On a strong minimum condition of a fractal variational principle

He, Ji‐Huan; Qie, Na; He, Chun‐Hui; Saeed, Tareq

doi:10.1016/j.aml.2021.107199

Cited by 67 publications

(42 citation statements)

References 29 publications

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“…String axial vibrations arise in various fields, e.g., suspension bridge, drill string, and ring spinning; the vibration property will significantly affect the safety and the life span, and it can be described by nonlinear differential equations with possible two-fractal derivatives, which are difficult to solve exactly. The scientists have made great efforts to provide information about the approximate solutions of these equations [1][2][3][4][5][6]. Of the nonlinear vibration systems that have been fascinating to engineers and scientists, by far the most perplexing is the frequency-amplitude relationship.…”

Section: Introductionmentioning

confidence: 99%

The Enhanced Homotopy Perturbation Method for Axial Vibration of Strings

El‐Dib

2021

FU Mech Eng

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103

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A governing equation is established for string axial vibrations with temporal and spatial damping forces by the Hamilton principle. It is an extension of the well-known Klein-Gordon equation. The classical homotopy perturbation method (HPM) fails to analyze this equation, and a modification with an exponential decay parameter is suggested. The analysis shows that the amplitude behaves as an exponential decay by the damping parameter. Furthermore, the frequency equation is established and the stability condition is performed. The modified homotopy perturbation method yields a more effective result for the nonlinear oscillators and helps to overcome the shortcoming of the classical approach. The comparison between the analytical solution and the numerical solution shows an excellent agreement.

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Section: Introductionmentioning

confidence: 99%

The Enhanced Homotopy Perturbation Method for Axial Vibration of Strings

El‐Dib

2021

FU Mech Eng

Self Cite

103

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“…In this work, the variational principle 35–38 is successfully developed by using the fractal semi‐inverse method, 39–44 which helps to analyze the structure of the solution and establish the conservation law. Based on the variational formula, we propose a new analytical technique to solve the fractal model, which perfectly combines the advantages of both the variational theory and the fractal two‐scale transform method 45–49 . The fractal two‐scale transform method is an excellent transform, which converts the fractal space into continuous space.…”

Section: Introductionmentioning

confidence: 99%

A study of the fractal foam drainage model in a microgravity space

Wang

2021

Math Methods in App Sciences

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The foam drainage mainly depends on the influence of gravity and foam surface tension. In this work, we consider the foam drainage in a microgravity condition, and its fractal modification is established by using the fractal derivative. The variational principle of the novel fractal model is successfully obtained by the semi‐inverse method. Finally, an analytical solution is obtained by the variational iteration method; the effect of the fractal order on the solution property is illustrated.

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“…The homotopy perturbation method [4][5][6][7][8][9][10][11]15,[24][25][26][27]43,44,47,55] constructs an effective iteration algorithm by guaranteeing its iteration path from the initial guess to the exact solution. The variational iteration method [29,32,35,51] adopts the variational theory [33,34] to find an optimal iteration formulation for differential equations. Taylor's series method [20,23] is simple, but only an infinite series solution is useful for analysis of a complex nonlinear problem.…”

Section: Introductionmentioning

confidence: 99%

New optimal fourth-order iterative method based on linear combination technique

Nadeem

Ali

2021

Hacettepe Journal of Mathematics and Statistics

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Newton's iteration method is widely used in numerical methods, but its convergence is low. Though a higher order iteration algorithm leads to a fast convergence, it is always complex. An optimal iteration formulation is much needed for both fast convergence and simple calculation. Here, we develop a two-step optimal fourth-order iterative method based on linear combination of two iterative schemes for nonlinear equations, and we explore the convergence criteria of the proposed method and also demonstrate its validity and efficiency by considering some test problems. We present both numerical as well as graphical comparisons. Further, the dynamical behavior of the proposed method is revealed.

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On a strong minimum condition of a fractal variational principle

Cited by 67 publications

References 29 publications

The Enhanced Homotopy Perturbation Method for Axial Vibration of Strings

The Enhanced Homotopy Perturbation Method for Axial Vibration of Strings

A study of the fractal foam drainage model in a microgravity space

New optimal fourth-order iterative method based on linear combination technique

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