Chebyshev Trigonometric Collocation Method for Sturm–Liouville Eigenvalue Problems: A Comparative Study

Abstract

A Chebyshev collocation scheme formulated in trigonometric coordinates — the Chebyshev Trigonometric Collocation (CTC) method — is presented for computing eigenvalues of Sturm–Liouville boundary value problems. The method uses the identity T_k (cosθ)=cos(kθ) to express the Chebyshev basis in trigonometric form, enabling all collocation matrix entries to be assembled from closed-form analytic expressions rather than numerically approximated differentiation weights. This yields a generalized algebraic eigenvalue problem at Chebyshev–Gauss–Lobatto (CGL) nodes. A conditional convergence result is established: for regular problems with analytic coefficient data, the k-th eigenvalue error is expected to satisfy |λ_k-λ_k^((N) ) |≤C_k ρ^(-2N) for a Bernstein-ellipse parameter ρ>1 determined by the domain of analyticity of the coefficients, subject to two technical hypotheses (collective compactness and operator consistency) that are stated explicitly but whose verification for the CTC scheme is deferred to future work. The method is benchmarked against the Hermite-interpolation method and the sinc-collocation/ differential quadrature methods. For problems where independent exact eigenvalues are available, the CTC method at larger basis sizes achieves substantially smaller errors than competing methods.