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

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran

    P. Rahimkhani

  2. Sama Technical and Vocational Training College, Islamic Azad University, Arak Branch, Arak, Iran

    R. Moeti

  3. Farhangian University, Tehran, Iran

    R. Moeti

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  1. P. Rahimkhani
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  2. R. Moeti
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Correspondence to P. Rahimkhani.

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Rahimkhani, P., Moeti, R. Numerical Solution of the Fractional Order Duffing–van der Pol Oscillator Equation by Using Bernoulli Wavelets Collocation Method. Int. J. Appl. Comput. Math 4, 59 (2018). https://doi.org/10.1007/s40819-018-0494-x

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  • DOI: https://doi.org/10.1007/s40819-018-0494-x

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