@Article{cmes.2010.056.085, AUTHOR = {Chein-Shan Liu}, TITLE = {The Lie-Group Shooting Method for Computing the Generalized Sturm-Liouville Problems}, JOURNAL = {Computer Modeling in Engineering \& Sciences}, VOLUME = {56}, YEAR = {2010}, NUMBER = {1}, PAGES = {85--112}, URL = {http://www.techscience.com/CMES/v56n1/25465}, ISSN = {1526-1506}, ABSTRACT = {We propose a novel technique, transforming the generalized SturmLiouville problem: w'' + q(x,λ)w = 0, a1(λ)w(0) + a2(λ)w'(0) = 0, b1(λ)w(1) + b2(λ)w'(1) = 0 into a canonical one: y'' = f, y(0) = y(1) = c(λ). Then we can construct a very effective Lie-group shooting method (LGSM) to compute eigenvalues and eigenfunctions, since both the left-boundary conditions y(0) = c(λ) and y'(0) = A(λ) can be expressed explicitly in terms of the eigen-parameter λ. Hence, the eigenvalues and eigenfunctions can be easily calculated with better accuracy, by a finer adjusting of λ to match the right-boundary condition y(1) = c(λ). Numerical examples are examined to show that the LGSM possesses a significantly improved performance. When comparing with exact solutions, we find that the LGSM can has accuracy up to the order of 10−10 . }, DOI = {10.3970/cmes.2010.056.085} }