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

Chein-Shan Liu1
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 .

Keywords

Generalized Sturm-Liouville problem, Eigenvalue, Eigenfunction, Lie-group shooting method, Eigen-parameter dependence boundary condition

Cite This Article

Liu, C. (2010). The Lie-Group Shooting Method for Computing the Generalized Sturm-Liouville Problems. CMES-Computer Modeling in Engineering & Sciences, 56(1), 85–112.



This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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