Improving Nonlinear Van der Pol Equation Numerical Solutions using a Hybrid GA-PSO Algorithm and Dynamic Adaptive Collocation Method

Abstract

In this research, a novel hybrid framework, DAC-GA-PSO, is proposed to solve the nonlinear Van der Pol equation, which overcomes the stiffness and rapid oscillations of the equation by introducing the Dynamic Adaptive Collocation (DAC) mechanism, which treats collocation points as shifting choice variables that migrate toward high-gradient zones, combined with Genetic Algorithms (GA) for global exploration and Particle Swarm Optimization (PSO) for local exploitation. Simulation results show that this approach can greatly reduce the Mean Squared Residual (MSR) and improve numerical stability with fewer evaluation points and lower computational costs than fixed-grid methods.