A New Wavelet-based Galerkin Method of Weighted Residual Function for The Numerical Solution of One-dimensional Differential Equations

In this paper, we developed a new wavelet-based Galerkin method of weighted residual function.In order to achieve this, we considered the wavelet transform as it relates to orthogonal polynomials, developed new wavelets using the Mamadu-Njoseh Polynomials, and formulated a base function with the newly developed wavelets.We considered the method of implementing solutions with the newly developed wavelet-based Galerkin method of weighted residual function, and applied it in obtaining approximate solutions of some one-dimensional differential equations having the Dirichlet boundary conditions.The results obtained from the newly developed method were compared with the results obtained from the exact solution and that from the classical Finite Difference Method (FDM) in literature.It was observed that the newly developed wavelet-based Galerkin method of weighted residual function demonstrated a high efficiency in providing approximate solutions to differential equations.The study revealed that the newly developed wavelet-based Galerkin method of weighted residual function converges at a good pace to the exact solution, and iterated the accuracy and effectiveness of its solutions.We used the MAPLE 18 software in carrying out all computations in this work.