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

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

Abstract

In this paper, an effective numerical algorithm based on shifted Chebyshev polynomials is proposed to solve the fractional partial differential equations applied to polymeric visco-elastic problems in the time-space domain under quasi-static loads. The governing equations using local fractional rheological models based on visco-elastic properties with fractional derivatives are established. The integer and fractional differential operator matrices of polynomials are derived according to the properties of shifted Chebyshev polynomials. The fractional order governing equation is rewritten into the form of matrix product by using the polynomial to approximate the unknown function. The collocation method is used to discretize the variables and transform the original problem into an algebraic equation system. The numerical solutions of the governing equations are obtained directly in the time-domain. In addition, an error analysis including the correction method is performed. The numerical examples have been performed to identify the sensibility of the proposed governing equations and to evaluate the efficiency and accuracy of the proposed algorithm.

Suggested Citation

  • Wang, Lei & Chen, Yiming & Cheng, Gang & Barrière, Thierry, 2020. "Numerical analysis of fractional partial differential equations applied to polymeric visco-elastic Euler-Bernoulli beam under quasi-static loads," Chaos, Solitons & Fractals, Elsevier, vol. 140(C).
    • Handle: RePEc:eee:chsofr:v:140:y:2020:i:c:s0960077920306512
    DOI: 10.1016/j.chaos.2020.110255
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    References listed on IDEAS

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    1. Meng, Zhijun & Yi, Mingxu & Huang, Jun & Song, Lei, 2018. "Numerical solutions of nonlinear fractional differential equations by alternative Legendre polynomials," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 454-464.
    2. Xie, Jiaquan & Yao, Zhibin & Gui, Hailian & Zhao, Fuqiang & Li, Dongyang, 2018. "A two-dimensional Chebyshev wavelets approach for solving the Fokker-Planck equations of time and space fractional derivatives type with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 197-208.
    3. Yu, Chunxiao & Zhang, Jie & Chen, Yiming & Feng, Yujing & Yang, Aimin, 2019. "A numerical method for solving fractional-order viscoelastic Euler–Bernoulli beams," Chaos, Solitons & Fractals, Elsevier, vol. 128(C), pages 275-279.
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    Cited by:

    1. Li, Yiqun & Wang, Hong, 2023. "A finite element approximation to a viscoelastic Euler–Bernoulli beam with internal damping," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 212(C), pages 138-158.
    2. Cui, Yuhuan & Qu, Jingguo & Han, Cundi & Cheng, Gang & Zhang, Wei & Chen, Yiming, 2022. "Shifted Bernstein–Legendre polynomial collocation algorithm for numerical analysis of viscoelastic Euler–Bernoulli beam with variable order fractional model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 200(C), pages 361-376.

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