Skip to main content
SpringerLink
Log in

On representation and interpretation of Fractional calculus and fractional order systems

  • Research Paper
  • Published:
Fractional Calculus and Applied Analysis Aims and scope Submit manuscript

Abstract

In this work a relationship between Fractional calculus (FC) and the solution of a first order partial differential equation (FOPDE) is suggested. With this relationship and considering an extra dimension, an alternative representation for fractional derivatives and integrals is proposed. This representation can be applied to fractional derivatives and integrals defined by convolution integrals of the Volterra type, i.e. the Riemann-Liouville and Caputo fractional derivatives and integrals, and the Riesz and Feller potentials, and allows to transform fractional order systems in FOPDE that only contains integer-order derivatives. As a consequence of considering the extra dimension, the geometric interpretation of fractional derivatives and integrals naturally emerges as the area under the curve of a characteristic trajectory and as the direction of a tangent characteristic vector, respectively. Besides this, a new physical interpretation is suggested for the fractional derivatives, integrals and dynamical systems.

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. A. S. Balankin, J. Bory-Reyes, M. Shapiro, Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric. Physica A. 444 (2016), 345–359; DOI: 10.1016/j.physa.2015.10.035.

    Article  MathSciNet  Google Scholar 

  2. R. Courant, D. Hilbert, Methods of Mathematical Physics, Volume II. Wiley, USA (1962).

    MATH  Google Scholar 

  3. K. Diethelm, N. Ford, Analysis of fractional differential equations. J. of Math. Anal. Appl. 265, No 2 (2002), 229–248; DOI: 10.1006/jmaa.2000.7194.

    Article  MathSciNet  Google Scholar 

  4. M. Duff, M theory (The theory formely known as strings). Int. J. Mod. Phys. A. 11, No 32 (1996), 5623–5641; DOI: 10.1142/s0217751x96002583.

    Article  Google Scholar 

  5. H. A. Fallahgoul, S. M. Focardi, F. J. Fabozzi, Fractional Calculus and Fractional Processes with Applications to Financial Economics. Elsevier, USA (2017).

    MATH  Google Scholar 

  6. S. Hadid, J. Alshamani, Lyapunov stability of differential equations of non-integer order. Arab. J. Math. 7/, No 1–2 (1986), 5–17.

    MathSciNet  MATH  Google Scholar 

  7. C. Ionescu, A. Lopes, D. Copot, J. Machado, J. Bates, The role of fractional calculus in modeling biological phenomena: A review. Commun. Nonlinear Sci. 51 (2017), 141–159; DOI: 10.1016/j.cnsns.2017.04.001.

    Article  MathSciNet  Google Scholar 

  8. F. Izsák, B. J. Szekeres, Models of space-fractional diffusion: A critical review. Appl. Math. Lett. 71 (2017), 38–43; DOI: 10.1016/j.aml.2017.03.006.

    Article  MathSciNet  Google Scholar 

  9. M. Lighthill, G. Whitham, On kinematic waves I. Flood movement in long rivers. P. Roy. Soc. A-Math. Phys. 229, No 1178 (1955), 281–316; DOI: 10.1098/rspa.1955.0088.

    MathSciNet  MATH  Google Scholar 

  10. M. J. Lighthill, G. B. Whitham, On kinematic waves II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. A - Math. Phys. 229, No 1178 (1955), 317–345; DOI: 10.1098/rspa.1955.0089.

    MathSciNet  MATH  Google Scholar 

  11. J. A. T. Machado, A probabilistic interpretation of the fractional-order differentiation. Fract. Calc. Appl. Anal. 6, No 1 (2003), 73–80.

    MathSciNet  MATH  Google Scholar 

  12. J. A. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus. Commun. Nonlinear Sci./ 16, No 3 (2011), 1140–1153; DOI: 10.1016/j.cnsns.2010.05.027.

    Article  MathSciNet  Google Scholar 

  13. I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5, No 4 (2002), 367–386; available at: https://arxiv.org/pdf/math/0110241.pdf.

    MathSciNet  MATH  Google Scholar 

  14. P. Shah, S. Agashe, Review of fractional PID controller. Mechatronics 38/(2016), 29–41; DOI: 10.1016/j.mechatronics.2016.06.005.

    Article  Google Scholar 

  15. L. Susskind, The world as a hologram. J. Math. Phys. 36, No 11 (1995), 6377–6396; DOI: 10.1063/1.531249.

    Article  MathSciNet  Google Scholar 

  16. V. Uchaikin, R. Sibatov, Fractional derivatives on cosmic scales. Chaos Soliton Fract. 102/ (2017), 197–209; DOI: 10.1016/j.chaos.2017.04.023.

    Article  MathSciNet  Google Scholar 

  17. T. Vistarini, Holographic space and time: Emergent in what sense? Stud. Hist. Philos. M. P. 59/ (2016), 126–135; DOI: 10.1016/j.shpsb.2016.07.002.

    Article  MathSciNet  Google Scholar 

  18. Y. Zhang, H. Sun, H. H. Stowell, M. Zayernouri, S.E. Hansen, A review of applications of fractional calculus in earth system dynamics. Chaos Soliton Fract. 102 (2017), 29–46; DOI: 10.1016/j.chaos.2017.03.051.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Ingeniería Química, Universidad de Guadalajara, Blvd. M. García Barragán 1451 Guadalajara, 44300, Jalisco, México

    Juan Paulo García-Sandoval

Authors
  1. Juan Paulo García-Sandoval
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Paulo García-Sandoval.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

García-Sandoval, J.P. On representation and interpretation of Fractional calculus and fractional order systems. FCAA 22, 522–537 (2019). https://doi.org/10.1515/fca-2019-0031

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1515/fca-2019-0031

MSC 2010

Key Words and Phrases

Search

Navigation