| Citation: |
Yuanlu Wang, Jiantao Jiang, Jing An. SPECTRAL-GALERKIN APPROXIMATION BASED ON REDUCED ORDER SCHEME FOR FOURTH ORDER EQUATION AND ITS EIGENVALUE PROBLEM WITH SIMPLY SUPPORTED PLATE BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 61-83. doi: 10.11948/20230018
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SPECTRAL-GALERKIN APPROXIMATION BASED ON REDUCED ORDER SCHEME FOR FOURTH ORDER EQUATION AND ITS EIGENVALUE PROBLEM WITH SIMPLY SUPPORTED PLATE BOUNDARY CONDITIONS
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Abstract
We develop in this paper a high-order numerical method for fourth-order equation with simply supported plate boundary conditions in a circular domain. By introducing an auxiliary function and using the dimension reduction technique, we reduce the fourth-order problem to a one-dimensional second-order coupled problem. Based on the one-dimensional second-order coupled problem, we prove the uniqueness of the weak solution and approximation solutions and the error estimation between them. Moreover, we extend the approach to fourth-order eigenvalue problem with simply supported plate boundary conditions in a circular domain. Finally, we carry out some numerical experiments to validate the theoretical analysis and algorithm.
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References
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Yuanlu Wang, Jiantao Jiang, Jing An. SPECTRAL-GALERKIN APPROXIMATION BASED ON REDUCED ORDER SCHEME FOR FOURTH ORDER EQUATION AND ITS EIGENVALUE PROBLEM WITH SIMPLY SUPPORTED PLATE BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2024, 14(1): 61-83. doi: 10.11948/20230018
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Figure 1.
Comparison figures of exact solutions (left) and approximation solutions (right) with
$ N = 30 $ $ M = 15 $ -
Figure 2.
The error figures of exact solutions
$ \hat{w}(x,y) $ $ \hat{w}_{MN}(x,y) $ $ N = 30 $ $ M = 14 $ $ N=45 $ $ M = 25 $ -
Figure 3.
Comparison figures of exact solutions (left) and approximation solutions (right) with
$ N = 35 $ $ M = 15 $ -
Figure 4.
The error figures of exact solutions
$ \hat{u}(x,y) $ $ \hat{u}_{MN}(x,y) $ $ N = 30 $ $ M = 15 $ $ N=45 $ $ M = 25 $ -
Figure 5.
Error curves between the numerical solutions
$ \hat{w}_{MN}(x,y) $ $ \hat{u}_{MN}(x,y) $ $ N $ $ M = 12 $ -
Figure 6.
Comparison figures of exact solutions (left) and approximation solutions (right) with
$ N = 35 $ $ M = 15 $ -
Figure 7.
The error figures of exact solutions
$ \hat{w}(x,y) $ $ \hat{w}_{MN}(x,y) $ $ N = 30 $ $ M = 14 $ $ N=35 $ $ M = 15 $ -
Figure 8.
Comparison figures of exact solutions (left) and approximation solutions (right) with
$ N = 35 $ $ M = 15 $ -
Figure 9.
The error figures of exact solutions
$ \hat{u}(x,y) $ $ \hat{u}_{MN}(x,y) $ $ N = 30 $ $ M = 14 $ $ N=40 $ $ M = 25 $ -
Figure 10.
The error tendency curves between numerical solutions and reference solutions with
$ m=0 $ $ m=1 $ -
Figure 11.
Errors between the numerical solutions and the reference solutions on log-log scale with
$ m=0 $ $ m=1 $ -
Figure 12.
The error tendency curves between numerical solutions and reference solutions with
$ m=0 $ $ m=1 $