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

Document Type : Research Paper

Authors

1 Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.

2 School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada.

Abstract

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.

Keywords

Main Subjects

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 13 April 2024

Files

History

  • Receive Date: 23 July 2023
  • Revise Date: 04 February 2024
  • Accept Date: 27 March 2024

Share

How to cite

Statistics