Numerical Solution of the Fractional Order Duffing–van der Pol Oscillator Equation by Using Bernoulli Wavelets Collocation Method
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.
KeywordsFractional order Duffing–van der Pol oscillator equation Collocation method Bernoulli wavelets Caputo derivative Numerical solution
Mathematics Subject Classification34A08 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.
- 1.Abd-Elhameed, W.M., Doha, E.H., Youssri, Y.H.: New spectral second kind Chebyshev wavelets algorithm for solving linear and nonlinear second order differential equations involving singular and Bratu type equations. Abstr. Appl. Anal. 2013 (2013)Google Scholar
- 3.Doha, E.H., Bhrawy, A.H., Ezz-Eldien, S.S.: An efficient Legendre spectral tau matrix formulation for solving fractional subdiffusion and reaction subdiffusion equations. J. Comput. Nonlinear Dyn. 10 (2015)Google Scholar
- 10.Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics. In: Carpinteri, A., Mainardi, F. (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York (1997)Google Scholar
- 15.Rahimkhani, P., Ordokhani, Y., Babolian, E.: Müntz–Legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numer. Algor. (2017). https://doi.org/10.1007/s11075-017-0363-4
- 21.Saeed, U., ur Rehman, M.: Haar wavelet operational matrix method for fractional oscillation equations. Int. J. Math. Math. Sci. 2014 (2014)Google Scholar
- 23.Sajadi, H., Ganji, D.D., Shenas, Y.V.: Application of numerical and semianalytical approach on van der Pol Duffing oscillators. J. Adv. Res. Mech. Eng. 1(3), 136–141 (2010)Google Scholar
- 27.Youssri, Y.H., Abd-Elhame, W.M., Doha, E.H.: Ultraspherical wavelets method for solving Lane–Emden type equations. Rom. J. Phys. 60, 1298–1314 (2015)Google Scholar