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Computing the eigenvalues of the generalized Sturm-Liouville problems based on the Lie-group \(SL(2,\mathbb R)\). (English) Zbl 1247.65102

Summary: For the generalized Sturm-Liouville problems we can construct an \(SL(2,\mathbb R)\) 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 \(\lambda \), the present method provides an easy numerical implementation and has a cheap computational cost. Numerical examples are examined to show that the present \(SL(2,\mathbb R)\) Lie-group shooting method is effective.

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
34L16 Numerical approximation of eigenvalues and of other parts of the spectrum of ordinary differential operators
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