The Lie-Group Shooting Method for Computing the Generalized Sturm-Liouville Problems

We propose a novel technique, transforming the generalized Sturm-Liouville problem: w '' + q(x, lambda)w = 0, a(1)(lambda)w(0) + a(2)(lambda)w'(0) = 0, b(1)(lambda)w(1) + b(2)(lambda)w'(1) = 0 into a canonical one: y '' = f, y(0) = y(1) = c(lambda). 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(lambda) and y'(0) = A(lambda) can be expressed explicitly in terms of the eigen-parameter lambda. Hence, the eigenvalues and eigenfunctions can be easily calculated with better accuracy, by a finer adjusting of lambda to match the right-boundary condition y(1) = c(lambda). 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).