Mathematical Researches
پژوهش های ریاضی
mmr
Basic Sciences
http://mmr.khu.ac.ir
1
admin
2588-2546
2588-2554
10.61186/mmr
fa
jalali
1399
9
1
gregorian
2020
12
1
6
4
online
1
fulltext
fa
یک مسئلۀ جدید مقدار ویژۀ معکوس برای ماتریسهای ژاکوبی و سیستم جرم-فنر متناظر
A New Inverse Eigenvalue Problem for Jacobi Matrices and Corresponding Mass-Spring System
جبر
alg
مقاله مستقل
Original Manuscript
<span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">ارتعاشات سیستمهای مختلف مانند جرم-فنر، نخ کشسان، میله و غیره بهصورت مسئلۀ مقدار ویژۀ ماتریس ژاکوبی مدلبندی میشوند. </span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">مسئلۀ تعیین ماتریس ژاکوبی با استفاده از دادههای طیفی معلوم، مسئلۀ مقدار ویژۀ معکوس ماتریس ژاکوبی گفته میشود. در این مقاله، ماتریس ژاکوبی</span></span><span dir="LTR" style="position:relative;top:3.0pt;"><span style="font-size:10.0pt;"> <img alt="" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.wmz" > </span></span><span dir="LTR"><span style="font-size:10.0pt;"></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> با استفاده از دو طیف و یک دادۀ اضافی</span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> بازسازی میشود. یکی از طیفها مقادیر ویژۀ ماتریس</span></span><span dir="LTR" style="position:relative;top:3.0pt;"><span style="font-size:10.0pt;"> <img alt="" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.wmz" > </span></span><span dir="LTR"><span style="font-size:10.0pt;"></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> و طیف دیگر مقادیر ویژۀ زیرماتریس حاصل از حذف همزمان دو سطر و ستون ماتریس</span></span><span dir="LTR" style="position:relative;top:3.0pt;"><span style="font-size:10.0pt;"> <img alt="" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.wmz" > </span></span><span dir="LTR"><span style="font-size:10.0pt;"></span></span> <span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">است. شرایط لازم و کافی روی دادههای طیفی را برای حلپذیری مسئلۀ معکوس ارائه کرده و الگوریتمهایی برای تعیین ماتریس ژاکوبی</span></span><span dir="LTR" style="position:relative;top:3.0pt;"><span style="font-size:10.0pt;"> <img alt="" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.wmz" > </span></span><span dir="LTR"><span style="font-size:10.0pt;"></span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> ارائه میگردد. سرانجام با ارائۀ چند مثال عددی ماتریس ژاکوبی و سیستم جرم- فنر متناظر بازسازی میشوند</span></span><strong><span style="font-family:B Nazanin;"><span style="font-size:14.0pt;">.</span></span></strong><br>
<br>
<strong>Introduction</strong><br>
Many problems in sciences and engineering can be studied by mathematical models. These models are classified as direct problems and inverse problems. In the structural vibrations, analysis and estimation of the behavior of system e.g. response of the system for an external force and natural frequencies from the known physical parameters is called a direct problem. Determination or estimation of the system physical parameters such as density, mass, stiffness and cross sectional area from the behavior of the system is called an inverse problem. A class of inverse problems which physical parameters determined from the spectral data (eigenvalues, eigenvectors, or both) is called inverse eigenvalue problem. There are many systems such as mass- spring system, vibrating Rods and Beams which are modeled as an eigenvalue problem. Free vibrations of a mass- spring system and discretization of a rod and Sturm-Liouville equations lead to Jacobi matrix eigenvalue problem. Inverse eigenvalue problem for Jacobi matrix is determination of entries using some spectral data. Different algorithms have been presented for constructing a Jacobi matrix. In this paper, we construct a Jacobi matrix and the corresponding mass-spring system using some new spectral data.<br>
<strong>Material and methods</strong><br>
We try to construct a Jacobi matrix from two spectra and one extra data. For this purpose, using given spectral data, we find the required data of well-known Lancsoz method. Then applying Lancsoz method, we construct a positive definite Jacobi matrix. Finally, according to the relations between Jacobi matrix, mass and stiffness matrices, we obtain corresponding mass-spring system.<br>
<strong>Results and discussion</strong><br>
Necessary and sufficient conditions on given spectral data for solvability of the inverse eigenvalue problem are presented.<br>
We find two algorithms for constructing positive definite Jacobi matrix and the corresponding mass-spring system.<br>
We solve some examples using the given algorithms. There is a good agreement between the spectral data of constructed matrix and initial given data.<br>
<strong>Conclusion</strong><br>
The following results are obtained from this research.
<ul>
<li>We find two algorithms for constructing a Jacobi matrix using two spectra and one extra data.</li>
<li>It is observed that, for a set of spectral data, there might be exist more than one solution.</li>
<li>It seems that, one may extend the method of this paper for matrix eigenvalue problem which arise in discretization of vibrating rod using finite element method.<a href="./files/site1/files/64/16.pdf">./files/site1/files/64/16.pdf</a></li>
</ul>
واژههای کلیدی: مسئلۀ مقدار ویژۀ معکوس, ماتریس ژاکوبی, دادههای طیفی, سیستم جرم-فنر.
Inverse eigenvalue problem, Jacobi matrix, Spectral data, Mass-Spring system.
671
686
http://mmr.khu.ac.ir/browse.php?a_code=A-10-704-1&slc_lang=fa&sid=1
Hanif
Mirzaei
حنیف
میرزائی
h_mirzaei@sut.ac.ir
10031947532846004205
10031947532846004205
Yes
Sahand University of Technology
دانشگاه صنعتی سهند، دانشکدۀ علوم پایه