Numerical Solution of Second-Order Differential Equations Using the Haar Wavelet Technique

Abstract

This paper studied the solution of second order differential equations numerically by using the Haar wavelet operational matrix approach. Haar wavelet characterized by their piecewise constant properties and orthogonality are used for this work due to their computational efficiency, minimal memory requirements and ease of evaluation. Unlike higher order wavelets such as Daubechies or Legendre wavelets, Haar wavelets offer a simple but yet powerful approach particularly well-suited for problems with discontinuities or sharp gradients. The methodology involves transforming the given differential equation into a system of algebraic equations through the use of Haar wavelet operational matrices. This transformation significantly simplifies the numerical computation by reducing the problem to the determination of a finite set of wavelet coefficients. These coefficients are then solved manually. The approach is not only efficient, but also extremely accurate. The numerical experiments carried out in this study reveal that the Haar wavelet-based solutions strongly agree with the corresponding exact solutions, thereby confirming the validity and robustness of the methods