Open Access

ARTICLE

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

Chein-Shan Liu¹

1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan. E-mail:ucs@ntu.edu.tw

Computer Modeling in Engineering & Sciences 2010, 56(1), 85-112. https://doi.org/10.3970/cmes.2010.056.085

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 .

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Keywords

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