| Citation: |
Yi Gong. A HERMITE FINITE ELEMENT METHOD FOR THE VIBRATION PROBLEM OF THE RAYLEIGH-BISHOP BEAM[J]. Journal of Applied Analysis & Computation, 2025, 15(2): 1134-1145. doi: 10.11948/20240285
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A HERMITE FINITE ELEMENT METHOD FOR THE VIBRATION PROBLEM OF THE RAYLEIGH-BISHOP BEAM
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Abstract
In this paper, a Hermite finite element method is proposed for the Rayleigh-Bishop equation which describes the vibration problem of the Rayleigh-Bishop beam. We first present the semi-discrete Galerkin finite element form for the Rayleigh-Bishop equation. Then by means of the cubic Hermite element, a full-discrete finite element scheme is established. Furthermore, a numerical algorithm based on the Hermite finite element method is proposed to solve the fourth-order Rayleigh-Bishop equation. Finally, a numerical example is given to illustrate the effectiveness of the proposed method. The Hermite finite element method is potentially applied to other vibration problems.
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References
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Yi Gong. A HERMITE FINITE ELEMENT METHOD FOR THE VIBRATION PROBLEM OF THE RAYLEIGH-BISHOP BEAM[J]. Journal of Applied Analysis & Computation, 2025, 15(2): 1134-1145. doi: 10.11948/20240285
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Figure 1.
Schematic of a beam with fixed ends.
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Figure 2.
The exact solution with
$ h=0.01 $ $ \tau=0.0001 $ -
Figure 3.
The numerical solution and the error of the fully discrete scheme with
$ h=\frac{1}{10^2} $ $ \tau=\frac{1}{10^4} $ -
Figure 4.
The exact solution with
$ h=0.01 $ $ \tau=0.0001 $ -
Figure 5.
The numerical solution and the error of the fully discrete scheme with
$ h=\frac{1}{10^2} $ $ \tau=\frac{1}{10^4} $