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In this paper, we first develop a novel power series method (PSM) by introducing the characteristic length R 0 into the power term (t/R 0 ) k to solve nonlinear oscillatory problems. We demonstrate that a suitable value of R 0 being selected can avoid the numerical instability due to a huge value times a tiny value appeared in the conventional power series method with t k as an expansion term. Then, we use a sequential method to continue the series solution to a large time span, in order to investigate the chaotic behavior and the Poincaré sections of Duffing oscillator. We also use the modified sequential power series method (MSPSM) to construct the frequency response curves which exhibit hysteresis loops, and within which the multiple solutions occur in a sub-interval of exciting frequency.

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Liu (2001) derived the first augmented Lie-group S O o (n, 1) symmetry for the nonlinear ordinary differential equations (ODEs):ẋ = f(x, t), and developed the corresponding group-preserving scheme (GPS). However, the earlier formulation did not consider the rotational effect of nonlinear ODEs. In this paper, we derive the second augmented Lie-group S O o (n, 1) symmetry by taking the rotational effect into account. The numerical algorithm exhibits two solutions of the Lie-group G ∈ S O o (n, 1), depending on the sign of f 2 x 2 − 2(f · x) 2 , which means that the algorithm may be switched between two states, depending on x. We give numerical examples to assess the new algorithm GPS2, which upon comparing with the GPS can raise the accuracy about three orders. It is interesting that for the chaotic system the signum function sign( f 2 x 2 − 2(f · x) 2 ) is frequently switched between +1 and −1 in time.

When adjoint eigenfunctions are adopted as the test functions in Green's second identity for the Euler‐Bernoulli beam equation, we can develop a quite simple noniterative numerical algorithm to recover an unknown space‐dependent external force H(x) exerted on the beam. The spatial parts of the adjoint eigenfunctions are used as the bases to expand the unknown function H(x), where we view the two end values of H(x) as two unknown coefficients for the simply supported and hinged‐clamped beams, and the left end value of H(x) as an unknown coefficient for the cantilevered beam. We can derive closed‐form solutions of the expansion coefficients, and thus closed‐form series solutions of H(x). Consequently, we have a noniterative method to recover the unknown force H(x) supplemented by the noisy final time displacement data. Numerical examples demonstrate the accuracy, efficiency, and robustness of the novel methods in the recovery of unknown forces on the simply supported, cantilevered, and hinged‐clamped beams.