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.

This manuscript presents a novel variant of the classical Szász-Mirakyan operators. The principal motivation is to obtain explicit closed-form expressions for the approximation error bounds, which are typically unavailable in general exponential preservation frameworks. In contrast to the standard operators, our proposed construction is specifically designed to reproduce the test functions $h_{0}(\chi )=1$ h 0 ( χ ) = 1 and $h^{*}(\chi )=e^{2\chi }$ h ∗ ( χ ) = e 2 χ precisely. We examine the approximation characteristics of these operators within the space of continuous functions defined on compact intervals. A uniform convergence result is established through Korovkin-type theory, and the convergence rate is characterized via the classical modulus of continuity. Furthermore, a Voronovskaja-type asymptotic expansion is derived to elucidate the asymptotic behavior. We also investigate simultaneous approximation of derivatives alongside shape-preserving characteristics, particularly monotonicity and convexity preservation. Numerical experiments accompanied by graphical visualizations substantiate the theoretical findings, revealing that for the benchmark function $\varphi (\chi )=e^{3\chi }$ φ ( χ ) = e 3 χ , the proposed operators achieve function-dependent improvement ratios: approximately 3 for $\varphi (\chi ) = e^{3\chi }$ φ ( χ ) = e 3 χ and approximately 7 for $\psi (\chi ) = \chi ^{2} e^{\chi }$ ψ ( χ ) = χ 2 e χ , in excellent agreement with the Voronovskaja-type asymptotic predictions.