A numerical analysis of the normalized time-fractional Keller–Segel equations

Abstract We propose and investigate a normalized time-fractional Keller–Segel (KS) model that incorporates logarithmic chemotactic sensitivity and a non-diffusive chemical response. The normalized time-fractional derivatives maintain a constant total memory effect across different fractional orders. This formulation enables clear interpretation of the effect of the fractional order and facilitates fair comparisons across varying fractional orders. An implicit-explicit finite difference scheme is applied to the density equation, and an analytical solution combined with the frozen coefficient technique is used to solve the chemoattractant equation. The numerical scheme achieves second-order accuracy in space and first-order accuracy in time, as demonstrated by convergence tests. Numerical experiments demonstrate that the fractional order significantly influences the blow-up behavior of the solution; specifically, smaller fractional orders result in stronger memory effects and lead to faster blow-up. Furthermore, the chemotactic sensitivity exponent and the fractional order play critical roles in determining the dynamics and the blow-up time of the system. The study confirms the validity and efficiency of the proposed algorithm and provides insights into the influence of the fractional process on chemotactic aggregation. The proposed method offers a promising direction for further investigation of biological systems influenced by nonlocal and memory-driven processes.