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Direct and inverse problems of rod equation using finite element method and a correction technique | ||
| Computational Methods for Differential Equations | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 25 فروردین 1403 اصل مقاله (1.35 M) | ||
| 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؛ finite element method؛ direct problem؛ inverse problem؛ Sturm-Liouville | ||
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آمار تعداد مشاهده مقاله: 32 تعداد دریافت فایل اصل مقاله: 50 |
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