A Promising Method to Approximate Fractional Derivatives Under Uncertainty

  • Ali AhmadianEmail author
  • Norazak Senu
  • Fudziah Ismail
  • Soheil Salahshour
Conference paper


In this work, we apply a new and promising method base on tau method for solving a variety of differential equations of fractional order under fuzzy concept. We employ a linearization method to approximate the fractional derivative of the Caputo-type under uncertainty, then, we get to a fuzzy algebraic linear system and we solve it using any type of numerical technique to achieve the solution. The algorithm handles the problem in a direct manner without any need to restrictive assumptions. We emphasize the power of the method by applying it to an example.


Fractional differential equations Caputo derivative Tau method Fuzzy settings theory 


  1. 1.
    Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35, 5662–5672 (2011)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bhrawy, A.H., Alofi, A.S., Ezz-Eldien, S.S.: A quadrature tau method for fractional differential equations with variable coefficients. Appl. Math. Lett. 24, 2146–2152 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order Legendre functions for solving fractional- order differential equations. Appl. Math. Model. 37, 5498–5510 (2013)Google Scholar
  4. 4.
    Esmaeili, S., Shamsi, M.: A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 3646–3654 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A new Jacobi operational matrix: An application for solving fractional differential equations. Appl. Math. Model. 36, 4931–4943 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bede, B., Gal, S.G.: Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy differential equations. Fuzzy Sets Syst. 151, 581–599 (2005)CrossRefGoogle Scholar
  7. 7.
    Ahmadian, A., Suleiman, M., Salahshour, S., Baleanu, D.: A Jacobi operational matrix for solving fuzzy linear fractional differential equation. Adv. Differ. Equations 2013, 104 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math Appl. 62, 2364–2373 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Ali Ahmadian
    • 1
    Email author
  • Norazak Senu
    • 1
  • Fudziah Ismail
    • 1
  • Soheil Salahshour
    • 2
  1. 1.Institute for Mathematical Research, Universiti Putra MalaysiaSelangorMalaysia
  2. 2.Young Research and Elite Club, Mobarakeh Branch, Islamic Azad UniversityMobarakehIran

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