-II-RU-TE-2014-4-0270.


  1. 1.
    Baban A, Bonchiş C, Fikl A, Roşu F (2016) Parallel simulations for fractional-order systems. In: SYNASC 2016, pp 141–144Google Scholar
  2. 2.
    Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2016) Fractional calculus: models and numerical methods, vol 5. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonchiş C, Kaslik E, Roşu F (2017) Improved parallel simulations for fractional-order systems using hpc. In: CMMSE 2017Google Scholar
  4. 4.
    Cafagna D, Grassi G (2008) Bifurcation and chaos in the fractional-order chen system via a time-domain approach. Int J Bifurc Chaos 18(7):1845–1863MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cottone G, Paola MD, Santoro R (2010) A novel exact representation of stationary colored gaussian processes (fractional differential approach). J Phys A Math Theor 43(8):085002.
  6. 6.
    Daftardar-Gejji V, Jafari H (2005) Adomian decomposition: a tool for solving a system of fractional differential equations. J Math Anal Appl 301(2):508–518MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Daftardar-Gejji V, Sukale Y, Bhalekar S (2014) A new predictor-corrector method for fractional differential equations. Appl Math Comput 244:158–182MathSciNetzbMATHGoogle Scholar
  8. 8.
    Deng W (2007) Short memory principle and a predictor-corrector approach for fractional differential equations. J Comput Appl Math 206(1):174–188MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Deng W, Li C (2012) Numerical schemes for fractional ordinary differential equations. In: Numerical Modelling. InTechGoogle Scholar
  10. 10.
    Diethelm K (2011) An efficient parallel algorithm for the numerical solution of fractional differential equations. Fract Calc Appl Anal 14(3):475–490MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Diethelm K, Ford N, Freed A (2002) A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn 29(1–4):3–22MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Duan JS, Rach R, Baleanu D, Wazwaz AM (2012) A review of the adomian decomposition method and its applications to fractional differential equations. Commun Fract Cal 3(2):73–99Google Scholar
  13. 13.
    Ford NJ, Simpson AC (2001) The numerical solution of fractional differential equations: speed versus accuracy. Numer Algorithms 26(4):333–346MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Galeone L, Garrappa R (2009) Explicit methods for fractional differential equations and their stability properties. J Comput Appl Math 228(2):548–560MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Garrappa R (2010) On linear stability of predictor-corrector algorithms for fractional differential equations. Int J Comput Math 87(10):2281–2290MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gong C, Bao W, Tang G, Yang B, Liu J (2014) An efficient parallel solution for caputo fractional reaction–diffusion equation. J Supercomput 68(3):1521–1537CrossRefGoogle Scholar
  17. 17.
    Palanivel J, Suresh K, Sabarathinam S, Thamilmaran K (2017) Chaos in a low dimensional fractional order nonautonomous nonlinear oscillator. Chaos Solitons Fractals 95:33–41MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pedas A, Tamme E (2011) Spline collocation methods for linear multi-term fractional differential equations. J Comput Appl Math 236(2):167–176MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Redbooks I (2009) IBM System Blue Gene Solution: Blue Gene/P Application Development. VervanteGoogle Scholar
  20. 20.
    Song L, Wang W (2013) A new improved adomian decomposition method and its application to fractional differential equations. Appl Math Model 37(3):1590–1598MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhang W, Cai X (2012) Efficient implementations of the adams-bashforth-moulton method for solving fractional differential equations. In: Proceedings of FDA12Google Scholar
  22. 22.
    Zhang W, Wei W, Cai X (2014) Performance modeling of serial and parallel implementations of the fractional adams–bashforth–moulton method. Fract Cal Appl Anal 17(3):617–637MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute e-Austria Timişoara RomaniaTimisoaraRomania
  2. 2.Department of Mathematics and Computer ScienceWest University of TimişoaraTimisoaraRomania

Personalised recommendations