On the Locally Compact Hausdorff Property in the Analytical Solutions and Stability Analysis of the Caputo Fractional Belousov–Zhabotinsky System

The fractional Belousov–Zhabotinsky system (BZS) is an important reaction–diffusion model used to describe oscillatory chemical reactions, pattern formation, and memory-dependent dynamical processes arising in chemistry and related applied sciences. The main objective of this study is to investigate the analytical properties of the fractional BZS and to construct accurate approximate solutions for the corresponding nonlinear system. To achieve this goal, the locally compact Hausdorff property together with the compact-open topology is employed to establish the existence and uniqueness of solutions within an appropriate Banach space framework. In addition, the Hyers–Ulam stability of the proposed fractional model is analyzed, demonstrating the robustness of the obtained solutions under small perturbations of the initial data. Furthermore, a hybrid analytical technique combining the double Laplace transform and the Adomian decomposition method, called the double Laplace Adomian decomposition method (DLADM), is developed to derive approximate analytical solutions. The obtained results show that the proposed approach produces rapidly convergent solution series and yields excellent agreement with the corresponding exact solutions. The theoretical findings confirm the well-posedness and stability of the fractional BZS, while the numerical results demonstrate the effectiveness and accuracy of the DLADM in handling the nonlinear fractional model. The proposed framework may be useful for the analysis of nonlinear fractional reaction–diffusion systems arising in chemical kinetics, biological processes, and other applied mathematical models.