<p>This study investigates the heat and mass transfer characteristics and flow behaviour of a non-Newtonian Maxwell fluid over a vertical plate embedded in a porous medium with non-Darcy permeability. The analysis incorporates viscous dissipation, activation energy, and entropy generation, capturing the interplay between thermal, chemical, and rheological processes. Such models are relevant in engineering and industrial applications, including catalytic reactors, biomedical flows, geothermal systems, nuclear heat transfer, and enhanced oil recovery. To solve the nonlinear ordinary differential equations governing boundary layer flow, we develop an operational matrix approach for Jacobi wavelet integration, enabling solutions of highly nonlinear ODEs. The accuracy of the method is validated through a benchmark problem with a known solution, showing close agreement with the Runge–Kutta–Fehlberg—45 method. The approach is then employed to analyze the boundary layer problem under various parametric influences, revealing how rheological behaviour, chemical reactions, thermal effects, and porous medium properties impact velocity, temperature, concentration, and entropy generation. The findings show that temperature increases with thermal radiation, Eckert number, and heat generation, whereas velocity decreases with magnetic field and permeability parameters. Entropy generation rises with Brinkman number and magnetic field, while the Bejan number exhibits the reverse trend. Comparative results indicate excellent agreement with existing literature for selected parameter sets, confirming the robustness and precision of the proposed scheme.</p>