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

Mirzaei, Hanif; Ghanbari, Kazem; Abbasnavaz, Vahid; Mingarelli, Angelo

doi:10.22034/cmde.2024.57676.2417

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