A study of Nonlinear Duffing Equations using the Haar Wavelet Method

The main goal of this study is to solve Duffing equations, which are nonlinear differential equations (DEs), using the Haar wavelet technique. The solutions to this equation for an externally pushed and damped oscillator show intriguing dynamic behavior. Mathematicians have recently become more interested in wavelet analysis because of its real-world applications. A few real-world problems have been considered to assess the current methodology. Here, the (OMI) Operational Matrix of Integration and collocation points transform the non-linear ODEs into a system of nonlinear algebraic equations. The resulting system of algebraic equations is then solved using the Newton-Raphson technique to retrieve the unknown Haar coefficients. The DEs are then solved using the unknown Haar coefficients that were obtained. Numerical tables and graphical representations compare the outcomes of the current method with exact solutions and other numerical approaches. This indicates that the current process yields more accurate findings. Theorems related to convergence analysis are examined. Software called Mathematica was used to obtain numerical computations.