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Using the Galerkin method to compute the eigenvalues and eigenelements of the second-order Sturm–Liouville problems

In this paper, an efficient method based on the Galerkin technique for computing the eigenvalues and eigenfunctions for the second-order Sturm-Liouville problems. The first kind of Chebyshev polynomials ( ) is used as basis functions to solve this problem. The Chebyshev-Galerkin method is applied to reduce an ordinary differential equation into a system of algebraic equations using the orthogonality of Chebyshev polynomials and new relations driven from the orthogonality property. Numerical examples show that the proposed method is an easy method to implement and introduce accurate results.