@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: <i>w'' + q(x,λ)w = 0, a<sub>1</sub>(λ)w(0) + a<sub>2</sub>(λ)w'(0) = 0, b<sub>1</sub>(λ)w(1) + b<sub>2</sub>(λ)w'(1) = 0</i> into a canonical one:  <i>y'' = f, y(0) = y(1) = c(λ)</i>. Then we can construct a very effective Lie-group shooting method (LGSM) to compute eigenvalues and eigenfunctions, since both the left-boundary conditions <i>y(0) = c(λ)</i> and <i>y'(0) = A(λ)</i> can be expressed explicitly in terms of the eigen-parameter <i>λ</i>. Hence,
the eigenvalues and eigenfunctions can be easily calculated with better accuracy,
by a finer adjusting of <i>λ</i> to match the right-boundary condition <i>y(1) = c(λ)</i>. 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<sup>−10</sup>
.
},
DOI = {10.3970/cmes.2010.056.085}
}