Solution of fractional order Brusselator chemical model by homotopy analysis method

The Brusselator chemical model is a well-known theoretical framework for describing autocatalytic reactions, oscillatory behavior, and pattern formation in chemical and biological systems. It has been widely employed to investigate nonlinear phenomena such as morphogen distribution, biological pattern selection, and periodic dynamics arising from reaction–diffusion mechanisms. Owing to its ability to capture complex temporal and spatial behaviors, the Brusselator model also finds applications in diverse scientific areas, including plasma physics and laser dynamics.

In this study, a fractional-order Brusselator chemical model is considered to incorporate memory and hereditary effects inherent in many real-world processes. The governing fractional differential equations are analyzed using two distinct approaches: Homotopy Analysis Method (HAM) and Euler’s Modified Method (EMM). HAM is employed as a semi-analytical technique to construct approximate solutions in the form of convergent polynomial series, facilitated by an appropriate selection of the convergence-control parameter. In parallel, EMM is used as a numerical scheme to obtain approximate solutions of the model for comparison purposes.

The obtained semi-analytical and numerical solutions are validated by comparing them with the solutions generated using the NDSolve routine in Mathematica. A detailed investigation of the model behavior is carried out for different values of the system parameters and fractional orders. The comparative analysis demonstrates excellent agreement among the results obtained by HAM, EMM, and NDSolve, confirming the accuracy and reliability of the proposed approaches. This comparative study highlights the rapid convergence and flexibility of HAM, compared to the numerical method EMM.

Finally, this work provides new insights into the fractional Brusselator model and establishes an effective framework for analyzing fractional nonlinear dynamical systems. The findings contribute to the numerical analysis of fractional chemical models and are relevant to broader applications in physics, biology, and engineering.