Numerical analysis of fractional partial differential equations applied to polymeric visco-elastic Euler-Bernoulli beam under quasi-static loads

Wang, Lei; Chen, Yiming; Cheng, Gang; Barrière, Thierry

doi:10.1016/j.chaos.2020.110255

Cited by 14 publications

(5 citation statements)

References 36 publications

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“…Take the partial derivative of Eq. 12 with respect to , yields ∑ ∑ (14) The relationship between the coefficients of the two expansions and are given by the following recurrence relation ∑ ∑…”

Section: Derivative Of With Respect Tomentioning

confidence: 99%

“…In [18,7], the computation of approximate solution for nonlinear singular initial value problems was considered using Hermite wavelets operational matrix of integration and Chebyshev wavelets operational matrix of integration respectively while in [9], the optimal control problems was solved with the aid of Legendre mother wavelets operational matrix of integration. A general expression for the operational matrix of integration of Bessel functions was derived in [16] to solve some examples in optimal control, for more works, see [8,14,15,16,17,18,19,22,24,25,26,27,28,29,30,31,32,33,34,35,36,37].…”

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confidence: 99%

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Some Results on a Two Variables Pell Polynomials

Sarhan

Shihab

Rasheed

2021

Al-Qadisiyah Journal of Pure Science

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In this study, Pell polynomials in two variables, and their properties are investigated. Some formulas for two variables Pell polynomials are derived by matrices. By defining special formula for Pell polynomials in one variable, new important properties of Pell polynomials in two variables can be enabled to derive. A new exact formula expressing the partial derivatives of Pell polynomials explicitly of any degree in terms of Pell polynomials themselves is proved. A novel explicit formula, which constructs the two explicit formulas, which construct the two-dimension Pell polynomials expansion coefficients of a first partial derivative of a differentiable function in terms of their original expansion coefficients, is also included in the present article. The main advantage of the presented formulas is that the new properties of Pell polynomials in two variables greatly simplify the original problems and the result will lead to easy calculation of the coefficients of expansion. A direct spectral method along with the presented two variables Pell polynomials is proposed to solve the partial differential equation. Illustrated examples are included to demonstrate the validity of the technique.

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Section: Derivative Of With Respect Tomentioning

confidence: 99%

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confidence: 99%

Some Results on a Two Variables Pell Polynomials

Sarhan

Shihab

Rasheed

2021

Al-Qadisiyah Journal of Pure Science

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“…For more details on the FC, one can see [17,23,24]. In fact, many researchers have devoted themselves to investigate fractional order differential equations and systems with different boundary conditions, for more details, the reader can see the works [1,3,4,6,9,10,14,20,22,29,[37][38][39]41].…”

Section: Introductionmentioning

confidence: 99%

Beam deflection coupled systems of fractional differential equations: existence of solutions, Ulam–Hyers stability and travelling waves

Bensassa,

Dahmani,

Rakah

et al. 2024

Anal.Math.Phys.

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In this paper, we study a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo fractional derivatives. Under flexible/fixed end-conditions, two main theorems on the existence and uniqueness of solutions are proved by using two fixed point theorems. Some examples are discussed to illustrate the applications of the existence and uniqueness of solution results. Another main result on the Ulam–Hyers stability of solutions for the introduced system is also discussed. Some examples of stability are discussed. New travelling wave solutions are obtained for another conformable coupled system of beam type that has a connection with the first considered system. A conclusion follows at the end.

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“…It captures the nonlinear and fractional order properties of the materials more accurately, while maintaining high numerical stability and reducing the accumulation of numerical errors. Wang and Dang et al 30,31,32,33 used the shifted Chebyshev polynomials algorithm to solve the control equations of arches, plates and beams of structural mechanics directly in the time domain and to simplify the solution process. Qu et al 34 used shifted Chebyshev wavelets to approximate the deflection function of the governing equations of a viscoelastic axially moving plate and provided the numerical solution of the governing equations.…”

Section: Introductionmentioning

confidence: 99%

Dynamic vibration analysis of cantilever beams on nonlinear fractional viscoelastic foundation under Gaussian white noise

Qu,

Cui,

Cui

et al. 2024

Preprint

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This paper presents the dynamic response of cantilever beam on fractional-order nonlinear viscoelastic foundation subjected to Gaussian white noise. The control equations of the cantilever beam on viscoelastic foundation are established using Hamilton’s principle. The problem is solved using the shifted Chebyshev polynomial algorithm, and the control equations are transformed into a system of nonlinear algebraic equations. Numerical examples analyse the correction error and the second norm error, confirming the effectiveness and accuracy of the algorithm in solving such problems. Furthermore, the algorithm’s robustness was verified by comparing the responses of Gaussian white noise and non-Gaussian white noise cantilever beam on the viscoelastic foundation. The study examined the impact of various loads and parameters on the cantilever beam, as well as the effect of different harmonic loads on its stress. The research results are in line with the existing literature. These studies offer valuable guidance for practical engineering and enhance comprehension of the dynamic response of cantilevers on foundation in complex environments.

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Numerical analysis of fractional partial differential equations applied to polymeric visco-elastic Euler-Bernoulli beam under quasi-static loads

Cited by 14 publications

References 36 publications

Some Results on a Two Variables Pell Polynomials

Some Results on a Two Variables Pell Polynomials

Beam deflection coupled systems of fractional differential equations: existence of solutions, Ulam–Hyers stability and travelling waves

Dynamic vibration analysis of cantilever beams on nonlinear fractional viscoelastic foundation under Gaussian white noise

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