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On the Homotopy Perturbation Method for Nonlinear Oscillators

Thapa, Chandra B

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2019, Master of Arts (MA), Bowling Green State University, Mathematics.
We study a family of nonlinear oscillators governed by the equation (d2x)/(dt2)= -βxp, β > 0 (1) Here β is a constant and p=1,3,5,... is necessarily odd to be an oscillator (the phase portrait needs to be closed and bounded). The initial condition is x(0) = A, x'(0) = 0. Our goal is to find the frequency of the oscillator and its trajectory for each p. We apply the Lindstedt-Poincaré method to a convex homotopy with the parameter λ: x''(t) + x(t) = λ[x(t) − βx(t)p], 0 ≤ λ ≤ 1. (2) Substituting the expansion x(t) = ∑i=0 λ ixi(t) and the expression 1 = ω2 − λα1 − λ2α2 − ... (3) into 1, the coefficient of x(t) on the left side of (2), we obtain the first few ODE’s for xi(t) such as Eq’s (4.3.4)-(4.3.6) below after equating the coefficients of the powers of λ. The final target solution x(t) and the frequency ω are obtained through a sequence of xi(t), αi, i = 1, 2, ... and setting λ to 1. It is necessary to kill the secular terms to have oscillatory behavior in the model. To achieve this goal, we study about the homotopy perturbation method for nonlinear oscillators which have been demonstrated and discussed in chapter 2. Then we apply He’s homotopy per-turbation method for conservative truly nonlinear oscillators and find their improved approximate solutions [9]. By this approach, we can find a truly periodic solution and the period of the motion as a function of the amplitude of oscillation. We take p = 3 in (1), so-called cubic oscillator and find that this approach works very well for the whole range of parameters and obtain excellent result match of frequencies between approximate and an exact one [9].
So-Hsiang Chou, Ph.D. (Advisor)
Tong Sun, Ph.D (Committee Member)
58 p.

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Citations

  • Thapa, C. B. (2019). On the Homotopy Perturbation Method for Nonlinear Oscillators [Master's thesis, Bowling Green State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1570793395580841

    APA Style (7th edition)

  • Thapa, Chandra. On the Homotopy Perturbation Method for Nonlinear Oscillators. 2019. Bowling Green State University, Master's thesis. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1570793395580841.

    MLA Style (8th edition)

  • Thapa, Chandra. "On the Homotopy Perturbation Method for Nonlinear Oscillators." Master's thesis, Bowling Green State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1570793395580841

    Chicago Manual of Style (17th edition)