Numerical computation of nonlinear boundary value problems in heat transfer and chemical engineering: a robust Lucas–Fibonacci series approach

This paper focuses on a general form of nonlinear boundary value problems, encompassing three of the most significant ordinary differential equations (ODEs) in physics and chemical sciences: Troesch's problem, Bratu's problem, and an elliptic equation with a nonlocal term. The proposed model is rooted in the context of heat transfer and chemical engineering. A hybrid collocation scheme is developed to approximate the solution using a truncated Lucas expansion series. Differentiation matrices are computed based on Fibonacci polynomial functions within this framework, reducing the solution process to solving a system of nonlinear algebraic equations. An efficient and accurate numerical approach is employed to resolve the resultant nonlinear system. Comparative analyses of the proposed techniques and computational results are provided. The proposed scheme is simple, stable, and offers a benchmark technique for numerically solving this class of problems.