TY - EJOU AU - Liu, Chein-Shan TI - The Lie-Group Shooting Method for Computing the Generalized Sturm-Liouville Problems T2 - Computer Modeling in Engineering \& Sciences PY - 2010 VL - 56 IS - 1 SN - 1526-1506 AB - 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 . KW - Generalized Sturm-Liouville problem KW - Eigenvalue KW - Eigenfunction KW - Lie-group shooting method KW - Eigen-parameter dependence boundary condition DO - 10.3970/cmes.2010.056.085