Skip to main content
SpringerLink
Log in

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

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Baban A, Bonchiş C, Fikl A, Roşu F (2016) Parallel simulations for fractional-order systems. In: SYNASC 2016, pp 141–144

  2. Baleanu D, Diethelm K, Scalas E, Trujillo JJ (2016) Fractional calculus: models and numerical methods, vol 5. World Scientific, Singapore

    Book  MATH  Google Scholar 

  3. Bonchiş C, Kaslik E, Roşu F (2017) Improved parallel simulations for fractional-order systems using hpc. In: CMMSE 2017

  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):1845–1863

    Article  MathSciNet  MATH  Google Scholar 

  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. http://stacks.iop.org/1751-8121/43/i=8/a=085002

  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):508–518

    Article  MathSciNet  MATH  Google Scholar 

  7. Daftardar-Gejji V, Sukale Y, Bhalekar S (2014) A new predictor-corrector method for fractional differential equations. Appl Math Comput 244:158–182

    MathSciNet  MATH  Google Scholar 

  8. Deng W (2007) Short memory principle and a predictor-corrector approach for fractional differential equations. J Comput Appl Math 206(1):174–188

    Article  MathSciNet  MATH  Google Scholar 

  9. Deng W, Li C (2012) Numerical schemes for fractional ordinary differential equations. In: Numerical Modelling. InTech

  10. Diethelm K (2011) An efficient parallel algorithm for the numerical solution of fractional differential equations. Fract Calc Appl Anal 14(3):475–490

    Article  MathSciNet  MATH  Google Scholar 

  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):3–22

    Article  MathSciNet  MATH  Google Scholar 

  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):73–99

    Google Scholar 

  13. Ford NJ, Simpson AC (2001) The numerical solution of fractional differential equations: speed versus accuracy. Numer Algorithms 26(4):333–346

    Article  MathSciNet  MATH  Google Scholar 

  14. Galeone L, Garrappa R (2009) Explicit methods for fractional differential equations and their stability properties. J Comput Appl Math 228(2):548–560

    Article  MathSciNet  MATH  Google Scholar 

  15. Garrappa R (2010) On linear stability of predictor-corrector algorithms for fractional differential equations. Int J Comput Math 87(10):2281–2290

    Article  MathSciNet  MATH  Google Scholar 

  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):1521–1537

    Article  Google Scholar 

  17. Palanivel J, Suresh K, Sabarathinam S, Thamilmaran K (2017) Chaos in a low dimensional fractional order nonautonomous nonlinear oscillator. Chaos Solitons Fractals 95:33–41

    Article  MathSciNet  MATH  Google Scholar 

  18. Pedas A, Tamme E (2011) Spline collocation methods for linear multi-term fractional differential equations. J Comput Appl Math 236(2):167–176

    Article  MathSciNet  MATH  Google Scholar 

  19. Redbooks I (2009) IBM System Blue Gene Solution: Blue Gene/P Application Development. Vervante

  20. Song L, Wang W (2013) A new improved adomian decomposition method and its application to fractional differential equations. Appl Math Model 37(3):1590–1598

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang W, Cai X (2012) Efficient implementations of the adams-bashforth-moulton method for solving fractional differential equations. In: Proceedings of FDA12

  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):617–637

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Institute e-Austria Timişoara Romania, Timisoara, Romania

    C. Bonchiş, E. Kaslik & F. Roşu

  2. Department of Mathematics and Computer Science, West University of Timişoara, Timisoara, Romania

    C. Bonchiş, E. Kaslik & F. Roşu

Authors
  1. C. Bonchiş
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Kaslik
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. F. Roşu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Roşu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bonchiş, C., Kaslik, E. & Roşu, F. HPC optimal parallel communication algorithm for the simulation of fractional-order systems. J Supercomput 75, 1014–1025 (2019). https://doi.org/10.1007/s11227-018-2267-z

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-018-2267-z

Keywords

Search

Navigation