Wavelet-Based Integer and Fractional Reaction–Diffusion Modeling of Immobilized Enzyme Systems

Abstract Accurate modeling of immobilized enzyme reactors requires the integration of nonlinear reaction kinetics with internal mass transfer effects. This study presents a Lucas wavelet method (LWM) for solving reaction–diffusion models describing simple Michaelis–Menten, uncompetitive substrate inhibition, and reversible Michaelis–Menten kinetics in both integer-order and fractional-order formulations. An operational matrix framework transforms the governing equations into algebraic systems, enabling efficient and accurate numerical computation. Conformable and Grünwald–Letnikov fractional derivatives are employed to investigate local fractional dynamics and phenomenological nonlocal transport behavior, respectively. Substrate concentration profiles and effectiveness factors are analyzed over a range of Thiele modulus values, substrate concentrations, and inhibition parameters. The proposed method shows excellent agreement with MATLAB, the fourth-order Runge–Kutta method, and the Modified Adomian Decomposition Method (MADM), while resolution studies and error analyses confirm its accuracy, convergence, and computational efficiency. The fractional formulation also exhibits the correct limiting behavior as the fractional order approaches unity. The results demonstrate the significant influence of the Thiele modulus, inhibition parameters, and fractional order on reactor behavior. Overall, the proposed framework provides an accurate and efficient computational tool for analyzing nonlinear integer- and fractional-order immobilized enzyme reaction–diffusion systems.