A Boundary Shape Function Method for Computing Eigenvalues and Eigenfunctions of Sturm–Liouville Problems

Abstract

In the paper, we transform the general Sturm–Liouville problem (SLP) into two canonical forms: one with the homogeneous Dirichlet boundary conditions and another with the homogeneous Neumann boundary conditions. A boundary shape function method (BSFM) was constructed to solve the SLPs of these two canonical forms. Owing to the property of the boundary shape function, we could transform the SLPs into an initial value problem for the new variable with initial values that were given definitely. Meanwhile, the terminal value at the right boundary could be entirely determined by using a given normalization condition for the uniqueness of the eigenfunction. In such a manner, we could directly determine the eigenvalues as the intersection points of an eigenvalue curve to the zero line, which was a horizontal line in the plane consisting of the zero values of the target function with respect to the eigen-parameter. We employed a more delicate tuning technique or the fictitious time integration method to solve an implicit algebraic equation for the eigenvalue curve. We could integrate the Sturm–Liouville equation using the given initial values to obtain the associated eigenfunction when the eigenvalue was obtained. Eight numerical examples revealed a great advantage of the BSFM, which easily obtained eigenvalues and eigenfunctions with the desired accuracy.