In this paper, the theoretical formulation of best Hermite wavelet approximation is presented, followed by a detailed methodology for its implementation. This paper presents a novel Hermite wavelet collocation framework for the optimal approximation of functions and the numerical solution of the nonlinear Duffing–Van der Pol oscillator equation. A rigorous theoretical formulation of the best Hermite wavelet approximation is developed, accompanied by detailed convergence analysis and explicit error estimates for functions belonging to the Hölder class. New theorems are established that quantify the rate of convergence in terms of wavelet resolution level, smoothness parameter, and order of differentiability. The derived error bounds demonstrate rapid decay of approximation error with increasing resolution and regularity, confirming the efficiency of the proposed framework.

Building on this theoretical foundation, a new Hermite wavelet-based collocation method is introduced to approximate solutions of the unforced Duffing–Van der Pol oscillator. The proposed approach transforms the nonlinear differential equation into a system of algebraic equations, significantly reducing computational complexity while maintaining high accuracy. Numerical experiments are conducted to validate the theoretical results and to assess performance. The obtained solutions are compared with classical numerical methods, including the fourth-order Runge–Kutta method, Euler’s method, and ODE45. The results show excellent agreement and demonstrate that the proposed method achieves high precision with strong convergence characteristics.