Some new finite difference methods for computing eigenvalues of two-point boundary-value problems (Q1085955)

The second order homogeneous linear differential equation \(y''+\{\lambda q(x)-p(x)\}y=0\) with the homogeneous boundary conditions \(\alpha y'(a)+\beta y(a)=0\) and \(\gamma y'(b)+\delta y(b)=0\) is considered. Under natural assumptions for p(x) and q(x) of this eigenvalue problem, some finite difference schemes of fourth and sixth order are derived to have band-structured coefficient matrices. The convergence of the schemes with respect to the discretization step-size h is shown to be quadratic or cubic in each case. Three numerical examples are given to illustrate the convergence.