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Research Papers

Green’s Function Iterative Approach for Solving Strongly Nonlinear Oscillators

[+] Author and Article Information
Marwan Abukhaled

Department of Mathematics and Statistics,
American University of Sharjah,
Sharjah, UAE
e-mail: mabukhaled@aus.edu

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 10, 2016; final manuscript received May 10, 2017; published online June 16, 2017. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 12(5), 051021 (Jun 16, 2017) (5 pages) Paper No: CND-16-1187; doi: 10.1115/1.4036813 History: Received April 10, 2016; Revised May 10, 2017

In this paper, a Green’s function based iterative algorithm is proposed to solve strong nonlinear oscillators. The method’s essential part is based on finding an appropriate Green’s function that will be incorporated into a linear integral operator. An application of fixed point iteration schemes such as Picard’s or Mann’s will generate an iterative formula that gives reliable approximations to the true periodic solutions that characterize these kinds of equations. The applicability and stability of the method will be tested through numerical examples. Since exact solutions to these equations usually do not exist, the proposed method will be tested against other popular numerical methods such as the modified homotopy perturbation, the modified differential transformation, and the fourth-order Runge–Kutta methods.

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Figures

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Fig. 1

Approximate displacements y(t) against time t for IVPs (23) and (24)

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Fig. 2

Approximate displacements y(t) against time t for IVPs (26) and (27)

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Fig. 3

Approximate displacements y(t) against time t for IVPs (29) and (30)

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Fig. 4

Absolute residuals for IVPs (29) and (30): (a) two iterations of GFIM and (b) ten iterations of MHPM [5]

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Fig. 5

Approximate displacements y(t) against time t for IVPs (29) and (30) with ε=0.5

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Fig. 6

Absolute residuals for Eqs. (32) and (33) with ε=0.5: (a) two iterations of GFIM and (b) ten iterations of MHPM [5]

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