Spectral and Computational Analysis of Sturm–Liouville Eigen value Problems

Abstract

A spectral and computational study of SturmLiouville eigenvalue problems, which are central to both mathematical physics and engineering applications, is presented in this paper. The analytical properties of eigenvalues and eigenfunctions are then first introduced and proved (orthogonality, completeness and spectral ordering) in self-adjoint cases. To overcome the drawbacks of traditional numerical methods, hybrid computational framework in terms of finite difference discretization and spectral weighting methods is constructed to obtain a good approximation of the eigenvalues. The suggested procedure converts the governing differential equation into a generalized matrix eigenvalue problem, which can be effectively solved numerically, yet retaining the spectral structure. An analysis of convergence and stability is conducted rigorously, and shows second-order accuracy and better numerical robustness than classical schemes. Benchmark Sturm-Liouville problems whose analytic solutions are known are used to validate the effectiveness of the approach. Comparative findings reveal the proposed method has lower approximation errors and increased stability as compared to the shooting method and conventional finite difference methods. The framework especially proficiently tackles problems with variable coefficients and boundary conditions. These results point to the promise of hybrid spectral and computational approaches to further develop numerical eigenvalue analysis, and further extensions to higher-dimensional systems and fractional operators.