A computer program for approximating a linear operator equation using a generalized Fourier series

Abstract

Many important engineering problems fall into the category of being linear operators, with supporting conditions. In this paper, an inner-product and norm is used which enables the numerical modeler to approximate such by developing a generalized Fourier series. The resulting approximation is the “best” approximation in that a least-squares (L²) error is minimized simultaneously for fitting both the problem's boundary conditions and satisfying the linear operator relationship (the governing equations) over the problem's domain (both space and time). Because the numerical technique involves a well-defined inner-product, error evaluation is readily available using Bessel's inequality. Minimization of the approximation error is subsequently achieved with respect to a weighting of the inner components, and the addition of basis functions used in the approximation. A computer program source code is provided (see Appendix A) to implement the procedures.