The purpose of this study is to present a simple and efficient approach to calculate the eigenvalues of the Sturm-Liouville problem. By expanding the unknown function as power series, we directly get the corresponding polynomial characteristic equations for kinds of boundary conditions, and the lower- and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for numerical computation used frequently in Sturm-Liouville problem of estimating eigenvalues are given to show that our method has fast convergence and the obtained numerical results have high accuracy.

approximate solution, Sturm-Liouville problem, eigenvalues, polynomial expansion.