Computing the eigenvalues of the generalized Sturm–Liouville problems based on the Lie-group SL(2,R)SL(2,R)

For the generalized Sturm–Liouville problems we can construct an SL(2,R)SL(2,R) Lie-group shooting method to find eigenvalues. By using the closure property of the Lie-group, a one-step Lie-group transformation between the boundary values at two ends of the considered interval is established. Hence, we can theoretically derive an analytical characteristic equation to determine the eigenvalues for the generalized Sturm–Liouville problems. Because the closed-form formulas are derived to calculate the unknown left-boundary values in terms of λλ, the present method provides an easy numerical implementation and has a cheap computational cost. Numerical examples are examined to show that the present SL(2,R)SL(2,R) Lie-group shooting method is effective.