A Numerical Framework for Solving Fractional Keller–Segel Systems in Chemotaxis Theory by Using the Expansion Homotopy Technique: Reliability and Analytical Confirmation

In this paper, we propose a robust numerical scheme for solving a nonlinear fractional Keller–Segel system (KSDS) arising in chemotaxis theory. The KSDS describes the aggregation and movement of biological cells in response to chemical signaling. The system is formulated using the Erdélyi–Kober (EK) fractional operator, which incorporates memory and nonlocal temporal effects relevant to complex biological processes. To obtain approximate solutions, we develop an expansion homotopy perturbation method (EHPM). The proposed approach transforms the governing system into an equivalent integral formulation and employs fractional power-series expansions and He’s polynomials to derive the solution components recursively. The theoretical validity of the model and the proposed scheme is established through a fixed-point analysis, which provides sufficient conditions for the existence and uniqueness of solutions. Convergence of the constructed series is proved, and stability is investigated in the Hyers–Ulam sense. An error analysis is also presented to assess the reliability of the approximation. To provide an independent validation, the EHPM results are compared with those obtained by the Adomian decomposition method (ADM). The two methods generate identical solution components and consistent numerical results when the same truncation order is used, while EHPM avoids the separate construction of Adomian polynomials. Numerical results further illustrate the ability of the fractional KSDS to capture complex spatiotemporal aggregation patterns and demonstrate the important roles of chemotactic interactions and memory effects in the formation of structured biological patterns.