Numerical approach for solving variable-order space–time fractional telegraph equation using transcendental Bernstein series

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


This paper presents the transcendental Bernstein series (TBS) as a generalization of the classical Bernstein polynomials for solving the variable-order space–time fractional telegraph equation (V-STFTE). An approximation method using optimization techniques and the TBS is introduced. The solution of the problem under consideration is expanded in terms of TBS with unknown free coefficients and control parameters. The new corresponding operational matrices of variable-order fractional derivatives, in the Caputo type, are derived. The proposed approach reduces the V-STFTE to a system of algebraic equations and, subsequently, to find the free coefficients and control parameters using the Lagrange multipliers technique. The convergence analysis of the method is guranteed by means of a new theorem concerning the TBS. The experimental results confirm the high accuracy and computational efficiency of the TBS method.


Variable-order space–time fractional telegraph equation Transcendental Bernstein series Optimization method Operational matrix Control parameters 

Mathematics Subject Classification

35R11 41A58 35B30 



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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  • H. Hassani
    • 1
    Email author
  • Z. Avazzadeh
    • 2
  • J. A. Tenreiro Machado
    • 3
  1. 1.Department of Applied Mathematics, Faculty of Mathematical SciencesShahrekord UniversityShahrekordIran
  2. 2.School of Mathematical SciencesNanjing Normal UniversityNanjingChina
  3. 3.Department of Electrical Engineering, Institute of EngineeringPolytechnic of PortoPorto 431Portugal

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