Numerical Solution of the Fractional Order Duffing–van der Pol Oscillator Equation by Using Bernoulli Wavelets Collocation Method

  • P. Rahimkhani
  • R. Moeti
Original Paper


In the present article, we propose a new numerical method to solve the fractional order Duffing–van der Pol oscillator equation. The proposed method is based on using collocation points and approximating the solution employing the Bernoulli wavelets. The Riemann–Liouville fractional integral operator for Bernoulli wavelets is introduced. This operator is then utilized to reduce the solution of the fractional order Duffing–van der Pol oscillator equation to a system of algebraic equations. In order to demonstrate the accuracy and efficiency of the proposed scheme, we have considered three problems namely: fractional order force-free Duffing–van der Pol oscillator, forced Duffing–van der Pol oscillator and higher order fractional Duffing equations.


Fractional order Duffing–van der Pol oscillator equation Collocation method Bernoulli wavelets Caputo derivative Numerical solution 

Mathematics Subject Classification

34A08 34K28 65L05 



Authors are very grateful to one of the reviewers for carefully reading the paper and for his(her) comments and suggestions which have improved the paper.


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Copyright information

© Springer (India) Private Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesAlzahra UniversityTehranIran
  2. 2.Sama Technical and Vocational Training CollegeIslamic Azad University, Arak BranchArakIran
  3. 3.Farhangian UniversityTehranIran

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