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

Abstract

In this paper, Legendre wavelet is considered. The convergence analysis of Legendre wavelet series for functions in the Holder’s class is studied. Two new moduli of continuity and two estimators of a functions in Holder’s class  by 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 may be expressed physically in three ways: single well, double well, and double hump. The Legendre wavelet collocation approach was utilised to solve the Van der Pol-Duffing equations in all three physical scenarios, and the comparison of the resulting Legendre wavelet solutions to numerical data (ODE45 solution) shows that this method is better, effective and easy for handling this issue. This method is an effective tool for dealing with nonlinear problems of this nature. This is a significant achievement of this research paper in wavelet analysis. Majority of computations are performed in MATLAB R2015a programming platform.