Numerical Investigation of Scale-2 and Scale-3 Haar Wavelet Approaches for Solving Elliptic Partial Differential Equations

This study presents a numerical investigation of Scale-2 and Scale-3 Haar wavelet methods for solving elliptic partial differential equations (PDEs) that describe steady-state heat distribution. The spatial derivatives are discretized using Scale-2 and Scale-3 Haar wavelet expansions, which are then integrated and extended to a 2D solution via Kronecker tensor product, incorporating boundary conditions through integration constants. The error analysis and convergence rate are performed to evaluate the numerical precision of the results. Computational simulations are carried out using MATLAB programming. Both the wavelet methods are compared with the existing finite difference method (FDM), and the results demonstrate that while all three approaches effectively solve elliptic PDEs, the Scale-3 Haar wavelet method outperforms the others by delivering more accurate approximate solutions with greater efficiency. The findings of this study highlight the potential and reliability of Haar wavelet methods for solving complex PDEs in various engineering applications.