Daubechies Wavelet-based Galerkin Method of Solving Partial Differential Equations

Abstract

In this work, we considered wavelet analysis and the application of
Daubechies wavelet-based Galerkin method (DWGM) in solving partial
Differential equations (PDEs). It is based on the Galerkin method as it
replaces the shape function in the residual equation with the scaling
function of the Daubechies wavelet, then the inner product operation is
performed in order to obtain a system of equations, which is obtained
after considering some properties of the scaling function. The method
was tested on some selected problems involving one dimensional
equation with Dirichlet boundary conditions. The resulting numerical
evidence showed that DWGM is an efficient and effective solver of
PDEs as the results exhibited a favourable and faster convergence to
the exact solution. Also, results were compared to those of Finite
Difference Method (FDM) as seen in literatures.