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

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)=۰, 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=۰ 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^۲i^۴) to O(h^۲i^۲) 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:

Rod equation , eigenvalue , finite element method , direct problem , Inverse problem , Sturm-Liouville