Solving time-varying nonlinear equations in real time is a significant challenge in modern computing. Dynamic Memristor-Inspired Zeroing Neural Networks (DMZNN) have shown strong performance in this field, but their convergence speed and robustness heavily rely on the design of the activation function. This paper proposes a novel hybrid activation function inspired by the nonlinear characteristics of memristors. By integrating cubic and sublinear terms, the proposed function facilitates multi-stage error decay, effectively addressing the slow convergence and poor noise resistance of traditional activation functions. Theoretical analysis shows that the DMZNN model, built upon this activation function, can converge in finite time. Robustness under parameter perturbations and additive noise is rigorously proven using Lyapunov theory. Simulation results demonstrate that the convergence speed of the DMZNN model is obviously faster than that of traditional ZNN models when solving second-order, third-order, and fourth-order time-varying nonlinear equations. Additionally, in the application of remote sensing image fusion, DMZNN outperforms traditional gradient-based methods in both fusion quality and processing speed, demonstrating its practical effectiveness and superiority in real-world applications.