Author(s)

Maha Hamed^1*, Ibrahim L. El-Kalla², Mohamed A. El-Beltagy³, Beih S. El-desouky¹

¹Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt.
²Department of Engineering Mathematics & Physics, Faculty of Engineering, Mansoura University, Egypt.
³Department of Engineering Mathematics & Physics, Faculty of Engineering, Cairo University, Giza, Egypt.

In this paper, we introduce the study of the general form of stochastic Van der Pol equation (SVDP) under an external excitation described by Gaussian white noise. The study involves the use of Wiener-Chaos expansion technique (WCE) and Wiener-Hermite expansion (WHE) technique. The application of these techniques results in a system of deterministic differential equations (DDEs). The resulting DDEs are solved by the numerical techniques and compared with the results of Monte Carlo (MC) simulations. Also, we introduce a new formula that facilitates handling the cubic nonlinear term of van der Pol equations. The main results of this study are: 1) WCE technique is more accurate, programmable compared with WHE and for the same order, WCE consumes less time. 2) The number of Gaussian random variables (GRVs) is more effective than the order of expansion. 3) The agreement of the results with the MC simulations reflects the validity of the forms obtained through theorem 3.1.

Van der Pol Oscillator, Wiener-Chaos Expansion, Wiener-Hermite Expansion, Limit Cycle, White Noise, Damping Force

Hamed, M. , El-Kalla, I. , El-Beltagy, M. and El-desouky, B. (2020) Solution of Stochastic Van der Pol Equation Using Spectral Decomposition Techniques. Applied Mathematics, 11, 184-202. doi: 10.4236/am.2020.113016.