An Adaptive Collocation Method for Solving Delay Fractional Differential Equations

  • M. MahmoudiEmail author
  • M. Ghovatmand
  • H. Jafari
Original Paper


In this article, an adaptive collocation method is investigated for solving delay fractional differential equations (DFDEs). The fractional derivative is described in the Caputo–Fabrizio sense, that is a new fractional derivative with non-singular kernel. This new definition has more advantages over the definition of Caputo fractional derivative that we consider in our numerical method. Our technique is based upon an adaptive pseudospectral method. First, we divide the interval of the problem into a uniform mesh and consider the Legendre polynomials on each subinterval then using the Chebysheve collocation points the given DFDE reduces to a system of algebraic equations. One of the reasons for using the adaptive methods is their superiority in solving the problem containing delay terms. The technique is simple to implement and yields precise results. The error approximation and convergence properties of the method are discussed. The proposed method in this investigation is easy and effective for solving DFDEs and can provide an accuracy approximate solution.


Caputo–Fabrizio fractional derivative Delay fractional differential equations Legendre–Gauss pseudospectral method System of algebraic equations 



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© Springer Nature India Private Limited 2019

Authors and Affiliations

  1. 1.Department of MathematicsShahrood University of TechnologyShahroodIran
  2. 2.Department of Mathematical SciencesUniversity of South AfricaPretoriaSouth Africa

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