Accurate Eigenvalues for the Sturm-Liouville Problems, Involving Generalized and Periodic Ones

Abstract

In the paper, the eigenvalues of Sturm-Liouville problems (SLPs), generalized SLPs and periodic SLPs are solved. First, we propose a new method to transform the SLP with mixed boundary conditions to a {generalized} SLP for a transformed variable, for which the Dirichlet boundary conditions occur on two-side, but the coefficients are nonlinear functions of eigenvalue. To computing the eigenvalue and eigenfunction, we further recast the transformed system to an initial value problem for a new variable. In terms of the relative residual of two consecutive terminal values of the new variable a nonlinear equation is solved for seeking the eigenvalue by the fictitious time integration method (FTIM), which monotonically converges to the exact eigenvalue. We solve a numerically characteristic equation by the half-interval method (HIM) and a derivative-free iterative scheme LHL {(Liu, Hong $\&$ Li, 2021)} to achieve high precision eigenvalues. Next, the generalized SLP is transformed to a {new} one, so that the Dirichlet boundary condition happens on the right-end. By using the boundary shape function method and the uniqueness condition of the transformed variable, a definite initial value problem is derived for the new variable. To match the right-end Dirichlet boundary condition a numerically characteristic equation is {obtained and} solved by the HIM and LHL. Finally, new techniques for solving the periodic SLPs with three types periodic boundary conditions are proposed, which preserve the periodic boundary conditions with the aids of boundary shape functions. Three iterative algorithms are developed, which converge quickly.  All the proposed iterative algorithms are identified by testing some examples.