{"title":"构建振动棒的逆矩阵特征值问题","authors":"Hanif Mirzaei, Vahid Abbasnavaz, Kazem Ghanbari","doi":"10.1515/cmam-2024-0001","DOIUrl":null,"url":null,"abstract":"The free longitudinal vibrations of a rod are described by a differential equation of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>P</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi>y</m:mi> <m:mo>′</m:mo> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>′</m:mo> </m:msup> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mi>P</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi>y</m:mi> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0062.png\"/> <jats:tex-math>{(P(x)y\\prime)^{\\prime}+\\lambda P(x)y(x)=0}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>P</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0397.png\"/> <jats:tex-math>{P(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the cross section area at point <jats:italic>x</jats:italic> and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mrow> <m:mi>𝐀</m:mi> <m:mo></m:mo> <m:mi>Y</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Λ</m:mi> <m:mo></m:mo> <m:mi>𝐁</m:mi> <m:mo></m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0437.png\"/> <jats:tex-math>{\\mathbf{A}Y=\\Lambda\\mathbf{B}Y}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐀</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0439.png\"/> <jats:tex-math>{\\mathbf{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝐁</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0440.png\"/> <jats:tex-math>{\\mathbf{B}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are Jacobi and diagonal matrices dependent to cross section <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>P</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0397.png\"/> <jats:tex-math>{P(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>P</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0397.png\"/> <jats:tex-math>{P(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>O</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msup> <m:mi>h</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_cmam-2024-0001_eq_0391.png\"/> <jats:tex-math>{O(h^{2})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"69 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod\",\"authors\":\"Hanif Mirzaei, Vahid Abbasnavaz, Kazem Ghanbari\",\"doi\":\"10.1515/cmam-2024-0001\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The free longitudinal vibrations of a rod are described by a differential equation of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>P</m:mi> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mi>y</m:mi> <m:mo>′</m:mo> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mo>′</m:mo> </m:msup> <m:mo>+</m:mo> <m:mi>λ</m:mi> <m:mi>P</m:mi> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mi>y</m:mi> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2024-0001_eq_0062.png\\\"/> <jats:tex-math>{(P(x)y\\\\prime)^{\\\\prime}+\\\\lambda P(x)y(x)=0}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>P</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2024-0001_eq_0397.png\\\"/> <jats:tex-math>{P(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the cross section area at point <jats:italic>x</jats:italic> and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mrow> <m:mi>𝐀</m:mi> <m:mo></m:mo> <m:mi>Y</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mi mathvariant=\\\"normal\\\">Λ</m:mi> <m:mo></m:mo> <m:mi>𝐁</m:mi> <m:mo></m:mo> <m:mi>Y</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2024-0001_eq_0437.png\\\"/> <jats:tex-math>{\\\\mathbf{A}Y=\\\\Lambda\\\\mathbf{B}Y}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>𝐀</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2024-0001_eq_0439.png\\\"/> <jats:tex-math>{\\\\mathbf{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mi>𝐁</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2024-0001_eq_0440.png\\\"/> <jats:tex-math>{\\\\mathbf{B}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are Jacobi and diagonal matrices dependent to cross section <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>P</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2024-0001_eq_0397.png\\\"/> <jats:tex-math>{P(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>P</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2024-0001_eq_0397.png\\\"/> <jats:tex-math>{P(x)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:mi>O</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:msup> <m:mi>h</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_cmam-2024-0001_eq_0391.png\\\"/> <jats:tex-math>{O(h^{2})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":48751,\"journal\":{\"name\":\"Computational Methods in Applied Mathematics\",\"volume\":\"69 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-08-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Computational Methods in Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/cmam-2024-0001\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Methods in Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/cmam-2024-0001","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
杆的自由纵向振动由一个微分方程描述,其形式为 ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 {(P(x)y\prime)^{\prime}+\lambda P(x)y(x)=0} 。 其中,P ( x ) {P(x)} 是 x 点的横截面积,λ 是特征值参数。本文首先用有限差分法对该微分方程进行离散化,得到一个矩阵特征值问题,其形式为 𝐀 Y = Λ 𝐁 Y {\mathbf{A}Y=\Lambda\mathbf{B}Y} 、其中,𝐀 {\mathbf{A}} 和 𝐁 {\mathbf{B}} 分别是与横截面 P ( x ) {P(x)} 相关的雅可比矩阵和对角矩阵。然后,我们通过修正所得矩阵特征值问题的特征值来估计杆方程的特征值。我们给出了一种基于修正思想的方法,通过求解逆矩阵特征值问题来构建横截面 P ( x ) {P(x)}。我们给出了一些数值示例来说明所提方法的效率。结果表明,该方法的收敛阶数为 O ( h 2 ) {O(h^{2})} 。
An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod
The free longitudinal vibrations of a rod are described by a differential equation of the form (P(x)y′)′+λP(x)y(x)=0{(P(x)y\prime)^{\prime}+\lambda P(x)y(x)=0}, where P(x){P(x)} is the cross section area at point x and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form 𝐀Y=Λ𝐁Y{\mathbf{A}Y=\Lambda\mathbf{B}Y}, where 𝐀{\mathbf{A}} and 𝐁{\mathbf{B}} are Jacobi and diagonal matrices dependent to cross section P(x){P(x)}, respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section P(x){P(x)} by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is O(h2){O(h^{2})}.
期刊介绍:
The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs.
CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics.
The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.