A Uniform Accuracy High-Order Finite Difference and FEM for Optimal Problem Governed by Time-Fractional Diffusion Equation

Cao, Junying; Wang, Zhongqing; Wang, Ziqiang

doi:10.3390/fractalfract6090475

Cited by 3 publications

(2 citation statements)

References 47 publications

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“…Yang et al [22] established a nonlinear fractional order model of viscoelastic microbeam by a new numerical algorithm, which simplifies the calculation. Cao et al [23] solved the time fractional diffusion equation optimal control problem by FEM and analyzed the stability and truncation error of the discrete scheme. In [24][25][26], several numerical theories for solving time fractional fourth-order partial differential equations based on mixed finite element method (MFEM) and mixed finite volume element method (MFVM) were presented and analyzed.…”

Section: Introductionmentioning

confidence: 99%

“…In [24][25][26], several numerical theories for solving time fractional fourth-order partial differential equations based on mixed finite element method (MFEM) and mixed finite volume element method (MFVM) were presented and analyzed. In [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], numerical examples are given and parameterization studies are carried out. The numerical results show that the fractional derivative order, viscoelastic coefficient and various damping coefficient have effects on the vibration of fractional viscoelastic beams under different boundary conditions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem

Sun,

Zhu,

Yin

et al. 2023

Fractal Fract

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In this paper, the vibration problem of a beam with a time fractional damping term is studied by the Hermite finite element method, and its fully discrete scheme is obtained. The stability and error estimation of the scheme are analyzed, and it was proved that it is unconditionally stable and has a convergence order of O(τ+τ3−α+h4). The validity of the scheme is verified by numerical examples, the effects of fractional derivative order and damping coefficient on beam vibration are analyzed and the superiority of the fractional order model has been demonstrated by comparing with the traditional damping model.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem

Sun,

Zhu,

Yin

et al. 2023

Fractal Fract

View full text Add to dashboard Cite

show abstract

Solving and Numerical Simulations of Fractional-Order Governing Equation for Micro-Beams

Yang

Zhang

et al. 2023

Fractal Fract

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This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the governing equations, the operator matrices of shifted Chebyshev polynomials are derived first. Shifted Chebyshev polynomials are used to approximate the deflection function, and the nonlinear fractional order governing equation is expressed in the form of operator matrices. Next, the collocation method is used to discretize the equations into the form of algebraic equations for solution. It simplifies the calculation. MATLAB software was used to program this algorithm to simulate the numerical solution of the deflection. The effectiveness and accuracy of the algorithm are verified by the numerical example. Finally, numerical simulations are carried out on the viscoelastic micro-beams. It is found that the viscous damping coefficient is inversely proportional to the deflection, and the length scale parameter of the micro-beam is also inversely proportional to the deflection. In addition, the stress and strain of micro-beam, the change of deflection under different simple harmonic loads, and potential energy of micro-beam are discussed. The results of the study fully demonstrated that the shifted Chebyshev polynomial algorithm is effective for the numerical simulations of viscoelastic micro-beams.

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A higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel

Wang,

Ma,

Cao

2024

MATH

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<abstract><p>In this paper, we proposed a higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel by using the modified block-by-block method. First, we constructed a high order uniform accuracy scheme method in this paper by dividing the entire domain into some small sub-domains and approximating the integration function with biquadratic interpolation in each sub-domain. Second, we rigorously proved that the convergence order of the higher order uniform accuracy scheme was $ O(h_{s}^{3+\sigma_{1} }+h_{t}^{3+\sigma_{2} }) $ with $ 0 < \sigma_{1}, \sigma_{2} < 1 $ by using the discrete Gronwall inequality. Finally, two numerical examples were used to illustrate experimental results with different values of $ \psi $ to support the theoretical results.</p></abstract>

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A Uniform Accuracy High-Order Finite Difference and FEM for Optimal Problem Governed by Time-Fractional Diffusion Equation

Cited by 3 publications

References 47 publications

Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem

Hermite Finite Element Method for Time Fractional Order Damping Beam Vibration Problem

Solving and Numerical Simulations of Fractional-Order Governing Equation for Micro-Beams

A higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel

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