Skip to main content
SpringerLink
Log in

Variable-order fractional derivative under Hadamard’s finite-part integral: Leibniz-type rule and its applications

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In this paper, we introduce a new definition of variable-order fractional derivative via the Caputo fractional derivative in the sense of Hadamard’s finite-part integral. Then, we give a Leibniz-type rule of the defined variable-order fractional derivative and present a generalized chain rule of the composite function, which are able to recover the corresponding rules of Caputo fractional derivative. Two applications of the rules including an analytical expression of the variable-order fractional derivative of the hyperbolic tangent function and some numerical solutions of the linear and nonlinear variable-order fractional partial differential equations are performed. Preliminary numerical results show that the proposed rule is promising.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Data availability

The codes written to run the simulations presented in this paper are available upon request to the authors.

References

  1. Leibniz, G.W.: Mathematische Schiften. Georg Olms Verlagsbuch-handlung, Hildesheim (1962)

    Google Scholar 

  2. Podlubny, I.: Fractional Differential Equations. Academic Press, California (1999)

    MATH  Google Scholar 

  3. Guo, B.L., Pu, X.K., Huang, F.H.: Fractional Partial Differential Equations and Their Numerical Solutions. World Scientific Publishing Co. Pte. Ltd, Singapore (2015)

    Book  Google Scholar 

  4. Metzler, R., Jeon, J.-H., Cherstvy, A.G., Barkai, E.: Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16, 24128–24164 (2014)

    Article  Google Scholar 

  5. Teodoro, G.S., Machado, J.A.T., de Oliveira, E.C.: A review of definitions of fractional derivatives and other operators. J. Comput. Phys. 388, 195–208 (2019)

    Article  MathSciNet  Google Scholar 

  6. Samko, S.G., Ross, B.: Integration and differentiation to a variable fractional order. Integr. Transf. Spec. Funct. 1(4), 277–300 (1993)

    Article  MathSciNet  Google Scholar 

  7. Lorenzo, C.F., Hartley, T.T.: Variable order and distributed order fractional operators. Nonlin. Dyn. 29, 57–98 (2002)

    Article  MathSciNet  Google Scholar 

  8. Li, Z., Wang, H., Xiao, R., Yang, S.: A variable-order fractional differential equation model of shape memory polymers. Chaos Solitons Fract. 102, 473–485 (2017)

    Article  MathSciNet  Google Scholar 

  9. Coimbra, C.F.M.: Mechanics with variable-order differential operators. Ann. Phys. 12, 692–703 (2003)

    Article  MathSciNet  Google Scholar 

  10. Patnaik, S., Hollkamp, J.P., Semperlotti, F.: Applications of variable-order fractional operators: a review. Proc. R. Soc. A 476, 20190498 (2020)

    Article  MathSciNet  Google Scholar 

  11. Pedro, H.T.C., Kobayashi, M.H., Coimbra, C.F.M.: Variable order modelling of diffusive–convective effects on the oscillatory flow past a sphere. J. Vib. Control 14, 1659–1672 (2008)

    Article  MathSciNet  Google Scholar 

  12. Moghaddam, B.P., Dabiri, A., Machado, J.A.T.: Application of variable-order fractional calculus in solid mechanics. In: Bǎleanu, D., Lopes, A.M. (eds.) Volume 7 Applications in Engineering, Life and Social Sciences, Part A, pp. 207–224. De Gruyter, Berlin (2019)

    Chapter  Google Scholar 

  13. Moghaddam, B.P., Machado, J.A.T.: Time analysis of forced variable-order fractional Van der Pol oscillator. Eur. Phys. J. Spec. Top. 226, 3803–3810 (2017)

    Article  Google Scholar 

  14. Tavares, D., Almeida, R., Torres, D.F.M.: Caputo derivatives of fractional variable order: numerical approximations. Commun. Nonlinear Sci. Numer. Simulat. 35, 69–87 (2016)

    Article  MathSciNet  Google Scholar 

  15. Herrmann, R.: Fractional Calculus: An Introduction for Physicists. World Scientific (2014)

  16. Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press 1923 (Dover Publications 1952)

  17. Elliott, D.: An asymptotic analysis of two algorithms for certain Hadamard finite-part integrals. IMA J. Numer. Anal. 13, 445–462 (1993)

    Article  MathSciNet  Google Scholar 

  18. Ioakimidis, N.I.: Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity. Acta Mech. 45, 31–47 (1982)

    Article  MathSciNet  Google Scholar 

  19. Macaskjll, C., Tuck, E.O.: Evaluation of the acoustic impedance of a screen. J. Aust. Math. Soc. Ser. B 20, 46–61 (1977)

    Article  MathSciNet  Google Scholar 

  20. Oldham, K.B., Spanier, J.: The Fractional Calculus. Mathematics in Science and Engineering. Academic Press (1974)

  21. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  22. Osler, T.J.: Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. Math. 18, 658–674 (1970)

    Article  MathSciNet  Google Scholar 

  23. Sayevand, K., Machado, J.T., Baleanu, D.: A new glance on the Leibniz rule for fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 62, 244–249 (2018)

    Article  MathSciNet  Google Scholar 

  24. Tarasov, V.E.: No violation of the Leibniz rule. No fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 18(11), 2945–2948 (2013)

    Article  MathSciNet  Google Scholar 

  25. Ortigueira, M.D., Machado, J.A.T.: What is a fractional derivative? J. Comput. Phys. 293, 4–13 (2015)

    Article  MathSciNet  Google Scholar 

  26. Leo, R.A., Sicuro, G., Tempesta, P.: A foundational approach to the Lie theory for fractional order partial differential equations. Fract. Calc. Appl. Anal. 20(1), 212–231 (2017)

    Article  MathSciNet  Google Scholar 

  27. Zhang, Z.Y.: Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation. Proc. R. Soc. A 476, 20190564 (2020)

    Article  MathSciNet  Google Scholar 

  28. Zhang, Z.Y., Zheng, J.: Symmetry structure of multi-dimensional time-fractional partial differential equations. Nonlinearity 34(8), 5186–5212 (2021)

    Article  MathSciNet  Google Scholar 

  29. Zhang, Z.Y., Lin, Z.X.: Local symmetry structure and potential symmetries of time-fractional partial differential equations. Stud. Appl. Math. 147(1), 363–389 (2021)

    Article  MathSciNet  Google Scholar 

  30. Osler, T.J.: The fractional derivative of a composite function. SIAM J. Math. Anal. 1, 288–293 (1970)

    Article  MathSciNet  Google Scholar 

  31. Almeida, R., Tavares, D., Torres, D.F.M.: The Variable-Order Fractional Calculus of Variations. Springer International Publishing AG, Cham (2019)

    Book  Google Scholar 

  32. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974)

    MATH  Google Scholar 

  33. Moghaddam, B.P., Babaei, A., Machado, J.A.T.: Highly accurate scheme for the Cauchy problem of the generalized Burgers–Huxley equation. Acta. Polytech. Hung. 13(6), 183–195 (2016)

    Google Scholar 

Download references

Funding

The paper is supported by the Beijing Municipal Natural Science Foundation (No. 1222014) and the National Natural Science Foundation of China (No. 11671014), the Yu Xiu Talent for North China University of Technology (207051360020XN140003), Fundamental Research Funds for Beijing Universities (No. KM201910009001) and the Cross Research Project for Minzu University of China.

Author information

Authors and Affiliations

  1. College of Science, Minzu University of China, Beijing, 100081, People’s Republic of China

    Zhi-Yong Zhang & Zhi-Xiang Lin

  2. College of Science, North China University of Technology, Beijing, 100144, People’s Republic of China

    Lei-Lei Guo

Authors
  1. Zhi-Yong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi-Xiang Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lei-Lei Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi-Yong Zhang.

Ethics declarations

Conflict of interest

The authors have not disclosed any conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, ZY., Lin, ZX. & Guo, LL. Variable-order fractional derivative under Hadamard’s finite-part integral: Leibniz-type rule and its applications. Nonlinear Dyn 108, 1641–1653 (2022). https://doi.org/10.1007/s11071-022-07281-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07281-1

Keywords

Search

Navigation