Stable fractional Chebyshev differentiation matrix for the numerical solution of multi-order fractional differential equations

Arman Dabiri, Eric Butcher

Research output: Contribution to journalArticlepeer-review

48 Scopus citations


This paper presents an algorithm to obtain numerically stable differentiation matrices for approximating the left- and right-sided Caputo-fractional derivatives. The proposed differentiation matrices named fractional Chebyshev differentiation matrices are obtained using stable recurrence relations at the Chebyshev–Gauss–Lobatto points. These stable recurrence relations overcome previous limitations of the conventional methods such as the size of fractional differentiation matrices due to the exponential growth of round-off errors. Fractional Chebyshev collocation method as a framework for solving fractional differential equations with multi-order Caputo derivatives is also presented. The numerical stability of spectral methods for linear fractional-order differential equations (FDEs) is studied by using the proposed framework. Furthermore, the proposed fractional Chebyshev differentiation matrices obtain the fractional-order derivative of a function with spectral convergence. Therefore, they can be used in various spectral collocation methods to solve a system of linear or nonlinear multi-ordered FDEs. To illustrate the true advantages of the proposed fractional Chebyshev differentiation matrices, the numerical solutions of a linear FDE with a highly oscillatory solution, a stiff nonlinear FDE, and a fractional chaotic system are given. In the first, second, and forth examples, a comparison is made with the solution obtained by the proposed method and the one obtained by the Adams–Bashforth–Moulton method. It is shown the proposed fractional differentiation matrices are highly efficient in solving all the aforementioned examples.

Original languageEnglish (US)
Pages (from-to)185-201
Number of pages17
JournalNonlinear Dynamics
Issue number1
StatePublished - Oct 1 2017


  • Caputo derivative
  • Chaos
  • Chebyshev polynomials
  • Fractional chaotic system
  • Fractional differential equations
  • Highly oscillatory
  • Nonlinear differential equation
  • Numerical solution
  • Numerical stability
  • Partial differential equation
  • Spectral collocation method

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics
  • Electrical and Electronic Engineering


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