Numerical solution of a class of space fractional nonlinear vibration equations with periodic boundary conditions by the Fourier spectral method

Zhu, Zhixin; Han, Ce; Liu, Haitao; Cao, Li Yun; Yulan, Wang

doi:10.1177/14613484211038781

Journal of Low Frequency Noise, Vibration and Active Control

2021

DOI: 10.1177/14613484211038781

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Numerical solution of a class of space fractional nonlinear vibration equations with periodic boundary conditions by the Fourier spectral method

Zhixin Zhu¹,

Ce Han

Haitao Liu

et al.

Abstract: Nonlinear vibration arises everywhere in engineering. So far there is no method to track the exact trajectory of a space fractional nonlinear oscillator; therefore, a sophisticated numerical method is much needed to elucidate its basic properties. For this purpose, a numerical method that combines the Fourier spectral method with the Runge–Kutta method is proposed. Its accuracy and efficiency have been demonstrated numerically. This approach has full physical understanding and numerical access; thus, it can be… Show more

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Cited by 2 publications

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Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Luo

Zhong

2023

Applied Mathematics and Nonlinear Sciences

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The oscillation theory of differential equations is an important branch of performance of differential equations, which is widely used in engineering control, vibration mechanics, mechanics, and industry. Therefore, the vibration performance of different parts has attracted people’s attention, and a lot of research work has been done. For a special class of delay differential equations - advanced piecewise continuous differential equations, the oscillation of numerical solution is discussed. The θ − method is used to discretize the equation, and the numerical method is obtained to keep the oscillation of the analytical solution of the equation, progressive conditions. At the same time, four different states of the dynamic behavior are discussed in detail for the analytical solution and the numerical solution respectively. Some numerical examples further verify the corresponding conclusions.

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Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Luo

Zhong

2023

Applied Mathematics and Nonlinear Sciences

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Discrete variational method for partial differential equations of functionally gradient beam vibration

2024

JCM

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The field of engineering is becoming increasingly complex. In order to adapt to the numerical simulation of solving the partial differential equation of functionally graded beam vibration, a higher order stable numerical algorithm has been constructed. Differential quadrature method is used in discrete space domain. The discrete variational method is constructed in the time domain. The index differential Algebraic equation are obtained by combining the two methods. The discrete variational scheme is constructed for simulation. The results indicate that under long-term simulation, both the velocity and displacement constraints of the Runge Kutta method have defaulted. Displacement constraint values differ by 5 × 10 - 10. The velocity, displacement and acceleration constraints of the discrete variational method are stable. Compared with the Runge Kutta method, the constraint magnitude is reduced. The speed constraint is maintained at within 2.5 × 10 - 15. The displacement constraint level is maintained at within 1 × 10 - 16. This indicates that the discrete variational method has high accuracy and good stability when solving problems such as the vibration equation of functionally graded beams. When the step sizes are h= 0.1 m and h= 0.01 m, the accuracy of the discrete variational method is close. The larger the step size h, the higher the computational efficiency of the discrete variational method. The discrete variational method can maintain structural and energy conservation, making it suitable for long-term simulations. This has a good effect on solving complex problems in the field of partial differential equations.

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Numerical solution of a class of space fractional nonlinear vibration equations with periodic boundary conditions by the Fourier spectral method

Cited by 2 publications

References 34 publications

Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Oscillation and Asymptotic Behavior of a Special Delay Third Order Nonlinear Neutral Functional Differential Equation

Discrete variational method for partial differential equations of functionally gradient beam vibration

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