Skip to main content
SpringerLink
Log in

Fast predictor-corrector approach for the tempered fractional differential equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

The tempered evolution equation describes the trapped dynamics, widely appearing in nature, e.g., the motion of living particles in viscous liquid. This paper proposes the fast predictor-corrector approach for the tempered fractional ordinary differential equations by digging out the potential ‘very’ short memory principle. Algorithms based on the idea of equidistributing are detailedly described. Error estimates for the proposed schemes are derived; and the effectiveness and low computation cost, being linearly increasing with time t, are numerically demonstrated.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Baeumera, B., Meerschaert, M.M.: Tempered stable Lévy motion and transient super-diffusion. J. Comput. Appl. Math. 233, 2438–2448 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cartea, Á., del-Castillo-Negrete, D.: Fractional diffusion models of option prices in markets with jumps. Phys. A 374, 749–763 (2007)

    Article  Google Scholar 

  3. Cartea, Á., del-Castillo-Negrete, D.: Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. Phys. Rev. E 041105, 76 (2007)

    Google Scholar 

  4. Deng, J.W., Zhao, L.J., Wu, Y.J.: Efficient algorithms for solving the fractional ordinary differential equations. Appl. Math. Comput. 269, 196–216 (2015)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Deng, W.H.: Numerical algorithm for the time fractional Fokker-Planck equation. J. Comput. Phys. 227, 1510–1522 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Deng, W.H., Li, C.: Numerical Schemes for Fractional Ordinary Differential Equations. In P. Miidla (Ed.) Numerical modelling 355–374, Intech, Rijeka, Chapter 16 (2012)

  8. Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dynam 29, 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Diethelm, K., Ford, N.J., Freed, A.D.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36, 31–52 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ford, N.J., Simpson, A.C.: The numerical solution of fractional differential equations: Speed versus accuracy. Numer. Algorithms 26, 333–346 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gajda, J., Magdziarz, M.: Fractional Fokker-Planck equation with tempered α-stable waiting times: Langevin picture and computer simulation. Phys. Rev. E 011117, 82 (2010)

    MathSciNet  Google Scholar 

  13. Hanyga, A.: Wave propagation in media with singular memory. Math. Comput. Model 34, 1399–1421 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, C., Deng, W.H.: High order schemes for the tempered fractional diffusion equations. Adv. Compt. Math. 42(3), 543–572 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, C., Deng, W.H., Zhao, L.J.: Well-posedness and numerical algorithm for the tempered fractional ordinary differential equations. arXiv:1501.00376v1 [math.CA]

  16. Magdziarz, M., Weron, A.: Competition between subdiffusion and Lévy flights: A Monte Carlo approach. Phys. Rev. E 056702, 75 (2007)

    Google Scholar 

  17. Meerschaert, M.M., Sabzikar, F., Phanikumar, M.S., Zeleke, A.: Tempered fractional time series model for turbulence in geophysical flows. Journal of Statistical Mechanics: Theory and Experiment 14, 1742–5468 (2014)

    Google Scholar 

  18. Meerschaert, M.M., Zhang, Y., Baeumer, B.: Tempered anomalous diffusion in heterogeneous systems. Geophys. Res. Lett. 35, L17403 (2008)

    Article  Google Scholar 

  19. Podlubny, I.: Fractional differential equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  20. Sabzikar, F., Meerschaert, M.M., Chen, J.H.: Tempered fractional calculus. J. Comput. Phys. 293, 14–28 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Samko, S., Kilbas, A., Marichev, O.: Fractional integrals and derivatives: theory and applications. Gordon & Breach, London (1993)

    MATH  Google Scholar 

  22. Schmidt, M.G.W., Sagués, F., Sokolov, I.M.: Mesoscopic description of reactions for anomalous diffusion: a case study. J. Phys. Condens. Matter 065118, 19 (2007)

    Google Scholar 

  23. White, A.B.: On selection of equidistributing meshes for two-point boundary-value problems. SIAM. J. Numer. Anal. 16, 472–502 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhao, L.J., Deng, W.H.: Jacobian-predictor-corrector approach for fractional differential equations. Adv. Comput. Math. 40, 137–165 (2014)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, People’s Republic of China

    Jingwei Deng, Lijing Zhao & Yujiang Wu

  2. School of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, 730030, People’s Republic of China

    Jingwei Deng

Authors
  1. Jingwei Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lijing Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yujiang Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jingwei Deng.

Additional information

Supported by NNSFC 11271173 and 11471150, FRF CU 31920150039 (Northwest University for Nationalities), and HSSYP ME 13YJCZH029.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Deng, J., Zhao, L. & Wu, Y. Fast predictor-corrector approach for the tempered fractional differential equations. Numer Algor 74, 717–754 (2017). https://doi.org/10.1007/s11075-016-0169-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-016-0169-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation