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Mirzaei, Hanif, Ghanbari, Kazem, Abbasnavaz, Vahid, Mingarelli, Angelo. (1403). Direct and inverse problems of ROD equation using finite element method and a correction technique. سامانه مدیریت نشریات علمی, 12(4), 651-668. doi: 10.22034/cmde.2024.57676.2417
Hanif Mirzaei; Kazem Ghanbari; Vahid Abbasnavaz; Angelo Mingarelli. "Direct and inverse problems of ROD equation using finite element method and a correction technique". سامانه مدیریت نشریات علمی, 12, 4, 1403, 651-668. doi: 10.22034/cmde.2024.57676.2417
Mirzaei, Hanif, Ghanbari, Kazem, Abbasnavaz, Vahid, Mingarelli, Angelo. (1403). 'Direct and inverse problems of ROD equation using finite element method and a correction technique', سامانه مدیریت نشریات علمی, 12(4), pp. 651-668. doi: 10.22034/cmde.2024.57676.2417
Mirzaei, Hanif, Ghanbari, Kazem, Abbasnavaz, Vahid, Mingarelli, Angelo. Direct and inverse problems of ROD equation using finite element method and a correction technique. سامانه مدیریت نشریات علمی, 1403; 12(4): 651-668. doi: 10.22034/cmde.2024.57676.2417

Direct and inverse problems of ROD equation using finite element method and a correction technique

Computational Methods for Differential Equations
مقاله 2، دوره 12، شماره 4، دی 2024، صفحه 651-668    اصل مقاله (625.48 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2024.57676.2417
نویسندگان
Hanif Mirzaei1؛ Kazem Ghanbari* 1؛ Vahid Abbasnavaz1؛ Angelo Mingarelli2
1Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.
2School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada.
چکیده
The free vibrations of a rod are governed by a differential equation of the form $(a(x)y^\prime)^\prime+\lambda a(x)y(x)=0$, where $a(x)$ is the cross sectional area and $\lambda$ is an eigenvalue parameter. Using the finite element method (FEM) we transform this equation to a generalized matrix eigenvalue problem of the form $(K-\Lambda M)u=0$ and, for given $a(x)$, we correct the eigenvalues $\Lambda$ of the matrix pair $(K,M)$ to approximate the eigenvalues of the rod equation. The results show that with step size $h$ the correction technique reduces the error from $O(h^2i^4)$ to $O(h^2i^2)$ for the $i$-th eigenvalue. We then solve the inverse spectral problem by imposing numerical algorithms that approximate the unknown coefficient $a(x)$ from the given spectral data. The cross section is obtained by solving a nonlinear system using Newton's method along with a regularization technique. Finally, we give numerical examples to illustrate the efficiency of the proposed algorithms.
کلیدواژه‌ها
Rod equation؛ Eigenvalue؛ Finite element method؛ Direct problem؛ Inverse problem؛ Sturm-Liouville
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