Thermal analysis of extended surfaces (fins) is critical for preventing overheating in a wide range of industrial systems, from microelectronics to aerospace engineering. Governing differential equations for fin models, particularly those incorporating temperature-dependent thermal conductivity, surface heat flux, or radiative effects, are inherently strongly nonlinear. Such complexity resists exact analytical solution and demands the development of accurate, efficient, and stable numerical methods.

To address this challenge, we aim to introduce a spectral collocation method utilizing a basis of Fibonacci polynomials. The technique is designed to solve nonlinear initial and boundary value problems governed by ordinary differential equations, under Dirichlet, Neumann, or mixed-type constraints. The proposed framework systematically transforms the governing differential equation into a tractable system of nonlinear algebraic equations. Solving this system for spectral coefficients yields a highly accurate approximate solution. A rigorous theoretical foundation is established, including a proof of convergence and the derivation of a priori error bounds, both of which are subsequently validated through comprehensive numerical tests.

The method’s practical efficacy and computational performance are demonstrated on two classic yet challenging benchmark problems from the heat transfer literature: (1) heat transfer in a longitudinal fin with temperature-dependent surface heat flux, and (2) the combined convecting–radiating cooling of a lumped system with variable specific heat. A detailed comparative analysis against established methods, including the Adomian decomposition method and variational iteration method, shows that the proposed Fibonacci-based spectral scheme delivers superior accuracy, faster convergence, and notable implementation simplicity. The obtained results confirm the method's potential as an accurate and efficient analytical-numerical tool, suggesting that further exploration and adaptation could extend its applicability to a broader range of strongly nonlinear problems in thermal engineering and applied mathematics.