Approximate Solutions of Differential Equations Using Galerkin's Method and Weighted Residuals

The usual textbook approach to Galerkin's method can be misleading, and fails to highlight all the possible errors that arise in an approximation scheme. A general weighted residual is presented that explicitly excludes all possible errors. Thus all the inherent approximations are made transparent to students. The conventional form of Galerkin's method is shown to be just one particular case of this method.