Moduli of continuity of functions in Hölder’s class and solution of Van der Pol-Duffing equations by Legendre wavelet

In this paper, Legendre wavelet is considered. The convergence analysis of Legendre wavelet series for functions in the Hölder’s class is studied. Two new moduli of continuity and two estimators of functions in Hölder’s class, using Legendre wavelets, have been determined. These moduli of continuity and estimators are novel, sharper and among the best possible in wavelet analysis. The Van der Pol-Duffing equation can be expressed physically in three ways: single well, double well, and double hump. The Leg-endre wavelet collocation approach is introduced to solve the Van der Pol-Duffing equations, and their solutions are obtained by this technique as well as Runge-Kutta method (ODE45). These solutions are compared and it is observed that the absolute errors between exact and Legendre wavelet solution are less than the absolute errors between exact and ODE45 solution of Van Der Pol-Duffing equation. Hence, the proposed method is more effective and accurate than ODE45 method.