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Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets. (English) Zbl 1152.65086
Summary: The paper presents a novel method for the computation of eigenvalues and solutions of Sturm-Liouville eigenvalue problems using truncated Haar wavelet series. This is an extension of the technique proposed by {\it C.-H. Hsiao} [Math. Comput. Simul. 64, No. 5, 569--585 (2004; Zbl 1039.65051)] to solve discretized version of variational problems via Haar wavelets. The proposed method aims to cover a wider class of problems, by applying it to historically important and a very useful class of boundary value problems, thereby enhancing its applicability. To demonstrate the effectiveness and efficiency of the method various celebrated Sturm-Liouville problems are analyzed for their eigenvalues and solutions. Also, eigensystems are investigated for their asymptotic and oscillatory behavior. The proposed scheme, unlike the conventional numerical schemes, such as Rayleigh quotient and Rayleigh-Ritz approximation, gives eigenpairs simultaneously and provides upper and lower estimates of the smallest eigenvalue, and it is found to have quadratic convergence with increase in resolution.

65L15Eigenvalue problems for ODE (numerical methods)
34L16Numerical approximation of eigenvalues and of other parts of the spectrum
65L20Stability and convergence of numerical methods for ODE
Full Text: DOI
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