A remark on Samuelson’s variational principle in economics

The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators

“…Nonlinear oscillations arise everywhere in our everyday life and engineering. As an exact solution might be too complex to be used for a practical application, many analytical methods have been used in open literature, for example, the variational iteration method, [1][2][3][4][5][6][7] the homotopy perturbation method, [8][9][10][11][12][13][14][15][16][17][18][19][20] He-Laplace method, [21][22][23] the variational approach [24][25][26][27][28][29] and the Hamiltonian approach. 30,31 The most important property of a nonlinear oscillator is the relationship between the frequency and its amplitude, the simplest method to estimate the frequency-amplitude relationship might be He's frequency formulation [32][33][34] and the max-min approach, 35,36 which are still under development and many modifications were proposed to improve the accuracy.…”

He’s frequency–amplitude formulation with average residuals for nonlinear oscillators

“…For large values of a 2 or a 3 , the relative errors of the approximate frequencies of equations (17), (18), (20), and (21) are, respectively, as follows lim x Exact À x Present j j x Exact ¼ 0:0018 (25) Comparison of the approximate frequencies (x Classic and x Present ) obtained by equations (18) and (21) with the exact frequencies x Exact is made in Table 1 for different values of a 1 , a 2 , and a 3 , where the exact frequencies x Exact are determined by equation (37) in the Appendix. All present solutions of x Present are in good agreement with the exact ones.…”

“…In general, it is very difficult to solve nonlinear problems accurately. After decades of continuous efforts, some excellent methods appeared in open literature, such as the homotopy perturbation methods, [1][2][3][4][5][6][7][8][9][10][11] the variational iteration method, [12][13][14][15][16][17] the parameter expansion method, 18 the max-min approach, 19 the variational approach, [20][21][22][23] the Hamiltonian approach, 24 the energy balance method, 25 the stability analysis, [26][27][28][29] and the simple formulae for fast prediction of frequency. 30 To solve nonlinear oscillators, Chinese mathematician Prof. Ji-Huan He proposed a simple but effect frequency-amplitude formulation.…”

Active controller design for precision computerized numerical control machine tool systems

“…The motor control card receives commands from the computer to drive the slides and move the X-Y platform. A standard first-order transfer function was chosen for the architecture of the controlled object, as shown in equation (1). Since the feedback data is position, the architecture of the controlled object is changed to use voltage as input and angular position as output so that the system model is more like an actual system.…”