<p>This paper presents an approximate solution framework for three distinct formulations of the two-dimensional Gelfand boundary value problem, a nonlinear elliptic partial differential equation characterized by exponential nonlinearity. Specifically, we investigate the classical Gelfand problem, its time-fractional extension, and the time-space fractional variant, each representing different states of complexity and generalization. These models are of considerable importance due to their wide applicability in describing diverse physical and engineering phenomena, including heat conduction, reaction-diffusion processes, and nonlinear membrane dynamics. To address the inherent challenges posed by strong nonlinearities, we employ the spectral collocation method, which provides a rational and highly accurate computational approach. The proposed methodology not only captures the essential features of the nonlinear behavior but also establishes a reliable tool for further analytical exploration and quantitative investigation. By extending the classical framework to fractional domains, this study contributes to a deeper understanding of fractional-order models via Atangana–Beleanu–Caputo (ABC) derivative and enhances the mathematical toolbox available for handling complex nonlinear systems. The effectiveness of the proposed method is demonstrated through numerical experiments, which show stable and consistent solution behavior for different values of the governing parameters. The results indicate that the method provides accurate approximations and captures the nonlinear and fractional effects efficiently.</p>