A Global Collocation Quasi-Linearization Method for solving uncoupled Van der Pol equation

In this research, the Global Collocation Quasi-Linearization Method (GCQLM) is employed to effectively address the nonlinear Uncoupled Van Der Pol equation, a significant model for studying nonlinear oscillatory systems. The GCQLM method combines the strengths of spectral techniques with the iterative linearization of nonlinear terms, resulting in solutions that are highly accurate and exhibit rapid convergence rates. A comprehensive validation process is conducted to verify the accuracy and reliability of GCQLM, comparing its numerical outcomes with results obtained from traditional methods, such as the Multistage Adomian Decomposition Method (MADM) was developed by Al-Bayati et al. [1], Adomian Decomposition Method (ADM) and Rungge–Kutta order four (RK45) methods. The quantitative assessments demonstrate excellent numerical agreement between GCQLM and MADM, ADM, and RK45, with results consistently aligning across the investigated parameter ranges, while the total CPU time required for GCQLM is also calculated, showing its computational efficiency compared to the other methods. This highlights the robustness and effectiveness of the proposed method. Overall, the findings position GCQLM as a powerful and efficient tool for addressing nonlinear differential equations, including the Uncoupled Van der Pol equation, across various scientific and engineering applications.