A Trigonometric Transformation–Based Chebyshev Collocation Method

Abstract

The Chebyshev collocation method with trigonometric transformation is a spectral numerical solution technique for initial and/or boundary value problems in computational mechanics. This approach leverages the trigonometric representation of Chebyshev polynomials to enable analytic differentiation, eliminating the need for numerical differentiation matrices. This technique has demonstrated superior accuracy and computational efficiency across diverse applications in engineering and physical sciences. This paper presents a comprehensive analysis of the trigonometric Chebyshev collocation method (TCM) and its application to four diverse problems: heat transfer in a triangular fin, torsion of a rectangular shaft, nonlinear heat conduction with temperature-dependent conductivity, and an integro-differential equation. Our results demonstrate that the TCM achieves spectral accuracy with significantly fewer computational nodes compared to conventional finite difference, and finite element methods, making it a viable alternative for problems requiring high-precision solutions.