An effective numerical method for solving nonlinear variable-order fractional functional boundary value problems through optimization technique

  • H. HassaniEmail author
  • J. A. Tenreiro Machado
  • Z. Avazzadeh
Original paper


An optimization method standing on a new basis formed by the transcendental Bernstein series (TBS) is proposed. The TBS includes the unknown free coefficients and control parameters for solving nonlinear variable-order fractional functional boundary value problems (NV-FFBVP). The corresponding operational matrices for variable-order fractional derivatives are derived for expanding the solution by means of TBS. The TBS is a generalization of the Bernstein polynomials (BP) and represent a superset of BP. In the particular cases where all the control parameters are zero, the TBS method is equivalent to the BP method. The proposed technique reduces the NV-FFBVP to a system of algebraic equations and, subsequently, to the problem of finding the free coefficients and control parameters using an optimization technique. Several numerical results reveal the computational performance and reliability of the method.


Nonlinear variable-order fractional functional boundary value problems Transcendental Bernstein series Variable-order fractional operational matrix Optimization method Control parameters 


Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  • H. Hassani
    • 1
    Email author
  • J. A. Tenreiro Machado
    • 2
  • Z. Avazzadeh
    • 3
  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesShahrekord UniversityShahrekordIran
  2. 2.Department of Electrical EngineeringInstitute of Engineering, Polytechnic of PortoPortoPortugal
  3. 3.School of Mathematical SciencesNanjing Normal UniversityNanjingChina

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