Lagrange crisis and generalized variational principle for 3D unsteady flow

He, Ji‐Huan

doi:10.1108/hff-07-2019-0577

Cited by 155 publications

(105 citation statements)

References 34 publications

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“…This paper, for the first time ever, applies the Elzaki transform to the variational iteration algorithm with great success, the identification of Lagrange multiplier, which was identified by the variational theory, becomes simplier. 34,35 An optimal variational iteration algorithm is obtained by the Elzaki transform, and the iteration algorithm converges fast and only one iteration results in a high accurate solution.…”

Section: Resultsmentioning

confidence: 99%

Numerical iteration for nonlinear oscillators by Elzaki transform

Anjum

Suleman

et al. 2019

Journal of Low Frequency Noise, Vibration and Active Control

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Iteration methods are widely used in numerical simulation. This paper suggests the Elzaki transform in the variational iteration method for simple identification of the Lagrange multiplier. The Elzaki transform is a modification of the Laplace transform, and it is extremely useful for treating with nonlinear oscillators as illustrated in this paper, a single iteration leads to a high accuracy of the solution.

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Section: Resultsmentioning

confidence: 99%

Numerical iteration for nonlinear oscillators by Elzaki transform

Anjum

Suleman

et al. 2019

Journal of Low Frequency Noise, Vibration and Active Control

Self Cite

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“…Based on the response time series u i , the single NNMs given in Table 1 are extracted by the OMA method based on GP. Generally, these natural frequencies are too close to tell apart, but the estimated responses are searched for by the OMA method based on GP, as shown in equations (13) to (15). In addition, in Figure 5, the estimated curves fit the reference curves well, and the magnitude of the absolute error is less than 1e À3 except for the initial segment as a shock.…”

Section: Numerical Resultsmentioning

confidence: 97%

“…Additionally, the response time series are obtained by GP on Eureqa. The function expressions of the response time series are given in equations (13) to (15), and the natural frequency of a single NNM can be easily identified using the coefficients in the function structure. Three criteria, including the absolute error and phase figure, are selected for comparison of the performance of the proposed nonlinear OMA method, and the results are listed in Table 2.…”

Section: Comparisonmentioning

confidence: 99%

Operational modal analysis using symbolic regression for a nonlinear vibration system

Jiang

2020

Journal of Low Frequency Noise, Vibration and Active Control

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The existence of nonlinearity is an inevitable frequent occurrence that should be considered to accurately identify the modal parameters of a vibration system using operational modal analysis. A problem is that the traditional operational modal analysis method based on the linear modal theory is not applicable to modal parameter identification of vibration systems with nonlinearity. A solution is as follows: this paper is aimed at solving the problem by proposing a new operational modal analysis method to carry out modal parameter identification for a nonlinear vibration system. The new operational modal analysis method, based on the forced response and symbolic regression method without assuming any pre-existing information and only using mathematical symbols, is introduced to solve the problem by automatically searching for the expression structure and modal parameters of a system in nonlinear normal modes. The simulation result of a three-degrees-of-freedom nonlinear system reveals the high accuracy of the proposed operational modal analysis method in extracting the modal parameters. Then, a rod fastening rotor model is considered, and the capability of the proposed operational modal analysis method to precisely extract its modal parameters is further evaluated.

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“…The values of λ 1 (ζ) and λ 2 (ζ) may be obtained most positively by the variational principle [24,25]. We obtain the estimation of λ 1 (ζ) and λ 2 (ζ), which is λ 1 (ζ) = λ 2 (ζ) = −1.…”

Section: Test Problemmentioning

confidence: 99%

Numerical Solutions of Coupled Burgers’ Equations

Ahmad

Khan

Cesarano

2019

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In this article, two new modified variational iteration algorithms are investigated for the numerical solution of coupled Burgers' equations. These modifications are made with the help of auxiliary parameters to speed up the convergence rate of the series solutions. Three numerical test problems are given to judge the behavior of the modified algorithms, and error norms are used to evaluate the accuracy of the method. Numerical simulations are carried out for different values of parameters. The results are also compared with the existing methods in the literature. Arora [5] proposed a scheme known as the Lai cubic B-spline collocation scheme for the solution of coupled viscous Burger equations, where the authors used a crank Nicholson scheme and cubic B-spline functions for time integration and space integration, respectively. Lai and Ma [6] proposed a new lattice Boltzmann model for the solution of coupled Burger equations, after selecting the distribution functions, Chapman-Enskog expansion was employed for the solution of coupled Burger equations. The Chebyshev-Legendre pseudospectral method [7] has been utilized by Rashid et al. for coupled viscous Burgers (VB) equations, where the leapfrog scheme and Chebyshev-Legendre Pseudo-Spectral method (CLPS) method were used for the time direction and space direction, respectively. Kumar and Pandit [8] used a composite numerical scheme for the numerical evaluation of coupled Burger equations, where the scheme was developed based on Haar wavelets and finite difference. Mohammadi and Mokhtari [9] used a reproducing Kernal method for an analytical solution in the form series of systems of Burger equations. At last, in 2019, Bak et al.[10] developed a new approach named a semi-Lagrangian approach for the numerical solution of coupled Burger equations. We compare our results with those of [10], and show the applicability of our proposed method to handle such problems as those that arise in applied science and engineering.This paper aims to solve three types of coupled Burgers equations by employing variational iteration algorithm-II and one of the examples to be solved by modified variational iteration algorithm-I. The organization of the rest of the paper is as follows; in Section 2, we elucidate the variational iteration algorithm-II, in the next Section 3, the semi-numerical method is applied to three test problems, and a comparison is made with some other methods; lastly, some conclusions are drawn in the last Section 4. Modified Variational Iteration Algorithm-IIConsider the nonlinear diffusion equation:where the terms L[w(x, t)] and N[w(x, t)] are linear and nonlinear terms in that order, while c(x, t) is the source term. For a given w 0 (x, t), the approximate solution w n+1 (x, t) of Equation (4) can be obtained as below:w n+1 (x, t) = w n (x, t) + t 0

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Lagrange crisis and generalized variational principle for 3D unsteady flow

Cited by 155 publications

References 34 publications

Numerical iteration for nonlinear oscillators by Elzaki transform

Numerical iteration for nonlinear oscillators by Elzaki transform

Operational modal analysis using symbolic regression for a nonlinear vibration system

Numerical Solutions of Coupled Burgers’ Equations

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