Approximation of the Problem on Eigenvibrations of a String with Attached Load

• Published in: Lobachevskii journal of mathematics Vol. 43; no. 4; pp. 996 - 1005
• Main Authors: Korosteleva, D. M., Koronova, L. N., Samsonov, A. A., Solov’ev, S. I.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 01-04-2022; Springer Nature B.V
• Subjects: Algebra; Analysis; Approximation; Eigenvalues; Eigenvectors; Finite element method; Geometry; Infinity; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; Strings; finite element method; string; load; ordinary differential equation; eigenvibrations; eigenfunction; eigenvalue; eigenvalue problem
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080222070150

The second-order ordinary differential eigenvalue problem governing eigenvibrations of a string with fixed ends and rigidly attached load at interior point is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We introduce limit differential eigenvalue problems and derive the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as a load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite element method on a non-uniform mesh. Error estimates for approximate eigenvalues and eigenfunctions are established.