Skip to Main Content
Frequently Asked Questions
Submit an ETD
Global Search Box
Need Help?
Keyword Search
Participating Institutions
Advanced Search
School Logo
Files
File List
thapa final.pdf (398.68 KB)
ETD Abstract Container
Abstract Header
On the Homotopy Perturbation Method for Nonlinear Oscillators
Author Info
Thapa, Chandra B
Permalink:
http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1570793395580841
Abstract Details
Year and Degree
2019, Master of Arts (MA), Bowling Green State University, Mathematics.
Abstract
We study a family of nonlinear oscillators governed by the equation (d
2
x)/(dt
2
)= -βx
p
, 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
i
x
i
(t) and the expression 1 = ω
2
− λα
1
− λ
2
α
2
− ... (3) into 1, the coefficient of x(t) on the left side of (2), we obtain the first few ODE’s for x
i
(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 x
i
(t), α
i
, 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].
Committee
So-Hsiang Chou, Ph.D. (Advisor)
Tong Sun, Ph.D (Committee Member)
Pages
58 p.
Subject Headings
Applied Mathematics
Keywords
Homotopy
;
Nonlinear Oscillators
;
Frequency
;
Trajectory
Recommended Citations
Refworks
EndNote
RIS
Mendeley
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)
Abstract Footer
Document number:
bgsu1570793395580841
Download Count:
341
Copyright Info
© 2019, all rights reserved.
This open access ETD is published by Bowling Green State University and OhioLINK.