Spectral Treatment of the Fractional Bratu Equation via Shifted Lucas Polynomials: A Precise Collocation Approach with Error Quantification

This paper introduces a novel spectral collocation method based on shifted Lucas polynomials for solving the fractional Bratu differential equation with nonhomogeneous Dirichlet boundary conditions. The method employs a homogenization strategy and an operational matrix formulation in the Caputo sense to transform the problem into a nonlinear algebraic system, which is efficiently solved via the Newton-Raphson method. Detailed error analysis confirms exponential convergence, and extensive numerical experiments demonstrate the method’s superior accuracy and efficiency compared to existing approaches, even for strongly nonlinear and fractional-order cases.