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**A finite extensibility nonlinear oscillator.**
*(English)*
Zbl 1211.65090

Summary: The dynamics of a finite extensibility nonlinear oscillator (FENO) is studied analytically by means of two different approaches: a generalized decomposition method (GDM) and a linearized harmonic balance procedure (LHB). From both approaches, analytical approximations to the frequency of oscillation and periodic solutions are obtained, which are valid for a large range of amplitudes of oscillation. Within the generalized decomposition method, two different versions are presented, which provide different kinds of approximate analytical solutions.

In the first version, it is shown that the truncation of the perturbation solution up to the third order provides a remarkable degree of accuracy for almost the whole range of amplitudes. The second version, which expands the nonlinear term in Taylor’s series around the equilibrium point, exhibits a little lower degree of accuracy, but it supplies an infinite series as the approximate solution. On the other hand, a linearized harmonic balance method is also employed, and the comparison between the approximate period and the exact one (numerically calculated) is slightly better than that obtained by both versions of the GDM.

In general, the agreement between the results obtained by the three methods and the exact solution (numerically integrated) for amplitudes \((A)\) between \(0 < A \leqslant 0.9\) is very good both for the period and the amplitude of oscillation. For the rest of the amplitude range \((0.9 < A < 1)\), an exponentially large \(L_{2}\) error demonstrates that all three approximations do not represent a good description for the FENO, and higher order perturbation solutions are needed instead. As a complement, very accurate asymptotic representations of the period are provided for the whole range of amplitudes of oscillation.

In the first version, it is shown that the truncation of the perturbation solution up to the third order provides a remarkable degree of accuracy for almost the whole range of amplitudes. The second version, which expands the nonlinear term in Taylor’s series around the equilibrium point, exhibits a little lower degree of accuracy, but it supplies an infinite series as the approximate solution. On the other hand, a linearized harmonic balance method is also employed, and the comparison between the approximate period and the exact one (numerically calculated) is slightly better than that obtained by both versions of the GDM.

In general, the agreement between the results obtained by the three methods and the exact solution (numerically integrated) for amplitudes \((A)\) between \(0 < A \leqslant 0.9\) is very good both for the period and the amplitude of oscillation. For the rest of the amplitude range \((0.9 < A < 1)\), an exponentially large \(L_{2}\) error demonstrates that all three approximations do not represent a good description for the FENO, and higher order perturbation solutions are needed instead. As a complement, very accurate asymptotic representations of the period are provided for the whole range of amplitudes of oscillation.

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

### Keywords:

finite extensibility; nonlinear oscillator; numerical examples; error bounds; generalized decomposition method; linearized harmonic balance procedure; periodic solutions
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\textit{M. Febbo}, Appl. Math. Comput. 217, No. 14, 6464--6475 (2011; Zbl 1211.65090)

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