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

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摘要

For the generalized Sturm–Liouville problems we can construct an 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 λ, 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) Lie-group shooting method is effective.

论文关键词:Generalized Sturm–Liouville problem,Eigenvalue,Eigenfunction,Eigen-parameter dependent boundary conditions,SL(2,R) Lie-group shooting method,Characteristic equation

论文评审过程:Received 8 April 2011, Revised 16 April 2012, Available online 21 May 2012.

论文官网地址:https://doi.org/10.1016/j.cam.2012.05.006