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An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod

  • Hanif Mirzaei EMAIL logo , Vahid Abbasnavaz and Kazem Ghanbari
Published/Copyright: August 2, 2024
Computational Methods in Applied Mathematics
From the journal Computational Methods in Applied Mathematics Volume 25 Issue 2

Abstract

The free longitudinal vibrations of a rod are described by a differential equation of the form ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 , where 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 𝐀 ⁢ Y = Λ ⁢ 𝐁 ⁢ Y , where 𝐀 and 𝐁 are Jacobi and diagonal matrices dependent to cross section 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 ) 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 ⁢ ( h 2 ) .

Keywords: Inverse Problem; Finite Difference Method; Jacobi Matrix; Sturm–Liouville; Rod Equation
MSC 2020: 34B24; 65F18; 15A29; 15A18

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Received: 2024-01-04
Revised: 2024-06-24
Accepted: 2024-07-02
Published Online: 2024-08-02
Published in Print: 2025-04-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. In Memoriam of Raytcho Lazarov
  3. A Finite Element Analysis in Balanced Norms for a Coupled System of Singularly Perturbed Reaction-Diffusion Equations
  4. Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation
  5. On Error Estimates of a Discontinuous Galerkin Method of the Boussinesq System of Equations
  6. Finite Element Formulations for Maxwell’s Eigenvalue Problem Using Continuous Lagrangian Interpolations
  7. Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability
  8. Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions
  9. Numerical Analysis of a Second-Order Algorithm for the Time-Dependent Natural Convection Problem
  10. A Space-Time Finite Element Method for the Eddy Current Approximation of Rotating Electric Machines
  11. A P 2 H-Div-Nonconforming-H-Curl Finite Element for the Stokes Equations on Triangular Meshes
  12. An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod
  13. Anisotropic Adaptive Finite Elements for a p-Laplacian Problem
  14. On an Optimal AFEM for Elastoplasticity
https://doi.org/10.1515/cmam-2024-0001
Keywords for this article
Inverse Problem; Finite Difference Method; Jacobi Matrix; Sturm–Liouville; Rod Equation

Articles in the same Issue

  1. Frontmatter
  2. In Memoriam of Raytcho Lazarov
  3. A Finite Element Analysis in Balanced Norms for a Coupled System of Singularly Perturbed Reaction-Diffusion Equations
  4. Analysis of a Combined Spherical Harmonics and Discontinuous Galerkin Discretization for the Boltzmann Transport Equation
  5. On Error Estimates of a Discontinuous Galerkin Method of the Boussinesq System of Equations
  6. Finite Element Formulations for Maxwell’s Eigenvalue Problem Using Continuous Lagrangian Interpolations
  7. Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability
  8. Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions
  9. Numerical Analysis of a Second-Order Algorithm for the Time-Dependent Natural Convection Problem
  10. A Space-Time Finite Element Method for the Eddy Current Approximation of Rotating Electric Machines
  11. A P 2 H-Div-Nonconforming-H-Curl Finite Element for the Stokes Equations on Triangular Meshes
  12. An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod
  13. Anisotropic Adaptive Finite Elements for a p-Laplacian Problem
  14. On an Optimal AFEM for Elastoplasticity
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