Skip to main content

Application to Ordinary Fractional Differential Equations

  • Chapter
  • First Online:
Book cover Numerical Methods for Fractional Differentiation

Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 54))

  • 1035 Accesses

Abstract

The numerical approximation of classical ordinary differential equations is relatively simple and, being a focus of mathematical studies for the last few decades, has been by now almost completely investigated. However, a fractional case is much less studied and is still poorly understood despite the fact that there has been a growing interest in the research of this area. In short, there has been just handful of research papers and books considering the numerical approximation of time-fractional ordinary differential equations.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. J. Al-Omari, S.A. Gourley, Monotone travelling fronts in an age-structured reaction-diffusion model of a single species. J. Math. Biol. 45, 294–312 (2002)

    Article  MathSciNet  Google Scholar 

  2. A. Atangana, R.T. Alqahtani, Modelling the spread of river blindness disease via the Caputo fractional derivative and the beta-derivative. Entropy 18, 40 (2018). https://doi.org/10.3390/e18020040

    Article  Google Scholar 

  3. A. Atangana, E.F. Doungmo Goufo, On the mathematical analysis of Ebola hemorrhagic fever: deathly infection disease in West African countries. BioMed Res. Int. Article ID 261383, 7 pages (2014). https://doi.org/10.1155/2014/261383

    Google Scholar 

  4. A. Atangana, A novel model for the lassa hemorrhagic fever: deathly disease for pregnant women. Neural Comput. Appl. 26, 1895–1903 (2015)

    Article  Google Scholar 

  5. A. Atangana, Derivative with a New Parameter: Theory, Methods and Applications (Academic Press, New York, 2016)

    Book  Google Scholar 

  6. A. Atangana, Fractional Operators With Constant and Variable Order with Application to Geo-Hydrology (Academic Press, New York, 2017)

    MATH  Google Scholar 

  7. J. Cao, C. Xu, A high order schema for the numerical solution of the fractional ordinary differential equations. J Comput. Phys. 238, 154–168 (2013)

    Article  MathSciNet  Google Scholar 

  8. M. Caputo, Linear models of dissipation whose \(Q\) is almost frequency independent II. Geophys. J. R. Astron. Soc. 13, 529–539 (1967)

    Article  Google Scholar 

  9. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1, 73–85 (2015)

    Google Scholar 

  10. G. Gao, H. Sun, Three-point combined compact alternating direction implicit difference schemes for two-dimensional time-fractional advection-diffusion equations. Commun. Comput. Phys. 17, 487–509 (2017)

    Article  MathSciNet  Google Scholar 

  11. T. Hoenen, A. Groseth, D. Falzarano, H. Feldmann, Ebola virus: unravelling pathogenesis to combat a deadly disease. Trends Mol. Med. 12, 206–215 (2006)

    Article  Google Scholar 

  12. I. Koca, A. Atangana, Analysis of a nonlinear model of interpersonal relationships with time fractional derivative. J. Math. Anal. 7, 1–11 (2016)

    MathSciNet  MATH  Google Scholar 

  13. J.H. Kuhn, S. Becker, H. Ebihara et al., Proposal for a revised taxonomy of the family Filoviridae: classification, names of taxa and viruses, and virus abbreviations. Arch. Virol. 155, 2083–2103 (2010)

    Article  Google Scholar 

  14. P. Kumar, O.P. Agrawal, An approximate method for numerical solution of fractional differential equations. Signal Process 86, 2602–2610 (2006). Special Section: Fractional Calculus Applications in Signals and Systems

    Google Scholar 

  15. T.A.M. Langlands, B.I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)

    Article  MathSciNet  Google Scholar 

  16. E.K. Leffel, D.S. Reed, Marburg and Ebola viruses as aerosol threats. Biosecurity Bioterrorism Biodefense Strat. Pract. Sci. 2, 186–191 (2004)

    Article  Google Scholar 

  17. E. Leroy, J.P. Gonzalez, X. Pourrut, Ebolavirus and other filoviruses. Curr. Top. Microbiol. Immunol. 315, 363–387 (2007)

    Google Scholar 

  18. Y. Lin, C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  Google Scholar 

  19. J. Liu, Z. Ma, Z. Zhou, Explicit and implicit TVD schemes for conservation laws with Caputo derivatives. J. Sci. Comput. 72, 291–313 (2017)

    Article  MathSciNet  Google Scholar 

  20. C. Lv, C. Xu, Improved error estimates of a finite difference/spectral method for time-fractional diffusion equations. Int. J. Numer. Anal. Model. 12, 384–400 (2015)

    MathSciNet  MATH  Google Scholar 

  21. J.B. McCormick, P.A. Webb, J.W. Krebs, K.M. Johnson, E.S. Smith, A prospective study of the epidemiology and ecology of Lassa fever. J. Infect. Dis. 155, 437 (1987)

    Article  Google Scholar 

  22. P.R. Murray, Medical Microbiology, 7th edn. (Elsevier Saunders, Philadelphia, PA, USA, 2013)

    Google Scholar 

  23. N. Ozalp, I. Koca, A fractional order nonlinear dynamical model of interpersonal relationships. Adv. Differ. Equ. 189, 7 pages (2012). https://doi.org/10.1186/1687-1847-2012-189

  24. J.R.C. Piqueira, V.O. Araujo, A modified epidemiological model for computer viruses. Appl. Math. Comput. 2, 355–360 (2018)

    MathSciNet  MATH  Google Scholar 

  25. S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, Amsterdam, 1993)

    MATH  Google Scholar 

  26. S.H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (Addison-Wesley, Reading, MA, 1994)

    MATH  Google Scholar 

  27. Z. Sun, X. Wu, A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    Article  MathSciNet  Google Scholar 

  28. B. Thylefors, M.M. Alleman, N.A. Twum-Danso, Operational lessons from 20 years of the Mectizan Donation Program for the control of onchocerciasis. Trop. Med. Int. Health 13, 689–696 (2008)

    Article  Google Scholar 

  29. C. Wang, J. Liu, Positivity property of second-order flux-splitting schemes for the compressible euler equations. Discret. Contin. Dyn. Syst. Ser. B 3, 201–228 (2003)

    Article  MathSciNet  Google Scholar 

  30. X. Zhao, Z. Sun, G.E. Karniadakis, Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys. 293, 184–200 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Owolabi, K.M., Atangana, A. (2019). Application to Ordinary Fractional Differential Equations. In: Numerical Methods for Fractional Differentiation. Springer Series in Computational Mathematics, vol 54. Springer, Singapore. https://doi.org/10.1007/978-981-15-0098-5_7

Download citation

Publish with us

Policies and ethics