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´)´ + λa(x)y(x) = 0, where a(x) is the cross sectional area and λ is an eigenvalue parameter. Using the finite element method (FEM) we transform this equation to a generalized matrix eigenvalue problem of the form (K − ΛM)u = 0 and, for given a(x), we correct the eigenvalues Λ 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 2 i 4 ) to O(h 2 i 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.